COLLEGE OF ENGINEERING
NEW YORK UNIVERSITY
DEPARTMENT OF METEOROLOGY
A UNIFIED MATHEMATICAL THEORY FOR THE
ANALYSIS, PROPAGATION, AND REFRACTION OF
STORM GENERATED OCEAN SURFACE WAVES
PART II
Prepared for
BEACH EROSION BOARD DEPARTMENT OF THE ARMY
Contract No. W 49-055-eng-1
OFFICE OF NAVAL RESEARCH DEPARTMENT OF THE NAVY
Contract No. N onr-285(08)
A UNIFIED MATHEMATICAL THEORY FOR THE ANALYSIS,
PROPAGATION, AND REFRACTION OF STORM GENERATED
OCEAN SURFACE WAVES
Part II
By
Willard J. Pierson, Jr.
Preliminary distribution
Prepared under contracts sponsored by the
Office of Naval Research and the Beach Erosion
Board, Washington, D. C.
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New York University
College of Engineering
Department of Meteorology
July 1, 1952
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Preface
The following pages represent part two of a book entitled
"A Unified Mathematical Theory for the Analysis, Propagation,
and Refraction of Storm Generated Ocean Surface Waves." They con-
tain three chapters which logically follow part one as presented in
March 1952. Chapters 11 and 12 complete the mathematical derivations
to be presented by giving additional properties of waves in deep
water and by deriving the procedures for the analysis of pressure
and wave records in waters of finite depth and for the refraction of
a Short crested Gaussian sea surface.
Chapter 13 is the beginning of that part of the book which
deals with the practical application of the theories presented in
the previous twelve chapters. It treats specific examples of wave
and pressure record analysis both by numerical and electronic methods.
Part three is still in preparation, and upon its publication,
this book will be complete. There will be two more chapters. One
chapter will deal with observations which confirm the forecasting
theory; and in it a hypothetically complete forecast will be carried
out. The last chapter will comment on current wave generation theory
and on the scope of the task which still needs to be done in order
to put these theoretical results on a firm practical basis. Part
three may be somewhat delayed because of a summer vacation for the
author.
July 1, 1952 Willard J. Pierson, Jr.
Department of Meteorology
New York University
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Index to Part II
Page
Chapter 11. Additional Properties of a Short Crested
Gaussian Sea Surface in Infinitely Deep
Water e e e e e e e e e e e e e e e e e e e e e e 1
Chapter 12. Wave Refraction in the Transition Zone ..... 24
Chapter 13. Examples of Pressure and Wave Record Analysis. . 79
Acknowledgements ea ge a eo ae ea ee ae ee eee
Continued Index to. the Ficures (ac 0< seen oie ae «es ve 40s se bee
Gontinued Index to the Plates: « <6. % 4% « ss #6 « «6. 123
Gonuinued inidax\toothe: Tables. Fo tate civeiusd a en «eae “eos tet-- ded.
Supplementary List. of References . sss « © os « « es « « « 225
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CHAPTER 11
ADDITIONAL PROPERTIES OF A SHORT CRESTED GAUSSIAN
SEA SURFACE IN INFINITELY DEEP WATER
introduction
In this chapter, the pressure, velocity fields and curvature of
the short crested sea surface will be studied. In addition, some of
the very important lines of future research which are possible by the
use of these new concepts will be suggested. Once [a5(u Saye has
been determined, all of the other desired properties of the sea sur-
face and the fluid motions can be determined to within the accuracy
of the linearization assumptions at the start of Chapter 2. Since
the sea surface is Gaussian, it follows that all of the other proper-
ties of the wave motion such as the fluid velocities, the pressure,
and the slope and curvature of the sea surface will be Gaussian. The
functions which describe the range of variability of these properties
are different from those which describe the sea surface. They are
various integrals and functional modifications involving the power
spectrum of the free surface which lead to some very important re-
sults about the nature of the power spectrun.
Pressure
In Chapter 4, equations (4.8) and (4.10) presented formulas
for the pressure at a point below the surface produced by a finite
wave group passing overhead. They are considered here only to show
how complex the problem can become when an attempt to solve it by
Fourier Integral Theory is made. Equation (4.10) shows that at x
equal to zero the period of the waves recorded by a sub-surface
TAT 9101d
0
“tH n = xDWg
(9 % Palla V]o, yz? 3
)
: nl 7)d - (r)9 dt =
(S"11) PLO ey] f= Ma (eA) Lv] 66,928 [trey]
7
0 0
(SII) (1)¢3pp[(7)4 +47] S09 Bd = mA] 64 22 ((7)A+47)so09 [Bd = (z'y)y
oO oo
2. 0
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pressure recorder becomes larger and larger with increasing depth.
In fact a little investigation shows that there will be crests ob-
served in the pressure at the depth, z = -h, when an actual trough
of the sea surface is passing overhead. Additional investigation
of these formulas will be left to the reader.
The pressure recorded at a depth, z, below the sea surface (z
is negative) can be found from the following arguments. The free
surface is given by equation (11.1), and it can also be represented
by the limit of a partial sum as in the second expression in equa-
tion (11.1). For each term in the partial sum, the pressure contri-
bution to the total pressure for that partial sum is found by simply
inserting pg exp(( Money) 2/8) for each cosine term in equation
(11.2). A term for the static pressure is also needed.
The limiting form is then given by the Gaussian Lebesgue Power
Integral in the second expression in equation (11.2). The pressure
at each point below the sea surface thus involves the contribution
of each of the elemental waves passing overhead modified by the
appropriate damping effect with depth.
Pressure is usually only recorded at one fixed point. From the
results of the first part of Chapter 10, the pressure at the point,
X,Y, at any fixed depth, z, is given by equation (11.3). Thus
the pressure as a function of time alone is Gaussian. A given
pressure record can be analyzed for its power spectrum in the same
way that a wave record of the sea surface can be analyzed for its
power spectrum. The pressure power spectrum, [ACH ie, is related
to the power spectrum for the free surface, [A(y age by equation
(11.4). Given either one, the other can be found from the
formula. EDfe ) is given by equation (11.5) and nt x iS given by
a
equation (11.6). Ey u Ds for z not zero, is always less than E(p )
point for point. Emax is always less than Enax*
There is always some depth below which the variation of the
pressure caused by the passage of a short period wave overhead is
undetectable due to the design of the pressure recorder. For ex-
ample a five foot high wave with a five second period produces a
pressure variation of only one one hundredth of a foot at a depth
of 125 feet. This variation is essentially undetectable. Any
variation in the power spectrum at the surface under the conditions
described above is undetectable for all » greater than 2nr/5.
These arguments also follow in a slightly modified way for
pressure recorders located in shallow water (see Chapter 12).
Ewing and Press [1949] have commented on the problem of the inter-
pretation of pressure records, and their explanation is correct in
that the correction must be applied to the whole power spectrum as
indicated in equation (11.4).
Everything that has been said about records of the sea surface
is true also about pressure records. The probability distribution
of points of a pressure record is Gaussian. An equation similar to
equation (7.33) can be written for the pressure distribution simply
by substituting P(t,) for 7 (t,) and E for E,ax° In addition,
pmax
the pressure record contains less of the non-linear effects which
cause an asymmetry of the distribution for the free surface.
The potential function and the velocity field
Given the pressure field and equation (2.7) and (2.9), then
by the methods of equation (11.1) and (11.2), the potential function
INAT 40ld
4
z" ow
(E111) gp 1939) amy] [- [a] au3a4ym
(201) 1) a oP
Zr )aP fe (¢ ea fea]
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(pi) giP OP (6 sos) lezn?v | Ye (neem | /=(t)4 pun
z oe
a7] ((n)p+47) soo /= (24)n
2 ne
(O11)
2 r4
Z “oo
F ‘of aie ni —(@ uls K +9 S09 x) E]s00 Pigs
(9 )h ' : 2” 2 “Pe ~
2. 6
7p Op tye a 571 509: | Sof + i7- uls K+ 9 SOd x #02 J ee
le” V]o/e,n2 z @ (9 )n 4 (@ ] Fi z [ pe n
lorrA+ 1 (6 uls K + @ Sod 08 Jus “eee
(611)
(8'11)
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7p @p eee pe ee lta" + jn -(Q ulS K + soo x) Sus / L.
seg g varenek ae ge aU Bae ale at
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9IDJINS DAS uUDISSNDy pajsasy foUS D 40 Saljsadoig JOUONIPpY
for the motion can be deduced immediately by integration of the
variable part of the pressure with respect to time. The potential
function is then given by equation (11.7). The u, v, and w velo-
city components can then be found immediately from the potential
function and they are given by equations (11.8), (11.9), and (11.10).
Note that pena Vy PMs > Pt Pyy ce QO, and the equation of
continuity and, consequently, the potential equation are automatically
satisfied for any functional form for [aj (u ,0)1°.
The kinetic energy integrated over depth and averaged over y
and t at any x is then given by equation (11.11). This is, of course,
to be expected from Lamb [1932]. The proof will be left to the reader,
The techniques of Chapter 9 can be employed. It can also be proved
that the kinetic energy integrated over depth and averaged over t at
any fixed point is also given by pg EB nax/* The proof follows by
the application of the methods of Chapter 10.
The u component of the velocity for a fixed x and y and for
any depth, z, can be written as a stationary Gaussian Integral as
a function of time as given by equation (11.12). The functions,
[D(p» 7° and F(u ), are given by equations (11.13) and (11.14).
The u velocities decrease in range with depth and change back and
forth more slowly with time at greater depths. A graph of u as
a function of time for some fixed depth, z, would look like a pres-
sure record as the velocity shifts back and forth. However the power
spectrum of the u velocity record would not have the same shape
as the power spectrum of the pressure record for the same short
crested wave system passing overhead. The interrelations are given
by equations (11.14) and (11.4).
The cumulative power density of the u component of the velo-
city must be bounded as stated by equation (11.15). Eauation (11.14)
shows that for z not equal to zero, the term, exp[2(y *)*2/g], can
cause F(y ) to be bounded for all » even if [a,(p ,e)]° is of a
form in which E,, is unbounded. Thus any admissible [a(n ,e)]°
which has a bounded Boat must also result in reasonable velocities
below the surface.
For z equal to zero in the equations for u, the equations give
values for the surface water velocities due to the waves. It is true
that the crest particle velocities occur at values of z greater than
zero and the trough particle velocities occur at values of z less
than zero, but such refinements are not justified in a linearized
theory.
In equation (11.14), consider the integration for the case where
z is zero. Suppose that the integration overyp and 96 for yu less
than pw K is bounded. Also suppose that [an(u ,0)1° can be expressed
in a series form for » greater than # K such that
[a(n O)]2 =f, +2,00)K 2+ 2,(e)/u 2+2,(0)/p * +
are? ae aed oe ene: head 0 a os
Then the integration over 6 of this series times (cose)* must yield
constant or zero values such that
er 2 4
[D(H] = Cy + Co © + Cop 3+ of *+....
2 2 2
It then follows that »“[D(yp )]° = CjH + Cy + C3/u + C,/ : eee
Now, integration of yu 2rp¢z 7° from , to infinity in the above
form would yield infinitely large values of Ee unless Ci» Coy and
C3 were zero. Therefore, they must be zero or else the power integral
will break down and predict infinitely strong u velocity components
(a0)
(O21)
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nee P66 ale” AC ig’ ‘6d
Oo
00 ~ wh SOK
ba bre ay Oe 1
AP4PZP(¥Z‘A'x) A (4'Z“A*x) wi] = ud
O7L +4 A +A A
a= [ters ve q wil
wo< 7!
=[(e'")?v] 6 nf wit
“ipaplie'”?v| (9809)((@'7)h +47 -( guish +9soox) = lf = (4x)
*pap|ia g‘H\y] 2 gsoo((g')h +4 9- (QUIS + gs0ox) Jus = (1x)
Wes
on!
(Stl) ie cn =(7)3 wi
Ja}0M daaeq Ayjayiuijuy
D8S UDISSNDD pajsasg 4sOUS D 40 Saljsadosg jOUOIIPpY
all of the time. The constants, Cis Coy and CS were determined from
f, (8), f,(e), and f,(0). These functions of © must be positive
everywhere or else [ay (pn ,0)1° can have negative values. Therefore
f,(e), f,(@), and f,(8) must be zero. Therefore for values of #
greater than p K? the power spectrum must be of the form f,(0)/u 4
(at least) such that when multiplied by» 3, it goes to zero asp
approaches infinity. A better way to state this requirement is
given by equation (11.16) because fractional powers in the series
expansion are then also possible.
The results that have just been obtained can be interpreted in
a very easy way by considering sea surfaces composed of purely per-
iodic ten second waves, purely periodic five second waves, purely
periodic two and one half second waves, and so on. If the various
separate wave trains are all of the same height then the particle
velocities at the surface are twice as great in the five second waves
as in the ten second waves and four times as great in the two and
one half second waves as in the ten second waves. If a condition
such as the one just derived is not imposed, very strong velocities
must result.
The slope and curvature of the sea surface
These power integrals can also be differentiated and integrated
with respect to the time and space wariables. The slope of the sea
surface in the x direction as a function of time and space variables
is given by equation (11.17). By the methods of Chapter 10, this
equation can be reduced to a function of time at any point. It then
follows that the slope in the x direction is given by an integral
of the form of equation (7.1) except that the power spectrum is
given by w*CD( p )1°/e" instead of FAC 2 17. The slopes are there-
fore distributed according to a normal distribution with a mean of
zero and a variance related to the integral of the function just
given above from zero to infinity.
The curvature of the sea surface in the x direction is given
by equation (11.18). The curvature as a function of time at a fixed
point is, from the same reasoning as used above, distributed ac-
cording to a normal distribution with a zero mean and a standard
deviation related to the integral from zero to infinity of
2 (p(n )1°/e?. |
Infinite values for the curvature of the sea surface mean that
at that point on the sea surface a sharp breaking angle occurs in
the wave profile. Equation (11.18) shows that these sharp curvature
changes are associated with the short waves (or the higher wave fre-
quencies). If the integral is to behave properly, the condition
given by equation (11.19) must be imposed.
Wave power and energy transfer
Consider the yz plane which results from picking a fixed value
of x. The work being done on this plane when averaged over y and t
and integrated over depth is the wave power or the flux of energy
in ergs/sec per centimeter of length along the y axis. The equa-
tion given in Lamb for section 237(equation 10) can be modified to
yield the first expression in equation (11.20). Substitution of
equations (11.2) and (11.8), followed by the indicated integrations
and limiting processes, then yields the average rate of transmission
of Wave energy across the yz plane per unit length of the y axis.
Without the cos@® term, equation (11.20) would represent the total
10
outflow of wave energy from the storm area. From arguments similar
to those which have been used above, the wave power will not be
bounded unless equation (11.21) holds.
Equation (11.20) has a particularly important application in
wave forecasting theory. At the forward edge of a storm at Sea,
it measures the energy which is being transmitted into the area of
calm by the waves as they leave the edge of the storm area. The
storm winds in the atmosphere by some mechanism transfer energy to
the waves in the generating area. The energy in the wave motion
in the generating area flows out of the generating area at a rate
given by equation (11.20) (plus a component in the y direction).
The important point is that in order to maintain the same amplitude
of the power spectrum near » equal to 27/10 that is maintained
near # equal to 27/5, the atmosphere must transmit twice as much
energy per unit time to the generating area near frequencies given
by p | to 27/10 than is required near # equal to 27/5.
Consider, for example, two power spectra. One is given by
[a,(# 40) ]° equals a constant over the area bounded by # equal to
2m/1l and 27/9 and by © equal to + 7/36. The other is given by
[anu 58)1* equals the same constant over the area bounded by “
equal to 27/5.24 and 27/4.74, and by © equal to + 1/36. The two
power spectra have the same band width and the same value of Emax?
but were such power spectra actually to exist over a generating
area, twice as much energy would have to be transmitted to the
sea surface by the atmosphere in order to maintain the waves for
the first power spectrum than would have to be transmitted to the
surface in order to maintain the second power spectrun. Energy
val
transmission for low values of # is much greater than for high
values of » and it is therefore more difficult for the storm winds
to maintain that part of a power spectrum which applies to low
values ofp.
a
considerations
It is dangerous to attempt to apply non-linear criteria to
linearized systems. The linearized theory presented so far has
gone a long way toward explaining the properties of actual storm
generated ocean waves, and it appears to give consistent results.
A linearized theory usually has one fault in that the theory in
itself seldom yields information on when it will fail.
For example, the requirement that ee ay be bounded was imposed
in Chapter 7 for the first time and equations such as equation
(11.15), (11.16), (11.19), and (11.21) have been deduced from this
property and other considerations. However [a,(u ,0)1° is still
undetermined to within a constant factor. That is, if a given
functional form for [a,(u ,6)]° satisfies all of the requirements
which have been deduced, it is still undetermined to within a con-
stant factor because it can be multiplied by a factor of 10 or 100
or 1000 and it would still satisfy all of these requirements.
This is, of course, against good sense and against the initial
assumption that the disturbance was small. There is no way to tell
when the theory will get seriously out of hand for large magnitudes
for the function from any of the previously given formulas.
It is possible to make an educated guess about when the theory
will certainly fail, and sometimes an educated guess is a very good
12
thing to have in lieu of actual knowledge. It is known from non-
linear wave theory (Lamb [1932], sec. 250, see footnote on the
work of Michell, and also Davies [1951]) that, for a purely periodic
wave of finite height, the ratio of the wave height to the wave
length cannot exceed one seventh. Michell obtained a value of 1/7.05,
and Davies' [1951] most recent results are given by a value of
1/6.914. Equation (11.22)uses the value 1/7 because the result
is to be only an approximation. Suppose that all of the power in
the power spectrum for w greater than p» K were concentrated in
one purely periodic wave with a wave length determined by pw K°
Then certainly the integral given on the left in equation (11.23)
is less than the integral given in the middle and there is reason
to believe that they both must be less than the value, 1/49. The
longest possible wave length has been taken on the left in this
equation, and the waves would certainly be very steep if [An(e co yds
had major contributions for values of # very much larger than p K°
Of course nothing can be said about how these short waves combine
with the lonzer waves for # less than # K in the non-linear case,
but if [A,( ey Ir were identically zero for # less than} x,
equation (11.23) would still have to hold. It would seem that any
added disturbance for # less than (ye would only serve to increase
the instability of the waves for # greater than Hye
If these arguments are valid, then equation (11.24) follows
from equation (11.23). It states that the power in [A(py TE from
wy to infinity must be less than some constant times p rae In
terms of T,, this power must be less than a200
For small values of pw a this result gives a larger possible
13
(6211)
(8271)
(22 0)
(92°11)
(S2'11)
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(C21)
(22 11)
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xp(d+x* A)U(x*A)L [TP wit = (d)Xx
xX+ xX
eS aie
t z Los
HP(d+4Z)q (4)4 [wry = (d)x
Ltys
SUOIJOUNS UOI}D]9I4I40H SSO0JIQ |BIdWOS awos
on
= [(a't)?u | 7 wi
Gb - 4, 8 5%,
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Wd 6b ene > an ee eer au
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Hof
» up 2x96e", 97 Ot. ty ye =
gwotLeze =e ep >[le'") = ap [lot) y] / 40
oO
3
nai
6b 224th 2[6 +2]
— > gp7p——_>——_ > ppt p———_—+___ ae sysaBbns yoium
z i a a
i
| (g'7)*y 7
foe]
6b oa
<= 8A0M JIpOolsad Ajaind Db 4103 Soohaleeitues JD9uUI| UOU WOI4
JO}0M daaq Ajayiurju|
JOUOILIPPY
BIDJINS DGS UDISSND paysas9d 4410YyS D JO Saijsadoud
value for the power present. For Tx equal to 10 seconds, the power
6 em (equivalent to a purely
could be of the order of 3.28 10
sinusoidal wave approximately eighteen meters high) between 277/10
and infinity. For Tx equal to 1 second, the power between 27 and
infinity could be of the order of 328 em (equivalent to a purely
sinusoidal wave 18 cm high).
Of course, for much smaller values of T,, these formulas begin
to lose significance because the elemental waves are no longer
gravity waves but capillary waves. The modification of these equa-
tions by the appropriate forms for capillary waves might yield ad-
ditional theoretical information about the high end of the spectrum.
The bound given on [Ace 1c by these considerations is most
likely an overestimate as to when the linear theory fails. That
is, if the inequality is not satisfied, then the linear theory
certainly fails, but if the inequality is satisfied, then the linear
theory may still fail in one or more theoretical aspects. In
addition, it would be fairly safe to predict that functional forms
for [aC uw 1° will never be found in nature which fail to satisfy
the requirements given in equation (11.24) because were the winds
to attempt to build such a wave system, the system would be destroyed
as fast as it is formed by breaking and turbulence at the crests.
The "outsize" waves predicted by The Gaussian distribution would
presumably be very unstable.
The shape and properties of the possible power spectra
Equations (11.16), (11.19), (11.21) and (11.24) taken together
yield a considerable amount of information on what power spectra
are possible, on the shape of the power spectra, and on the appear-
15
ance of the sea surface. The shapes of the power spectra will be
discussed first at the high frequency end and then at the low fre-
quency end in terms of what power spectra are possible. This will
also permit a discussion of the appearance of the sea surface as
a by-product.
Equation (11.16) and (11.24) combined with the discussion
given in the paragraphs on the potential function and the velocity
field show from equation (11.25) (if [a,( 4 ,0)1° has a series ex-
pansion) that the constant, C,, must be less than the value given
in (11.26). For wy, equal to 27, C, must be less than 117,6007r,
and the largest possible value for the term (when pw equals 27)
is equal to 117, 6001r/(2r)* or 236 cm* sec. With the above value
for Cay the power between 27 and 37 as computed irom (11.25) with
different limits of integration is very nearly 328 cm’, and there
is little power above the value 37.
At the high end of the frequency spectrum, then, the spectrum
must die down in amplitude at least as fast as C,/u 4. For moderate
values of » , the spectrum can get to be quite high but it must
always satisfy equation (11.24).
Equation (11.21) applies to the low end of the frequency spect-
rum. Especially in the source region, it states that the flow of
energy across the forward edge of the storm must be bounded. Some
results from the formulas given in Chapter 9 also apply here. For
a wave system over a fetch 250 km long, seventeen hours after the
winds cease, the power spectrum at the edge of the fetch will no
longer contain values of » less than 21/5. Waves therefore die
down in the storm area very rapidly as soon as the winds cease
16
(see figure 25). Conversely, tremendous amounts of energy have
to be supplied to the sea surface from the winds in the storm
overhead. For a steady state, the energy supplied per second
from the atmosphere to the waves must balance the energy dissi-
pated per second by the waves in breaking at the crests and the
energy per second which flows out of the forward edge of the storm
area as the waves propagate into the area of calm. The balance
must hold for each possible elemental area in a net of the p ,@
plane.
At the low end of the power spectrum, very large amounts of
energy are leaving the generating area every second. Consequently
the lower the value of » , the more difficult it is for the storm
to maintain a wave of any appreciable amplitude. Therefore, as #
is decreased the power spectrum must pass through some peak value
and then begin to decrease as # gets close to zero.
If the oceans were infinitely deep these considerations would
hold exactly and equation (11.21) would have to hold exactly. The
oceans are only about 3000 meters deep. A wave 6000 meters long
is still essentially in deep water. This corresponds to a period
of 61.7 seconds or a » of 27/61.7 seconds. Thus for » less than
21/61.7 seconds, these arguments do not hold exactly. However, the
rate of energy flow out of a generating area is still tremendous
for # less than 27/62 and the arguments are still qualitatively
valid since the ocean is not really shallow water (C = “gh ) until
the period of the waves becomes about 1330 seconds. The point,
B= 21/1330, is very close to the origin in all of the forecast-
ing curves which have been shown.
17
Sea surface "glitter"
So far no claim that equation (11.19) must hold has been put
forward. The sea surface could be covered by many small facets
and at many points the curvatures can be sharp. However, intui-
tively at least,equation (11.18) should have a meaning everywhere
and this is not the case unless equation (11.19) holds. If equation
(11.19) holds, then C, must be zero. In addition, in the same
series discussed above, Coy Ces Cos Coy and Co must all be zero
and for some # greater than p K the series must be of the form
» Ca, (#48) ]*(cose)® = £49(0)/ n° plus higher order terms.
The results show that there is a tendency for the high fre-
quency components to produce many sharp facets on the sea surface.
These facets can be observed in fresh waves from a generating area
and they are particularly noticeable in the photograph which has
been chosen for the frontispiece. Any light breeze can super-
impose a high frequency spectrum on a swell and it is believed
that these considerations account very nicely for the sea surface
VoJi1t ber. i
Final form for the power spectrum
If it is required that all derivatives of the sea surface,
the velocity field, and the pressure field have a defined power
spectrum and a defined power integral, then the requirement posed
by equation (11.27) must be fulfilled for any integer value of M,
no matter how large. No polynomial in (1/p )" can satisfy this
requirement. Therefore [a(m ,0)1° cannot be represented by a
fraction consisting of polynomials in » in the numerator and
denominator. The power spectrum must therefore be either some
18
entire transcendental function capable of satisfying equation
(11.27) and equation (11.24) for greater than pw, if it is to
have a value for all p or it must be identically zero for »
greater than some value. The functions given in the examples in
Chapter 9 satisfy equation (11.27) [and therefore (11.19) and
(11.16)]. When they were manufactured, condition (11.24) was not
known. It might be an interesting problem for the reader to see
if they satisfy equation (11.24) for all values of yp Ke
The use of autocorrelation functions
The non-normalized autocorrelation function given in eauation
(10.26) was used to find the power spectrum. In its own right
it is an extremely important function in wave theory because it
permits short range predictions of what the next few waves will
be like. The non-normalized autocorrelation function of a wave
record dies down to zero for large values of p and it is very small,
for example, for p equal to about 180 seconds for a power spectrum
from a "sea" record. This means that what occurs at the point of
observation three minutes after, say, a crest passes that point has
very little to do with the fact that a crest passed three minutes
ago. Stated another way, it is impossible to predict whether a
erest or a trough will be passing the point of observation three
Minutes after a given time of observation. Note that the power
spectrum of the wave system tells us a great deal about the whole
wave record, about the characteristics of the record, and about
the "sea" and "swell" properties. However nothing can tell us the
exact shape of the wave record three, ten, twenty or thirty minutes
into the future.
19
In contrast, if a wave record could be represented by any
number of discrete spectral components with, say, four place ac-
curacy for the spectral periods, then it is theoretically possible
to predict the wave records into the future for a long time. For
example, suppose that a wave record were actually composed of three
Sine waves of amplitudes A,, A>, and Aas with periods of 8.75,
10.35, and 14.10 seconds, respectively, and that the numbers actually
mean that the periods are between 8.745 and 8.755, 10.345 and 10.355,
and 14.095 and 14.105. Then after one thousand seconds (17 minutes),
the greatest possible predicted phase error would be 24 degrees.
At a point in the future one thousand seconds ahead at which, say,
theoretical positive cosinusoidal reinforcement is to occur the
predicted amplitude would have to be between Ay + Ay + A, and
A, cos 2325° + A,cos 16.8° + A,cos 9°, The autocorrelation function
implies that such accuracy is fallacious, that a wave record cannot
be predicted that far into the future for "sea" conditions, and that
the sea surface cannot possibly be composed of discrete spectral
components.
Wiener [1949] has given the mathematical procedure for predict-
ing the future behavior of a stationary time series given its past.
From the past, the first step is to find an estimate of the auto-
correlation function. The autocorrelation function can then be
used to determine the kernal of an integral equation such that when
the past of the record is multiplied by the kernal and integrated
over past time, a number results which is the best possible forecast
for the value which will occur, say, thirty seconds into the future.
The best possible forecast is in the least square sense; that is,
20
the difference between the forecasted value and the actual value
squared is a minimum over all forecasts. If the autocorrelation
function is essentially zero from lags of three minutes onward, then
the forecast would be a zero amplitude disturbance at all times be-
yond three minutes in the future. This forecast would be correct
in the least squares sense because the second moment about the mean
(zero) is the smallest second moment possible and in the sense that
the autocorrelation function implies that what will happen in three
minutes has nothing to do with what is happening.
If it ever becomes essential to know thirty seconds in advance
that a big wave is coming then it is possible to imagine an elect-
ronic circuit constructed along the lines of the one described by
Lee [1949] which will graph the wave record as it will occur 30
seconds in the future given the present wave record. Note also
figure 22 in Lee's paper. The random voltages shown look exactly
like wave records!! The machine described by Lee [1949], if one
imagines it applied to wave forecasts would only predict the records
about three seconds in advance.
A ship at sea is acted upon by a Gaussian wave system. There-
fore it pitches, rolls, and rises and falls according to a Gaussian
law. The continuous record of, say, the inclinometer is therefore a
temporarily homogeneous Gaussian record, and from the autocorrelation
function of the inclinometer record it is therefore possible to pre-
dict from the past when the next big roll of the ship will occur.
A very fruitful line of future research will be to apply the
methods given by John [1949] to a Gaussian sea surface and determine
the movement of floating objects on the sea surface in response to
21
the waves. John [1949] has solved the problem for a purely periodic
wave. His solution is given in the form of a Fredholm integral
equation (which may or may not be solvable itself). Stoker,in a re-
cent conference, suggested more direct methods which could yield
immediate practical results. By the principals of Chapters 9 and
10 these results can be extended to the Gaussian case.
Cross correlation functions
Another important tool for the study of ocean waves is the
cross correlation function. There are many possible cross corre-
lation functions which can be constructed. For example, X(p),
given by equation (11.28), gives the cross correlation between the
height of the free surface at a fixed point and the pressure re-
corded by a pressure recorder at some depth, z, below the surface
at that same point. As another example, equation (11.29) gives
the relationship between the free surface at two different values
of y at a fixed time.
With these cross correlation functions, many properties of the
sea surface, the velocity component fields, and the pressure fields
can be interrelated and studied. A detailed study of equation (11.28)
would probably show that the deeper the pressure recorder the less
it reflects the passage of high short "apparent" period waves over=
head and that it is easily possible for a pressure recorder to record
a crest when actually a trough is passing overhead (or conversely).
The cross correlation functions must be studied by carefully keeping
the same net and the same ¥(»,0) for each term in the net for the
two functions being studied. Although values are Gaussian, for
example, an accidental high crest is related to an accidental high
22
u velocity component at that same point and time of observation.
Lines for future research
All of the things suggested above on the autocorrelation
functions and the cross correlation functions cannot be treated here
in detail because they recuire very extensive mathematical abilities
and they are sidelights on the main problem of wave analysis, wave
propagation, and wave refraction. Their importance is obvious,
and they suggest many avenues for future research and investigation.
23
Chapter 12. WAVE REFRACTION IN THE TRANSITION ZONE
Introduction
The assumption that the oceans are infinitely deep have proved
very useful so far in the study of ocean waves. For all practical
purposes, the errors involved are not important. Sooner or later,
somewhere, the disturbance is dissipated by the breaking of the waves
on a coastline. Waves leave the deep parts of the oceans and travel
finally to the shallow waters bordering a coast of an island or a
continent. In the shallower waters, if the depth is constant over a
relatively large area, the wave crest speed of a purely sinusoidal wave
is given by equation (12.1). But wave refraction complicates the
problem, and it is necessary to treat the wave crest speed as if it
were a slowly varying function of position. There are varying degrees
of accuracy with which the problem of wave motion over an area where
the depth is less than, say, one half the wave length of the lowest
important spectral component, can be treated. These methods will
be discussed in this chapter.
As the waves advance into an area where the effect of depth is
important, a large area can be found such that the results of the
previous chapters can be extended to explain the observed patterns
and aerial photographs. Later, as the waves near the breaker zone,
a transformation often appears to occur which substantiates some
of the results of Munk [1949] on Solitary Wave Theory. Finally, the
waves peak up and break.
The breaking wave is a phenomenon of the non-linearity of the
original equations of motion. All methods of wave analysis and wave
24
refraction which are based upon the linear theory fail in one or
more important aspects in the breaker zone. Therefore the methods
developed in this paper cannot be applied to the breaker zone.
Between deep water and the coast there will first be found a
zone which will be referred to in this paper as the transition zone.
Between the transition zone and the coast there is a possibility
of a solitary wave zone, a shallow water wave zone, and a breaker
zone. If boundaries between the transition zone and the above three
zones can be defined, then in this paper the theory will apply to
the transition zone as marked by deep water on one side and the
boundary of that zone (of the above three zones) which is farthest
from the coast. Non-linear effects of great importance must be pre=
sent in these near-shore zones, and they will not be treated in this
paper.
It might also be noted that conditions can occur in which the
solitary wave zone, the shallow water zone, and the breaker zone
would not occur. Also any two of the above zones or any one of
the above zones might be missing. For example, waves approaching
a vertical cliff rising sheerly out of a depth of forty feet at the
edge of a bottom of variable depth could be reflected back out to
deep water without ever undergoing any of the above suggested modi-
fications.
The invariance of discrete spectral periods
Consider the following experiment in a very long deep wave tank.
Waves with a period of exactly two seconds are generated in a forty
foot depth at one end of the tank. The water for all practical
purposes is infinitely deep, and the waves can be expressed as a
25
function of x and t alone at that end of the tank. Twenty miles
away let the depth shoal gradually and linearly over a distance of
ten miles to a final depth of five feet. For another twenty miles
let the depth remain at five feet and then let the tank be ended by
a perfect wave absorber without any reflection. Suppose also that
the generator has been running for about two months so that all
transient effects can be ignored. Finally let the amplitude of the
waves at the generator be two inches so that the small height assump-
tion can be used as an approximation.
Now, at a distance of five miles from the generator, the waves
will have a speed given by c? = gor /4r@ = gL/2r. Exactly one sinu-
soidal crest will pass the point of observation every two seconds.
The wave record will be essentially a pure sine wave if observed at
a fixed point. The period of the wave will be exactly two seconds.
At a distance of forty five miles from the generator, the waves
will have passed over the sloping bottom, and at a distance of fifteen
miles from the slope, since the deep water wave length is only one
two hundredth and sixty fourth of a mile, the waves in the region
ought to be again nearly sinusoidal in form and the crests ought to
be traveling again with a constant speed. The crest speed ought to
be given by equation (12.1) from classical theory.
A long time ago in Chapter 2, under the assumption that the
motion was purely periodic with one discrete spectral period, a
periodicity factor in time for depth still variable was split off
from the potential equation. The above experiment has been designed
to show why this assumption is valid. Suppose that at this second
point of observation the period of the wave is recorded. The period
26
must _be exactly two seconds.
Suppose that the period is not exactly two seconds at the sec-
ond point of observation. Suppose, for example, that the period is
really 2.01 seconds. Near the generator the period is two seconds.
Each periodic motion at the first point of observation means that
one wave crest has progressed toward the second point of observa-
tion. In the next one hundred hours, then, 180,000 waves will pass
the first point of observation. At the second point of observation,
where the period is assumed (erroneously) to be 2.01 seconds only
179,104 waves will pass during the time of observation. Thus 896
But at the start, it was assumed that the motion had settled down
to a steady state; and now it is found that the number of wave crests
between the two points is continuously increasing. The assumption
that the period is not the same is therefore wrong. Therefore the
period at the second point of observation must be exactly the same
as at the first point of observation.
It might be remarked that a formal exact mathematical solution
to the experiment just described has never been obtained. The works
of Stoker [1947] and Eckart [1951] come close to solving the problem,
but Stoker's solution for a linear sloping beach although exact, as
far as the linear theory goes, is not quite a solution to this
problem and Eckart's methods would yield only an approximation to
the true solution.
Finally, though, the important point is that whatever solution
is found the period of the motion at the second point must be the
same as at the first point. Also the wave speed at the second point
27
will be essentially given by equation (12.1).
Waves in water of constant depth:
Consider a point in the transition zone where the depth is con-
stant over a rather large area. The problem is to represent the sea
surface and the other desired quantities in the vicinity of that
point. None of the previous representations are correct in the tran-
sition zone except that the wave record as a function of time is
still given by the same general function of time discussed in Chapter
7. In particular the methods given in Chapter 10 for the determi-
nation of power spectra as a function of w and 0, will not apply to
waves measured in the transition zone.
Eouation (12.1) gives the speed of the wave crests as a function
of the wave length for a pure sine wave in water of depth, H. Equa-
tion (12.2) relates the speed of the wave crests to the wave length
and the wave period. The period is independent of depth. From equa-
tion (12.1) and (12.2), an equation for the wave length of a wave in
water of depth H can be found in terms of the wave period. A con-
version of spectral periods to spectral frequencies then will permit
integrals over power spectra similar to those considered before.
If the expression for C in terms of L and # in equation (12.2)
is substituted into equation (12.1), equation (12.3) is the result.
Rearrangement then yields equation (12.4) in which the wave length
in water of depth, H, is given as a function of the spectral fre-
quency and the depth.
Usually (12.4) has been solved graphically (with a slightly
different notation). Sverdrup and Munk [1944] give graphs of L/L,
(i.e. Le/2npee) as a function of arH/L,- The Beach Erosion Board
28
Waves’ in Water of Constant Depth
2.9L. 27H
er tanh a
L _LHK
OF ot
2
LH. g tanh 274
27 E
fa
fT, & 27
i g coth H +
2
Or sf coth H+
Ee w ore) HT
2 2
Daimler ae or
<a- 7 coth (H 5 coth H ee
2 2
an. H coth (HAF coth (Hg coin bigs
a 2
27. coth (4 coth (HE coth (HE "coth H 22)))
2 2 2 2
27. coth (H = coth (H = coth (HA Lae )))
a7. He pe
Seca licotn (H Ey ene 5
pe pe p?
Itcoth Hirai coth(H-~- Itcoth H-- )
[ = coth (H = “Ty
me a | coth'!
g I =
Plate LXI
(12.1)
ie)
(i233)
(12.4)
(12.5)
(12.6)
(12.7)
(12.8)
(12. 9)
(12.10)
(2200)
(12.12)
(12.1 3)
gives complete tables of the same ratio. However, given T, and
H and a table of ordinary hyperbolic functions, it is possible to
find the above length without recourse to these graphs and tables.
Equation (12.5) is equation (12.4) written down again. Substi-
tute the expression for 27/L on the right in (12.5) for the 27r/L
under the hyperbolic cotangent on the right of equation (12.4). The
result is equation (12.6). Again substitute the value of 27/L in
(12.5) into the far right of (12.6). The result is equation (12.7).
Do it again. The result is equation (12.8). After an infinite
number of substitutions the result is that 27/L is given as a function
of # and H* alone on the right hand side of the equation. Thus, in
a sense, equation (12.4) has been solved for 2r/L in terms of H and
pe The new function suggested by equation (12.9) is defined by
equation (12.10) to be the Itcoth of Hu-/g, [or (Hp @/g)]. The
symbol, Itcoth(Hp “/z), is to be read as the iterated hyperbolic
cotangent of Hy /e. It can also be pronounced easily just as it
reads. The Itcoth appears to be a brand new function, never written
down before.
The point of the new function is that substitution of Hp °/g
for H2r/L at the far right in the iteration makes no error in the
value of the function. In fact only seven or eight iterations yield
three place accuracy for the Itcoth starting out with Hi /e instead
of H2r/L if Hp °/g is fairly large. Near zero values, many more
iterations are needed.
Table 17 illustrates this point. Let the depth be one eighth
*The usual notation for this symbol is h, but H is used here in
order to avoid confusion with the h of Chapter 10.
30
Table 17. Computation of the Itcoth by Iteration
2 arL
Let H = 9 ; ot = ae = aan = 5 = .785
Number
of
Iterations
a) eoth( .785) = 1.524
2 coth(.785) (1.524) = coth(1.12) = 1.238
3 coth(.785) (1.238) = coth(.972) = 1.333
4 coth(.785) (1.333) = coth(1.05) = 1.282
5 coth(.785) (1.282) = coth(1.006) = 1.309
6 coth(.785) (1.309) = coth(1.028) = 1.294
Y, coth(.785) (1.294) = coth(1.0158) = 1.302
8 coth(.785) (1.302) = coth(1.022) = 1.297
Therefore 1.297 <1(.785) <1.302;
ab L, were equal to 1000 feet and
if H were 125 feet, then L would be
1000/1.30 or L = 769.2 feet.
of the depth water wave length. Then » “H/g equals 2H/L, which in
turn yields 2rL,/8L, or the number 7/4.
The value of 7/4 to three figures is given by 0.785. The
hyperbolic cotangent has the value 1.524 as shown in the first row.
The second row gives the hyperbolic cotangent of 0.785 times 1.524.
The true value of the Itcoth lies between the two numbers given by
2.5924 and 1.238.
Eight iterations then yield the values given by row 7 and row
8. Within an error of one half of one percent the true value of
the Itcoth for p 2u/e equal to 7/4 is 1.30. Given that the wave
length in deep water is 1000 feet, the depth would then be 125 feet
Su
and the wave length at the depth of 125 feet would be 769.2 feet.
Note that no special tables or graphs were used.
The Itcoth has an additional property which is given as equation
(12.11). If the hyperbolic cotangent of the product of Hp/p and
the Itcoth of Hu “/g is formed, it will again equal the Itcoth of Hp-/e.
This is shown by Table 17. If the Itcoth is treated as the dependent
variable, I, equation (12.12) follows. The inverse of the equation
then yields Hp °/¢ as a function of I, and in equation (12.13), Hp ?/e
is given as a function of I. The function given by eouation (12.13)
is graphed in figure 31. Other relationships of a useful nature are
also given in the figure.
Since the wave length of a wave with a known spectral frequency
(or period) has now been given as a function of that spectral fre-
quency and the depth, H, of the water, it is now possible to write
down the expression for the free surface for one pure sine wave, in
water of constant depth, H. The free surface is given by equation
(12.14) in which the constant spectral frequency is given by I and
the depth is He It is easy to show that this expression reduces to
the forms given before if H becomes infinite.
Equation (12,15) then yields the potential function. It is
again easy to show that the potential function satisfies all required
properties and that it reduces to the appropriate form in water of
infinite depth.
The appropriate Gaussian systems then follow immediately from
previous considerations. The free surface is given by equation (12.16).
Boy (CH 98) is the cumulative power distribution function for waves in
water of depth, H. The function, ¥(,@) and the function, E5,(p ,6)
a2
“SUOIJOUN} Pa}DJad Ja4yZO puD OH yO UOlLOUN} D SD YJOD}T aUy yO YdoIg |{¢ aunbBi4
H27
6
o=||(2 | ae
aah
002 GL os'! ral
fr. al aa & T T T T T line [TI § T T T | Sa |
omnes 6
Hew
6
(AT 29
| Bra |
=| ef
°
sor &
J |
cis
Be
| g
| i
o n
an
24 fa si iS
uz n >|
aes ae
Hr ng He
6
vey oz
sz
oe
se
—33-—
TXT 240Id
H ysoo
(21°21) gP 7 [ptr a] ef oma a Bu &+ 9 s00 Sut S]os- __ lH parr Fuso
ysoo
Z2+H)-(—H yeaa
[ H)-(S-H)T al
(91°21) (gi) 3 | p [to A)p + dt — [@ uls K + 9 S09 x Ate H)I° Al
swajsks upissno9g
6 6
H (8,4) T= |ysoo I
(S121) e+ yin—['@uis K+ 'gsoo x tyra Sy ne - Le ger Sls ul
[s [é ] mul ul [(e4H) Ga H)T-ay | by
z rf
soo
4
& oo
effec (4)
(pr 2u) [e+ ir —'g uis f + ‘9509 x] (tah) I oy soo Vv
yuidag yuojsu0gd 40 JdJ0M~ UI SaADM
z
¢
A
Uy
x) ®
have the same properties as required in Chapter 9 for the analogous
functions in that chapter. The subscript H's have been added to
emphasize the fact that, given [E,(p ,©)] offshore in deep water,
then Ey ( # 99) is an unknown function unless the refraction proper-
ties of the transition zone are given.
The potential function is given by equation (12.17). [ioe CE oul
is the power spectrum of the waves in water of depth, H. It cannot
be found from the theories given in Chapter 10, although appropriate
modifications of the formulas given therein would yield correct re-
sults.
As a function of time at a fixed point, these equations can
be treated just as in Chapter 10. The record as a function of time
is Gaussian and the results of Chapter 7 again apply. Boyle ) is
by analogy equal to Boy(H 91/2). As before, [ALC WE is the inte-
gral over 0 of Berg arch ahr
Ti.e pressure at a depth, z, produced by a short crested Gaussian
sea surface on the surface of a layer of water of depth, H, is given
by equation (12.18). It reduces to the results given in Chapter
1l as the depth approaches infinity.
The pressure at a fixed point in the x,y plane as a function
of z and t is given by equation (12.19). The equation can be de-
rived by the use of the methods of Chapter 10. For a fixed value
of zy, a pressure record as a function of time is therefore Gaussian
and can be analyzed for its pressure power spectrum in the same way
that a wave record can be analyzed.
The power spectrum of the pressure record for a pressure re-
corder at any depth (not necessarily the bottom) is related to the
6S)
IUXT 940 ld
(22°21) ee
i231)
fore
as Zz) woO}}oq 4D
(71) "Vv __ Le [pred gee [rls as = Lor) Hay
a by SERRE fo
(61°21) zJd6 — 7 eee a PE [(7),A + 4]s09 Bd -(\'2)d
co
(8°21)
byt 5 |so9- LE _ HW T S| us00# 77 eee
(Gado yi! —[g us A + 9sS09 «|(GH) aw [24H 8 HTS uso, (Se WIL [49 Bo = (4 2°A‘x)d
Yid9Q jyudJsSUCD 40 4daJDM Ul Spsoday auNssaid
power spectrum of the wave record taken of the free surface by equa-
tion (12.20). Given either one and given the depth of the water,
and the depth of the instrument, the other can be computed except
for the high spectral components lost by filtering due to depth due
to the fact that the pressure recorder simply will not respond to
minute variations in the pressure field.
At the bottom, z equals minus H, and equation (12.20) becomes
equation (12.21). The pressure record recorded by a pressure re-
corder on the bottom is therefore some segment of one of the infin-
itely long records which result from the limit of a partial sum such
as those discussed in Chapter 7.
Wave refraction in the transition zone
The refraction of the short crested Gaussian waves which have
been derived in the previous chapters is an extremely complicated
problem. The basic theory which has been derived by Sverdrup and
Munk [1944], Johnson, O'Brien, and Isaacs [1948], arthur [1946],
Eckart [1951], and Arthur, Munk and Isaacs [1952], is correct, but
it applies only to one pure sine wave of constant period. The theory
needs to be placed upon a somewhat firmer theoretical basis as pointed
out by Pierson [195la], and the results of Eckart [1951] are a first
step in this direction.
The theory of wave refraction is at the level of theoretical
development which was attained by the theory of optics before the
work of Luneberg [1944, 1947] in optics. That is, wave refraction
theory has been derived not from the basic hydrodynamic equations,
but by a series of approximations and assumptions about the nature
of the motion of a pure sine wave over a bottom of variable depth.
BH
For example, Snell's Law is either assumed or proved from very
Simple considerations. Also the shrinking in the wave length as
the wave progresses into shallower water is not shown to be a con-
tinuous process; that is, the length in deep water is Ly and the
length in water of depth, H, is given by equation (12.10), but no-
where in the theory is the exact profile along an orthogonal given.
Luneberg started with Maxwell's equations and showed how the
theory of geometrical optics for light or any other form of electro-
magnetic radiation could be derived rigorously from the equations.
In addition, the systematic approach which he used has permitted
attempts to refine the theory to the level of physical optics. Con-
siderable success along these lines has been obtained by Keller,
Kline, and Friedman of New York University.*
Similarly, it ought to be possible to derive wave refraction
theory with the original hydrodynamic equations as a start. Were
this done, the results would possibly indicate better relations for
the wave height in the neighborhood of a caustic made possible by
the consideration of higher order effects.
One fundamental assumption of wave refraction theory is that
the dimensions of the refracting bottom contour systems must be
large compared to the wave length of the waves on the surface. As
has been pointed out by Pierson [195la], in many practical cases
*The author in this section is indebted to Professor Joseph Keller
for his series of lectures on geometrical optics given at the Math
Institute during the past year. Wave refraction theory for
Gaussian waves has an analogue in the problem of colored light
scattered in two dimensions passing through a medium with a con-
tinuously varying index of refraction such that the index of re-
fraction is a function of the wave length of the light and of
only two space variables.
38
this assumption is not fulfilled too well. Thus some numerical
results of wave refraction theory must not be taken too quanti-
tatively although they may be correct within 30 or 40 per cent.
Were the theory derived rigorously, it might then be possible to
estimate the amount of error introduced by the above assumption in
a practical case.
ae a a a -
From the results of the past chapters, it is possible to deter-
mine the two dimensional power spectrum at a point located offshore
in deep water from a point of interest in the refraction zone. For
example, the power spectrum could be determined by direct measure-
ment from stereo-aerial photographs and deep water wave records as
a function of time at a point a few miles from the coast under in-
vestigation. By the methods of Chapter 9, if the torm power spect-
rum were known, it would then be possible to forecast the power
spectrum offshore from the point of interest. Given these deep
water quantities, what can be said about the records which can be
obtained in the transition zone?
The problem can be solved to various degrees of accuracy.
Given a linear sloping beach, and the results obtained by Peters*
expressed in terms of the parameters, # and 9, and a deep water
wave of unit height, then it would be possible to find a represent-
ation for the sea surface in the transition zone by a Lebesgue
Power Integral in the Gaussian case over the power spectrum multi-
plied by Peters' solution. At any point the wave record as a function
of time would be Gaussian. As a function of x and y, the elemental
Feces ences to Part I. The paper has appeared in the publication
ed.
32
crests would be curves in the refraction zone. Such a solution
would be exact (in a linear sense) everywhere, and would agree
well with reality until non-linear effects near the breaker zone
caused it to fail. Apart from the difficulty of evaluating the
result, (and it is difficult enough for a pure sine wave), very
few linear sloping beaches are found in nature. As soon as the
depth becomes a complicated function, wave refraction theory must
be used.
The solution to the wave refraction problem in the transition
zone is found in practice by graphical methods. The orthogonal
method as presented by Johnson, O'Brien, and Isaacs [1948] and most
recently by Arthur, Munk and Isaacs [1952]* is the best procedure
because errors are not cumulative and the method discovers caustic
curves. It would now appear that it is possible at a sufficient
distance beyond the caustic to use the usual formulas for the value
of KD based on the separation of the orthogonals at the point of
In general, for a pure sine wave in deep water, the crests in
the transition zone are curved. All of the systems discussed so
far consist of elemental straight crests. The equations for the
crests in the transition zone are very complicated and they have
rarely been formulated mathematically except for extremely simple
bottom configurations. Some examples in which the crests can be
found explicitly (since the orthogonals are given) can be found in
papers written by Arthur [1946], Pierson [195la] and Pocinki [1950].
*The abstract of the paper by Arthur, Munk and Isaacs[1952] can be
found in the American Geophysical Union's vrogram for its May 5-7,
1952 meetings in Washington. A preliminary copy provided by the
authors shows that errors in previous methods can be eliminated
by a more refined application of Snell's law.
40
In general, the refraction problem is treated even less
specifically for practical purposes. Given the deep water wave
direction, amplitude and period of a pure sine wave, data are
usually provided which give the angle the crest makes with the
shore and the amplitude of the sine wave as observed at one point
of special interest. For example, the data presented by Pierson
(195la] for Long Branch apply only to one point, namely the point
where the wave recorder used to be. It was at a depth of 21 feet,
mean low water, offshore from latitude 40°18.2'. It is now at a
depth of 30.5 feet, mean low water, offshore from latitude 40°18.2'.
The slight change in location has negligible effects for this case
since most of the refraction occurs in deeper water.
In figure 32, consider the point B, in deep water just outside
of the transition zone. At the point B, Xp and Yp are zero and the
wave system will be referred to the Cartesian coordinate system in-
dicated on the figure. If a pure sine wave of spectral frequency,
Hy» were to exist in deep water and if it were traveling in the
direction, ey” (measured with respect to o,* equal to zero coincident
with the Xp axis), then the sea surface could be given by equation
(12.23). The equation would hold everywhere in deep water. In the
transition zone, equation (12.23) is not valid.
In figure 32, consider the point C in the transition zone. At
the point C, XR and Yp are zero. The XR axis is parallel to the
Xp axis (and not necessarily coincidental). The depth at that point
is H(xp,yp) = H = H(0,0) referred to this coordinate system. If the
assumptions of wave refraction theory hold, then the bottom is nearly
level at that point, The crests although slightly curved will have
41
The Transition Zone
In Deep Water near X=Xyp > Y=Yp
2 * ; a
[= Cos E & cos 6, ale Yo sin a, | a us| (12.23)
In Transition Zone, near Xx=XR, y=yp
n=Agy COS E ae HH) [Xr cos 8, + yr sin Or} — Hot +8] (12.24)
where Ary =KxyD A (2°29)
Problem; to generalize above to a short crested Gaussian
Sea Surface
Definitions of Terms
[a,(4,8)]” is the power spectrum at the edge of the fetch (12.26)
[Ane (H4,8e)) is the forecasted pOwer spectrum at the point offshore
from point of forecast given by x,,y,. (12.27)
6-=@-@), from equation (9.61) (12.28)
[Ave # (1,9) is the forecasted power spectrum at the point Xp
rotated to line it up with the refraction diagram (12.29)
OF = 6--B where B is the angle between the continuation of
R through 8B and the iine drawn out to sea perpendicular to
the coast through the point (xp, ye) (l2.30)
2
[Azan (Hr Oa) is the forecasted power spectrum in the Refraction
Zone at the point xg,yp and at the depth, H (2.31)
2
Problem; To find [Aran Hs OR) from refraction diagram data
and the forecasted power spectrum.
Plate LX1V
4SD09 04
4D|NDIpuadsad auiy
ater Zone
Breaker Zone and/or Shallow
a
i=
°
N
Cc
°o
=
”
c
°o
-
=
oe
‘Asoay, uo1pd04yas aaom Kq [ee] 04 [(+9'7!) 7y]
wos} 06 Of; MOY JNO PUly Of SI Wa|qdsg
juasaid
“WalGOig UOI}DDIJaY AADM JOY UOIVOJON ‘ZE “HI4
ey] [(ga'7) #2]
O=“ O=*x'9 4
O=% o=°x'g iv
| 2) 29
g-“9 =
woibo0ig vol 04jay
yyim wniyoads
JaMOg pajsooas04
uBijD Of uo1yOJOYy
(26 ‘r) w2y]
Deep Water
uolyOJOY
ajdwis Ag
29 — 'g
°9-@= 469
O=%‘O =9K 4D 4aJOmM
daap ui wnsayoads
JaMOdg pajsodas04
[(49'7)42y]
ainpad0ig
Buijsodas03 Ag
'g@-o—-V
S3ADM 40 a24N0S 40
wnij9adS JamM0g
[le'7)?v]
-—43-
a certain direction of forward progress at the point and a wave
length determined by pu I and H. Finally, the crests will have a
new amplitude and phase at that point, which can be determined from
tracing the family of orthogonals near the point of study. These
features are all incorporated in equation (12.24). Any is the new
height at the new point of observation which can be determined from
A by wave refraction theory and equation (12.25). Op is the new
direction of progress of the crests. 6 is a phase lag due to the
Slowing down of the crests.
Equation (12.24) does not hold everywhere in the transition
zone. In fact it holds only at one point; namely, Xp = 0; Ya = 0.
However, in the vicinity of the point, the equation approximates
the local state of affairs. The degree of approximation is somewhat
crude but actually to develop the formulas with curved crests which
would apply to greater distances away from the point of observation
would be far too difficult.
The problem of the refraction of a short crested Gaussian sea
surface can be solved by showing how it is possible to extend the
application of the refraction data already obtained for pure sine
waves to an infinite sum of infinitessimally high sine waves in ran-
dom phase. It can be done easily to the degree of approximation
just described above. In this way, the sea surface is approximated
in the vicinity of the point under study by a Lebesgue Power Integral
quite similar to the one discussed above and in previous chapters.
A wave record taken as a function of time at the point of interest
will be quite accurately given but the slight curvature of the
individual crests in the neighborhood of XpoVp will not be represented.
From the edge of the fetch to the transition zone
At the edge of a storm at sea, in connection with the fore-
casting problem, it is more convenient to line up the 9® equal to
zero axis with the direction of the winds in the storm. The distance
R, from the center of the forward edge of the storm to the edge of
the transition zone is essentially the same, as far as the magni-
tude of the parameters is concerned, as the distance to the point,
ae Vp O in the transition zone. Thus in figure 32, the distance
from A to B is essentially the same as the distance from A to Cc. In
the process of forecasting considerations for the point, C, then,
the procedures presented in Chapter 9 can be applied to reach the
point, B, and then refraction theory can be applied without considera-
tion of the added distance from B to C in order to find the effects
at C. This procedure also neglects some minor effects on the power
Spectrum since it varies slowly from point to point in deep water
and all rays arriving at C do not come from B. Various operations
must be applied to the power spectrum at the source in order to
find the power spectrum at B and in order to put it into a mathe-
matical form which permits the application of refraction theory to
the power spectrum at B. Then the problem of prime importance in
this chapter is to show how it is possible to go from the point B
to the point C.
The operations needed to proceed from point A to B and to orient
the forecasted spectrum at B so that it can be easily refracted are
Shown on the right of figure 32. The various terms are defined in
Plate LXIV.
At the forward edge of the storm, the power spectrum can be
a2
defined to be [a5 (@ ,0)1° as in equation (12.26). The angle 6 is
defined as zero along the x axis defined in relation to the storm
in connection with the forecasting problem. At B, the forecasted
power spectrum can be found from the results of Chapter 9. A new
angular variable can then be found, which will be called Ope In
terms of the forecasted power spectrum, by equation (12.28), the
power spectrum at B is given by [Bop ( H yO) 16 The direction 6,
equals zero is usually the dominant apparent direction of the short
crested waves at B. The line, ©, equal to zero, is shown on the
coordinate system labeled B, in figure 32.
The variable, 6,;, must be transformed to the variable, 6p *; in
order to align the forecasted power spectrum with the refraction
diagrams for the point C. The angle, On*; can best be picked to
be zero when the angle coincides with the Xp axis chosen perpendi-
cular to the coast through the point C. The angle £8, which defines
6," in terms of ep is the angle between the continuation of R through
B and the line xp equal to zero. The function [Aon *(# ,6n*)]°, is
thus the forecasted power spectrum at B aligned properly to the
Xp = O and xp = O axis. The line, On* equal to zero, is shown on
the polar diagram marked By in figure 32.
If now, (BE (Pare) is is defined to be the power spectrum at
the point, C, (that is, in the vicinity of the point Xp = 0, Yp = 0)
how can it be determined from [Ane (1 0p *) 1° given the usual re-
fraction data? It can be found by applying operations to the con-
tinuous spectrum which are analogous to those operations applied to
pure sine waves in the theory of wave refraction. The reader can
check each step of what follows and assure himself that each step applied
to a sum such as in equation (8.5) would yield correct results for
each discrete component.
46
Needed modifications of refraction diagrams _
The next step is to modify the usual refraction data so that
they can be easily applied to [A (iaeg irs One of the quanti-
ties evaluated in a study of refraction is the quantity Kie This
quantity is a value related to the ratio of the distance between
orthogonals at the point of observation to the distance between the
Same orthogonals in deep water. Peocedates for obtaining the quantity
are given by Johnson, O'Brien, and Isaacs [1948]. The value of Key
must be multiplied by a factor D which depends on the depth below the
point of observation and the period of the wave. It is essentially a
correction for the group velocity effect in order to maintain a con-=
stant energy flux between orthogonals. The product KD is usually then
plotted as a function of the period and deep water direction of the
wave. Such diagrams are given by Munk and Traylor [1947] and Pierson
[1949]. The isopleths are lines of constant K,D on a polar diagram.
To prepare such a diagram for application to the refraction of a Gaus-
sian short crested wave system, it is necessary to invert the diagram
and plot it as a function of / and 6," where H is the spectral fre-=-
quency ard Op” is the direction toward which an elemental crest is
moving just offshore in deep water (e,* is zero when the crest in deep
water is parallel to the coast). The values on the diagram must also
be squared point for point. The result is a considerably more rapid-
ly varying function which will be defined to be the function
[K,D(# y0_*) 1° as in equation (12.32) and which will be named the
spectrum amplification function. The function must approach unit
values as # approaches values of the spectral frequency such that
the depth is greater than one half of 2mg/p °.
a7
The other quantity usually evaluated in refraction data is
the angle the crests make with the shore at the point of observa-
tion near the shore. This angle is identically eaual to op which
is the direction toward which the elemental crest is traveling.
8p equal to zero designates a crest traveling directly toward the
coast. These values can be plotted as a function of # and Ope
The function, @p = O( 4 ,e,*), can thus be shown as isopleths of
8, as a function of w and On" As# becomes large, ©, equals
o,* asymptotically. These relations are defined in equation
(12.33). © (#,0,*) will be called the direction function.
The inverse of 0, = © CH en”) is also needed. That is, values
of 6, isoplethed on ap 4,0, polar coordinate system, are needed.
This inverse function is defined by equation (12.34) as 6,*= O*(,e,).
From the isoplethed values of equation (12.34) it is possible
to evaluate lr ( # yep) as given by equation (12.35). The function,
['(#,0,), is the change of @,* per unit of change of 6, expressed
as a dimensionless number in radians per radian or degrees per de-
ereee r(p ,Op) is a measure of the crowding together of the power
spectrum due to refraction and its significance will be discussed
later. It is the Jacobian of the inverse of the direction function.
Steps in wave refraction
Given the functions described above and their definitions, three
steps are required to find Eee Cae from [Ap Mt # 50,*) 1°. The
function, eae ils could also be the power spectrum of any
system observed immediately offshore in deep water. At this stage,
then, the functions defined by equations (12.29), (12.32), (12.33),
(12.34) and (12.35) are known.
48
The Transition Zone
2
[KyD(#,9F) | is the spectrum amplification function. It is the square
of the ordinary Refraction Diagram plotted in the 1,9¢. plane instead
of in the T,@ plane. (2,32)
Ae
Opx=O(p,8F)is the angle that the crest makes with the y axis at Yr,
Xp Plotted as a function of » and an, ie. the wave frequency and
the deep water direction with respect to a line perpendicular to the
coast at the forecast point. (12253)
g* = @"(u,8,) is the inverse of the functian given above (12.34)
06% JO"(udp)
Psd eA aa ct als a
Gone eee RD (12.35)
Steps in Wave Refraction
: 2 Dealia : :
Step I, Multiply [A.(,6%)| by [Ky D(H, 4F ) graphically to find
[aoa OP I” [Ku D(H 8]? (12.36)
Step I, Substitute equation (12.34) for oF to express(12.36) as a function
of 8 andfind [A,%u,9 (u,4a))|* [KyD(2,97%(1,8a))| (12.37)
StepII Correct, by multiplication by equation (12.35),for distortion
to find [Azpy(H,On)]°= [Aoet(H,0 (4,8R))]° [Ku D(#,07(1,9R))] °F (4,8R) (12.38)
Plate LXV
Step one is to multiply the power spectrum in deep water by
the spectrum amplification function. Graphically this can be done
by computing the value of the product point for point of (12.29) and
(12.32). For any finite net over [A5p *(H on ]*, as in equation
(9.22), the result of this operation is to predict the height of
each elemental wave in the partial sum for the new point of obser-
vation.
Step two is to substitute equation (12.34) for 0,* everywhere
it occurs. This converts the product given in (12.36) to the pro-
duct given in (12.37). The result is some function of # and Ope
For any partial sum the result is to assign the correct spectral
directions to each elemental wave at the new point of observation.
In general equation (12.36), is a continuous function and the effect
of this operation is to squeeze (12.36) into a more compact function
in the # 99, plane since elemental wave components with widely
different directions in deep water have more nearly the same direction
at the point of observation in the transition zone.
Graphically this step can be accomplished by plotting the value
of (12.36) at # =, and 6," = Orr in the BO," coordinate
system at the point OR = @ (eH 7en7) and # = By in the new H 58,
polar coordinate system. A line on which (12.36) is a constant
is thus mapped into a new line in the pw 29, plane on which the same
constant value is found.
The third step is to multiply (12.37) by T (#,6,) as given
by equation (12.35). The result is the desired power spectrum,
[asee( sea l=, as a function of # and ©, at the point of obser-
vation in the transition zone. This step is needed since the power
50
spectrum is treated as a continuous function. If the spectrum
were discrete jumps in E,(,0) as in equations (9.38b) and (9.39),
this step would not be needed.
['(#,0) could be called the distortion correction function.
It is the Jacobian or equation (12.34) and it corrects for the
squeezing together of (12.36) when it is changed to (12.37).
Consider an example to clarify this point. Suppose that (12.36)
is given by a constant value from # equal to 27/10 toH equal to
2r/9 and for en equal to -1/30 to +1r/30 and by zero otherwise, and
that (12.37) is the same except that 0, ranges from -7/60 to +1/60.
Both (12.36) and (12.37) represent the average potential energy at the
new point of observation, and yet the integral over # and 6p" for
the first case is not equal to the integral over M and Op in the
second case. But the value of 06,*/de, in this case is equal to
two radians per radian, and thus doubling the value of the second
spectrum corrects the value of the average potential energy.
The power spectrum of the waves in the vicinity of the point
under study in the transition zone is now known. It is given by
equation (12.38). In terms of the XpoVR coordinate system at the
point, C, in figure 32, and in terms of the p 99p» power spectrum
defined there by the above procedures, the short crested sea surface
near Xp and YR equal to zero is given by equation (12.39). In appear-=
ance, the sea surface will be different in many ways from the sea sur-
face at the point, B. The procedures described above predict many
properties of the waves at the point C which are verifiable by
aerial photographs and observational procedures. The properties
will be described later.
One property which follows from the derivation is given by
a
TAXI 90ld
a[(siea) ql 4809] , (7) Hatly]
(fv 21)
ea
SJaps0001 ainssaid ysow Aq padAsasqo uoljouUN} BU} JO WNIyoedS 4JamMOd aU]
[Greta 45-5
((71) + 471)s090/ Bd = (4'H-)d
foe)
HIB + rps
2 [7 )H4y]
(2p 21)
|
1 )HYy] (71 nf : o
(1b 21) Mal) My\(7) A+ in)soo/= (Ls
[ee]
n= 0
(Ov 2l) — ven} ipep yor =
(0)
pgp (ta) [ea], aH] [eg 3? = tpsep Its dp'z}g" “ [(:e'" 4
(62°21) Ap gpaltg'r ) Hee eee aes y A Oo) iy ast eheax) i
Z
DJDP Jau;O PUD WOIbDIP uUOlJOD4Ja4 Ppapesu puss “tgomngee] | vento
@UOZ UOlJISUDJ] 9YL
equation (12.40). After multiplication of the deep water power
spectrum by the spectrum amplification function, (12.32), the sub-
sequent change of variables does not affect the potential energy
of the waves at the point of observation. Nevertheless the potential
energy at the point in the transition zone may be completely dif-
ferent from the potential energy in deep water since the spectrum
amplification function in general does not leave the total volume
under [Ao*(p ,On*) 1° unaltered.
The spectrum amplification function can change markedly upon
the choice of different points, C, in the transition zone. In the
short distance of thirty miles along the coast of New Jersey, it
can vary tremendously. Consequently not only will the wave height
vary over a distance in the transition zone which is very short com-
pared to the deep water forecast parameters but also the "signifi-
cant" period will vary from place to place. These points will be
verified by examples in a later chapter.
The wave record at the point of observation
The wave record which will be observed by, say, a step resist-
ance guage at the point Xp = 0, Yn = O is given by equation (12.41)
where [Apy(# )]° is the integral over 6p of [Appu(# ,0,)]*. This
function has all of the properties of the one described in Chapter
7 and it can be derived from equation (12.37) by the exact same
arguments given in Chapter 10 for the deep water case. In Chapter
7, a wave record in the transition zone was shown to have the pro-
perty that points chosen at random from it were normally distributed.
The definition of the integral given in equation (7.1) and in sub-
sequent equations can just as easily be applied to equation (12.41)
ie}
and the results are thus to be expected. The reader, though, would
have been perfectly justified in objecting at that point in Chapter
7 where a transition zone wave record and transition zone pressure
records were used to prove the Gaussian property, and then the
Gaussian property was tacitly assumed for deep water waves. These
results now show that given that the waves have the Gaussian property
in deep water, it then follows that they have the Gaussian property
in the transition zone (and conversely since the wave refraction pro-
cess can be theoretically reversed).
From equations (11.3), (12.41), (12.18), and (12.21), it then
follows that the pressure record which will be recorded at the bot-
tom by a pressure guage at the point of observation in the transition
zone is given by equation (12.42). The pressure record is therefore
Gaussian. The power spectrum of the pressure record is related to
the power spectrum of the waves passing overhead by equation (12.43).
Given [Ap py ie the power spectrum for the surface record can be
computed from equation (12.43), and conversely. For those pressure
recorders which respond to different periods in different ways, the
calibration curve appropriately modified must be inserted as another
function at this point. An instrument with a completely flat response
curve is assumed in this derivation.
Ewing and Press [1949] are of course correct in their statement
of the problem of pressure record analysis. These formulas simply
formalize the procedures to be employed.
Equations (12.42) and (12.43) are extremely important to the
practical engineer. Nearly all of the wave records being taken at
the present time in the United States are made with a pressure recorder
54
on the bottom at a point in the transition zone at some depth, H.
for example as summarized most recently by Snodgrass [1951], are
for all practical purposes useless. For that matter any step in
current practices which involves the assumption that the "significant"
period can be treated as if it were a discrete spectral component
automatically introduces huge errors for "sea" records which com-
pletely invalidate all quantitative values which result from the
analysis.
In order to demonstrate this point, some statements will be
quoted from the paper by Snodgrass [1951]. Then the point at which
the error was made will be shown. Finally Snodgrass' analysis of
the inaccuracies which result will be interpreted in the light of
the results shown in this chapter. Selected quotations from the
paper referred to above follow.
eens. oie The following basic definitions have been accepted
(Folsom, 1949):
Ete Sieerere
3. Wave period is the time interval between the appearance
at a fixed point of successive wave crests.
4. Characteristic wave period is the average period for
the well-defined series of highest waves observed.
"Analysis of wave records for wave period. Analysis of
wave records for the characteristic period is accomplished by
measuring the average period of the larger, well-defined waves
appearing on the record........ The characteristic period of
the waves does not describe the period-distribution, as the
characteristic height describes wave-height distribution. .....
ora op information is needed to adequately describe wave
periods.
2D
SS eS
surface wave records. The records differ, however, in that the
short period waves are not registered to the same degree as the
long period waves by pressure recorders due to the hydrodynamic
pressure attenuation of the water. As a result, many of the
shorter period waves may not appear on the pressure record.
"If the technique of measuring the periods of only the
larger, well-defined waves of the record is followed (as de-
scribed in the above section), the measured period will be
approximately the same as would be obtained if the record were
made with a surface type gage. For locations on the exposed
coast, the short period waves, not recorded by pressure, generally
are generated by local wind. Irregular and of small amplitude,
these waves are neglected in the analysis of surface records.
"In several cases, attempts have been made to utilize the
hydrodynamic attenuation of short period waves by installing
gages in deep water (about 600 feet) so that only the waves of
long periods (the characteristic forerunners of storms) will
be recorded. These long period waves are recorded by pressure
heads installed in shallow water, but are "lost" in the record
of shorter period waves. Installations of this type of instru-
ment have been made, but due to instrument difficulties no
satisfactory records have been obtained.
"To obtain the surface wave heights from the pressure
record, two factors are required; (1) the calibration of the
instrument and (2) the pressure response factor relating the
subsurface pressure fluctuations to the surface wave. Thus,
hie
H = wave height at the surface (in feet);
Cy = calibration factor of the instrument (expressed
in feet of water pressure variation per chart
division);
K = pressure response factor based on the depth of
the instrument, the depth of the water and the
length (or period) of the wave being recorded;
R; = reading of the instrument;
the following equation is used to obtain the surface wave height:
H = C4/K (Ry) e e e e e e e e e e e e (1)
"The calibration factor for most instruments in use today
is a constant independent of wave period and depth of the
instrument. The instrument provides a record of the pressure
variations at the instrument which is accurate in amplitude
and wave form.
56
"The relation of the subsurface pressure fluctuations to
the surface wave has been determined theoretically for two
dimensional, irrotational motion of an incompressible fluid in
a relatively deep channel of constant depth (Folsom, 1947).
The response factor K has been shown to be:
cosh 2rd/L (1 - 2/d)
RS coshorasL ne ee pre ee
where
z = depth at which the pressure variation
is being measured (in feet),
d = depth of water at the instrument (in feet),
L = length of the surface wave (in feet).
"When z = d, the pressure variation is measured at the
bottom and equation 2a reduces to:
eee eee
K — cosh Or d L e e e e ° ° e e e ° e e e (2b)
Pressure records do not enable the direct measurement of wave
length; the wave length must be calculated from the wave period
using the following equation:
2
L = &-) tanh or d/L. ee ee ee (3)
Where T = wave period (in seconds).
"Suitable graphs and tables (Wiegel, 1948) are available for
the solution of these equations. Graphs have been prepared which
enable the response factor (K) to be determined if the water
depth (d), instrument depth (z) and wave period (T) are known.
Two errors arise when the above equations are used to determine
the response factor (K) for ocean waves; (1) an average or
characteristic period must be used in the equation while the
actual wave period is continuously varying and individual waves
are not sinusoidal in form, (2) wave heights computed from these
equations have been shown by several observers to be from six
to twenty-five percent too low.
"Considering the first of these two errors, greater accuracy
probably could be attained if the pressure response factor (K)
were determined for each wave and the equivalent surface wave
were individually computed. This procedure might be feasible
from a practical standpoint if the statistical distribution
of wave height and wave period could be established so that
fewer waves need to be analyzed to completely describe the state
of the waves. (See the above section on "Analysis of wave re-
cords for wave height".)
37
"The second of these two errors emphasizes the need to re-
consider the basic theory which does not agree with experiment.
Every observer who has simultaneously measured the surface waves
and the subsurface pressure fluctuations has *.: 4d the theo-
retical response factor determined from equativ” 2a to be too
small. Ten random measurements made at the Waterways Experiment
Station (Folsom, 1947) indicated an average correction of 1.07
should be applied to equation 1. Seventeen laboratory measure-
ments at the University of California, Berkeley, indicated an
average correction of 1.10 (1949). Field data reported by the
Woods Hole Oceanographic Institute (Admirality Research Labo-
ratory, 19473; Seiwell, 1947) indicated a correction factor in
excess of 1.20 while the three sets of field data obtained at
the University of California (Folsom, 1946) indicated values of
1.06, 1.00, and Lelde”
The basic fallacy occurs at the very beginning of the material
quoted when the statement is made that "The following basic defi-
nitons have been accepted" and that the "wave period is the time
interval between the appearance at a fixed point of successive wave
crests." What is measured are the time intervals between success-
ive relative maxima of a non-periodic* function. These time intervals
have absolutely nothing to do with the time intervals between success-
ive crests of a pure sine wave such as in equation (2.19). From the
measurement of this quantity, the error is compounded by averaging
a number of such measured quantities and calling the result the
"characteristic wave period." From then on, the "characteristic
wave period" is applied to the wave record as if it were actually
the true period of the wave record and as if the wave record had
one discrete spectral component. All wave records are thus treated
as if they were the one special case given in example one of Chapter
9. All of the subsequent formulas quoted are also based upon this
assumption.
*See the correct mathematical definition of period in Chapter 2
(equation (2.11) for a pure sine wave).
58
For a "swell" record with a narrow band width such as those
shown in Chapter 9, the fallacy of the method does not produce too
important a discrepancy between the theoretically computed values
of the surface quantities and the observed surface quantities, but
for a "sea" record, such as those shown in the appendix to part
one, the procedure effectively ignores a large part of the high
end of the power spectrum. The surface "significant" height (or
"characteristic" height) in "sea" conditions is always observed to
be greater than the value predicted erroneously from the pressure
record, and the surface "significant" period, (or "characteristic"
period), were it also measured, would be found to be lower than the
"significant" period (or the "characteristic" period) of the pressure
record.
Thus the fact that "wave heights computed from these equations
have been shown by several observers to be from six to twenty-five
percent too low" is not at all surprising. The error is not a con-
sistent error in that it varies from record to record depending on
the power spectrum and in that it varies as a function of the depth
of the pressure recorder. If the basic theory referred to in the
last paragraph of the quotation is the theory which accepts as a
basic definition the definition of wave period at the start of the
quotation, then these considerations have shown wherein the error
of the theory lies. ,
On the other hand, if the basic theory referred to in the last
paragraph of the quotation is the theory of purely sinusoidal waves
with one discrete period, then that basic theory is correct and the
theory has been misapplied to a pressure record which is not a purely
se]
sinusoidal variation with one discrete spectral period.
Finally, stated another way, most of the current theoretical
work on wave theory would be correct if ocean waves were actually
pure sine waves. Since ocean waves are not pure sine waves, the
theory has been misapplied to situations it cannot possibly adequately
describe. The derivations and considerations in this paper when they
refer to Gaussian systems apply exactly to ocean waves as they act-
ually are, except for non-linear effects. Ina later chapter a pres-
sure record will be correctly analyzed, and the correct values of
the surface wave record will be deduced from the analysis by the use
of equation (12.43).
To the reader, it may seem that the author is being unduly
harsh with the authors of other works using the incorrect methods
described above. The works of Wiener, Tukey, and Hamming did not
appear until 1949, and the methods and techniques based on the sig-
nificant height and period were undoubtedly the best that could be
employed at the time. The literature on practical wave theory is
full of such results, in particular, some of the results of Pierson
[195la] which use the concept of significant height and period to
obtain theoretical results are completely wrong and practically
useless.
The velocity field, kinetic energy, and energy flux in the
transition zone
From previous considerations, the u, v, and w velocities at the
point of observation are given by equations (12.44), (12.45), and
(12.46). The vertical velocity is zero at the bottom, and the functions
automatically satisfy the equation of continuity and consequently
60
the potential function. At z equal to zero the expressions simplify
considerably, and possibly some interesting properties about the
power spectra in the transition zone can be deduced by considerations
Similar to those of the previous chapter.
The kinetic energy integrated over depth and averaged over
time and the y, direction is given by equation (12.47). The Itcoth
of H 2H/e times the hyperbolic tangent of }# *n/elI( # 4H)] is equal
to one by virtue of equation (12.11). Thus the average kinetic energy
is equal to the volume under ec ee ils (that is, equation (12.48))
times pg/4. From previous considerations this is equal to the po-
tential energy averaged over YR and t. At a fixed point, say, Xp
and yp equal to zero, where the statement is exact these values also
hold and the potential energy and the kinetic energy (integrated over
depth) averaged over time are both equal to (pg E )/4. This
RHmax
statement can be proved by use of the results of Chapter 10.
The flux of energy toward shore in ergs/sec per centimeter of
length along the YR axis, is the average value of the work being
done on the Ypoz plane determined by setting XR equal to zero. The
wave power is then given by equation (12.49), and the results check
with the same result in Lamb [1932] where the flux is determined
for a pure sine wave.
If the short crested wave system is concentrated in a narrow
8p band width at the point of observation, and if the important
spectral components are all traveling in nearly the same direction,
Say On,, then it is possible to omit the cos 8, term in equation
(12.49). Then equation (12.49), as modified, is the flux of energy
in the Ont direction at the point of observation. It can then be
estimated (except for the short crested effect) from [Ang (H Ie as
61
ao
it
(8¢ 21) "PEP, [(ug'n) HB a] ems aJauM
elo
(Ly ai) ; er os oor: re
xpw ; t
oe oe = PUES (HH tT =) 4UOK(H Ne =Kp jp ZP(gm+2A+ 2n)—S Sait = zp» * a
OF 1 4yih a i Oo a
2
[HlH' W)T Flysos mw,
(9¢°21) ePp7p C yey) ee
2 8 Jez) wi Bluuis(H 1) yt
[6 1) + y7t-(4g usd + (ee uIS
uy HuzZ [Ht mM Slusoo ra
6 Se
uel ano |e v| (H+2MH'7/)q BlusootH )q7 “guis|( i+ ir! (49 uis4k +49 s098x)(H $09
Z
; ‘ 6 in
ee as (ot sey [H 1 a al 4g00|(g'n)A +47-(} soup bakes naa Sheer ie
zZ (HZ) B]usoo(H'7)17 gu! @ ails :
TZ 0
9UOZ UOIJISUD4] BU]
determined from either a pressure record and equation (12.43) or
from a record of the free surface. The computation of the energy
flux from the "significant" height and period is completely meaning-
less, especially for "sea" conditions.
If the beach has contours parallel to a straight shoreline, and
if the waves have infinitely long crests (as in equation (7.36))
which are parallel to the shore, then the wave power intezrated over
depth and averaged over time is given by equation (12.50) on the
left in the transition zone and on the right in deep water. The
energy flux in this case only, is equal at both points. Equation
(12.50) is the extension of usual refraction theory considerations
to Gaussian systems. Equations (12.51) and (12.52) are the analogues
to (12.50) for the discrete case. They are given by Sverdrup and
Munk [1944b] and Mason [1951].
One of the unsolved problems of wave forecasting and wave anal-
ysis theory in terms of "significant" height and period was the
problem of the combination of wave systems from two different storms
either in deep water or in the transition zone. In terms of power
spectra and the methods developed in this paper, the problem can
easily be solved. It is easy to prove that the power spectrum of
the sum of two different disturbances equals the sum of the power
spectra of the two different disturbances. From this, it follows
that all other properties combine in the same way, and the pressure,
velocity components, and energy flux of combined wave systems can be
found immediately. If the sum of the two power spectra yields a power
*This section is also a proof of the statement made on page 260. The
argument is given for two superimposed small spectra, but it would
also follow for two adjacent spectra, (page 260 of Part I).
63
TAXT 940!d
5
: pap 4's | @ *
(2G2i) Sas tt =
(ISI) 0 U2/_°H
20 H
/ (ater (ares 8
(OS 21) pepe LAs, alien A OA (aca oes a eg 1) ie
Pw) 26d "8 y) ay) | (H'7)Ia71H?e a eGo) pe l
@ oo rn
‘a4OUS Of Ja}]0410d
}a||;O40d et UaNAIO
S}S84190 pud'[i¢y]UOIyDNba ay]; SAaADM ‘“Sinoyu0d YyoDeq
(oo 6 _
(H‘r)q71 HW)larz2 Jyurs 6
a
ral —————
BigP 8 800 yHezy] [* eo ore +1]
(6¢21)
1 H7
ny a»? +
+k
Ap ¢p(zp( y‘z‘S‘A'4x) 0. (4'2°4R4x) ) | Fil = ZPdM/ = dM
Fy oetay
Of 1 tyi Kak 9
@uOZ UOl}ISUDI] BY]
spectrum which does not satisfy the conditions imposed in Chapter
11, then the sea surface in the region of superposition will break
up due to non-linear effects. However for swell arriving from a
distance this effect is usually of no importance and thus most re-
fraction problems are easily dealt with.
Consider equation (12.53). It states that given two power
spectra, [Ana (H 90)1° and ieee (Paseo for two different wave
systems present at the same point and time of observation, then the
total effect is obtained by adding them point for point and calling
the sum ees Ge) le If equation (12.53) is true, then a
similar equation holds for any number of power spectra, and the state-
ments made in the paragraphs above are proved. All of the steps are
valid for both deep water and the transition zone so that H can also
be infinite in any of the equations which follow.
Now, [Apu7(# ,0)]° substituted into an equation like (9.47)
would yield an expression for a sea surface which will be called
77, and similarly (Aouqtr( #59) 1° would yield 7,,. Consider the
power contributed to some one net element in the H ,® plane, upon
passing to the limit inside the one net element, and consider that
part of the total integral contributed by TAI and "ATI which
involves these power contributions. Let AE, be the power contri-
buted by Any # 9) to "AT and let AEty be the power contributed
by Bout! # 19) to 7 Ary as defined by C12 G54)
Then points chosen at random from mz, either as a function
of time at any fixed point or as points chosen from the whole x,y,t
space, will be distributed according to equation (12.55). Points
chosen at random from 1 AIT will be distributed according to equation
65
(12.56). From the derivations of the power integrals involved,
there is no correlation between 7 AI and 7 AID? and the two
distributions are independent. These statements follow from the
results of Chapter 7 and Chapter 10.
A theorem of statistics can now be used to prove equation
(12.57). If two independent random variables are distributed ac-
cording to the distributions given by equations (12.55) and (12.56),
then the sum of the two independent random variables is distributed
according to equation (12.57). For a proof of this theorem, see
Cramer [1946] (page 212).
These equations hold for any net element anywhere in the
and E
#H,® plane. They also hold for E Thus the total
Imax IImax°
power is the sum of the power of the two systems. Also the power
in any net element remains in that net element. It follows immed-
iately then that equation (12.58) holds and that the integrals
which represent 7, and 7,77 combine according to equation (12.59)
where the integration over © may have to be from -7 to 7. Then
from the definition given in equation (12.53), the desired pro-
perties are proved.
If equation (12.59) is approximated by a finite net, it will
be seen that the equation is not a true identity for the finite net.
The equation is valid only in the limit, and to prove the equation
for a finite net, it would be necessary to consider a sub net ap-
proaching infinitesimal areas inside of each net element.
No theoretical analysis or finite net is capable of resolving
the spectrum of 7 I+II into the spectrum of Nt and 737 if the power
spectra overlap. However, if a swell power spectrum is added to a
66
Additivity of Power Spectra
2 2 2
[Azur(#,4)] a Asante) = [Asucrem)(#,6)] (IZ53)
Let AE; and AEy be the power in the same net element for
rr and nee respectively Sh (12.54)
AE
P(E<,,<€+d€) = “ eEEe dé (i255)
Alea 6 dc): = ain Cena (12.56)
Z
| SAE, AE
p(E<nar+nan<é+ dé) = ———L_ e “(AE ,AEg), 12.57
(S<nArt+naAn<é ) mBE; + BED 3 ( )
War Ur + Oy (1258)
ee
[Jett I(u,H)[xcos@ + ysin@]—pt + ¥(u,8)|VfAant (4,8) ]* dudé@
OF Sor,
of | cos[ Pt aatscoses ysin8 ]—t+4(u,8)| J[Ar yn (0) dude
o/-1
© eo
of [oon sant cos@+ysin@]—pt + ¥(u,8)| V [Aau(re mz (#,4)]*dud8 (12.59)
0 4-1
Plate LXIX
low local chop power spectrum, then the methods of analysis pre-
sented in Chapter 10 will separate the two spectra.
Some properties of the refraction of short crested Gaussian waves
Consider the refraction of the most elementary short crested
wave system possible as given by equation (8.1) or by equation (8.4).
Let the angle in deep water between the two elemental crests be
given by, say, thirty degrees. Given the discrete spectral period,
it is then possible to find the apparent length of the crests in
the direction of the crests in deep water.
If the system is approaching an uncomplicated coastline without
crossed orthogonals for that discrete spectral component and without
caustics, then the closer to the shore the system is studied, the
more the angle between the elemental crests is decreased because the
crests are more nearly parallel to the shore. Thus nearer shore the
apparent crests are longer than they are in deep water.
For any power spectrum with discrete spectral components such
as the one treated in equation (8.5), the same thing occurs, and,
in the limit, for the continuous power spectrum the same results are
accounted for by 9( 4,0.) and [(p ,O,).
If in addition, the power spectrum varies over a wide range of
HM, the low # valiues are amplified in general more than the high
values of # by the effect of the factor, D, in wave refraction
theory. Consequently, as a short crested sea surface approaches a
coast in many cases, the crests become longer and more well defined,
and the "significant" period of a wave record near the shore becomes
longer than the "significant" period of a record taken at the same
time in deep water. The refraction of a short crested sea surface by
68
the use of the "Significant" height and period is therefore just as
much in error as the analysis of a pressure record in terms of
these values. For sea conditions, the results are meaningless.
"En echelon" waves can also be treated by these considerations.
Suppose that a given filter from some storm has a narrow ® band
width and a wide # band width. Then the waves in deep water will
have relatively long crests. Upon refraction, the long narrow
spectrum becomes an arc, and evaluation of the finite net would
then show the "en echelon" effect. The apparent crests would be
shorter along the crests after refraction than before refraction.
pomgese nad pho esnanhs
In this section two very fascinating aerial photographs and
some enlargements of parts of these photographs will be discussed
in detail. These photographs were furnished by Mr. Dean F. Bumpus
of Woods Hole Oceanographic Institution. They are both very clear-
cut examples of the refraction of a Gaussian short crested sea sur-
face. They were taken along the coast of North Carolina at Oracoke
and Swash Inlet by the Coast and Geodetic Survey on January 24,
1945. Figure 33 is an aerial photograph over Oracoke. Figure 34
is an aerial photograph over Swash Inlet. Figures 35, 36, and 37
are enlargements of parts of figures 33 and 34 for easily recog-
nized areas.
Both photographs show some very interesting features. In the
deeper water on the right, the longer crests are at about an angle
of forty-five degrees to the coast line. The lengths of the apparent
crests are quite short. Half way between the edge of the photo and
the coast, the crests are more nearly parallel to the coast and
69
Photograph over Oracoke.
Figure 33. Aerial
Photograph over Swash _ Inlet
34. Aerial
igure
F
Figure 35 Enlargement over Oracoke
Figure 36. Enlargement over Oracoke.
Inlet
Figure 37. Enlargement over Swash
the apparent crests are much longer. No individual crest can be
followed very far by the eye before it becomes lost in an area of
poor definition and low amplitudes.
A second interesting feature is the local chop which is super-
imposed on the longer apparent crests. At the far right, the crests
are at about an eighty-five degree angle to the coast. Even near
the coast, these short crested waves are only slightly affected by
the bottom, and they have only turned a few degrees more parallel
to the coast. The assumption of linearity, of course, assumes that
neither system has an effect on the other which is not true for the
higher order effects.
A third interesting feature is found by a careful study of
the zone between the coast and a line about one-eighth of the dis-
tance to the outside edge of the photo and of the triangular zone
at the base of the Oracoke photo. The crests in these zones do not
have the same profile as the crests outside of the zones. The
crests are higher and more peaked and the troughs are longer and
shallower. Outside of the zones discussed above, the crests and
troughs are equal in importance, and a graph (as a function of,
Say, distance along a dominant orthogonal) of the wave height would
look very much like a wave record except that the apparent crests
would be shorter near the coast. The outside edge of this zone and
some curve probably off the picture define the transition zone
studied in this chapter. Note that the local chop keeps on doing what
it had been doing before even after the longer crests have been
modified in profile (see figure 37).
(le)
The breaker zone
Between the curve defining the transition zone on the coast-
ward side and the coast, non-linear effects are apparently dominant.
From these photographs, Munk's Solitary Wave Theory [1949] may well
be a first step in a study of this zone. This near shore zone is
probably the location of the important beach erosion effects. In
this paper, these effects as far as they can be treated by the methods
used herein are covered in Plate LXX. In Plate LXX, only one point
is emphasized. That point is that important non-linear effects
cannot and must not be treated by the theories developed herein.
Summary of the past chapters
Methods and formulas which apply to storm generated ocean sur-
face waves from the time they leave the edge of a storm at sea until
they are just about to enter the zone where they break upon some coast
have been presented in this chapter and in the past chapters. The
procedures and techniques described apply realistically to waves as
they aree They can discriminate between sea and swell. They can
predict waves given data not currently available. They explain nearly
all of the observed facts about ocean waves within the linearity ap-
proximation.
Two important problems have not been treated. They are the
problem of the generation of waves and the problem of the breaking
of waves in the breaker zone. Some general comments on wave gene-
ration will be made in a later chapter, but breaking waves will not
be discussed.
76
The Solitary Wave Zone ?
The Breaker Zone ?
The Shallow Water Zone ?
NON-LINEAR
Plate DLXX
Plan for the rest of the paper
The techniques and equations for the description of the sea
surface have been presented in Chapter 5, and in Chapters 7 through
12. No more equations and derivations are needed as far as this
paper is concerned, and thus there will be no more plates presented
in the text.
In the next three chapters, these equations will be applied
to practical data. An example of an accurate wave analysis will
be given. The important numbers which can be obtained from such
data will be computed. The character of wave records will be
described in greater detail. A theoretical forecast will be carried
out which will show the strange effects of refraction on the waves
which reach the North Jersey coast. Wherever possible, published
data and observations will be used to substantiate the results.
It should be pointed out that the derivations presented and
the theoretical results obtained just scratch the surface of the
results which can be obtained by continued investigation along
the lines pursued herein. The problems of ship motion, radar
reflectivity, the relationship between wave and wind spectra,
capillary waves, circular storms, moving storms and very short
range wave prediction are all still unsolved.
78
Chapter 13. EXAMPLES OF PRESSURE AND WAVE RECORD ANALYSES
Introduction
In this chapter, a detailed and highly informative analysis of
a pressure record will be carried out according to the procedures
devised by Tukey and Hamming [1949]. The pressure power spectrum
will then be corrected for the effect of depth thus obtaining the
power spectrum of the free surface. The 10% to 25% error of the sig-
nificant height and period method will then be explained. Various
features of the free surface will be deduced.
The analyses of wave record correlograms carried out by Tukey
and Hamming will be discussed and interpreted in the light of some
of the papers published by Seiwell [1949, 1950]. A refutation of
the conclusion that wave records have one or more "discrete" periods
(or cyclic components) will be given by showing that such components
have not been proved to exist and that the available evidence cor-
rectly interpreted indicates the contrary.
The design criteria for wave analyzers as described by Tukey
and Hamming will be applied to known wave analyzers and their perform-
ance will be interpreted in the light of these design criteria.
A detailed analysis of a pressure record*
The twenty-five minute pressure record which was sampled in Chap-
ter 7 to see if it was Gaussian and which was taken on 18 December
1951 starting at 2258 EST can be analyzed and studied in great detail
* See also a paper to be published by Pierson and Marks [1952] in the
A.G.U. Mr. Wilbur Marks has done all of the numerical work for this
section and has written up the details of the procedures employed
in the A.G.U. paper.
lis
because it is long enough to yield reliable results. Before so doing,
however, interesting results can be deduced just on the basis of the
fact that the record is Gaussian.
The twenty-five minute record was recorded on ordinary chart
paper (such as is shown in Figure 12) at a fairly rapid speed of 6
inches equal to one minute. The range of the record covers from
extreme to extreme about seven or eight of the small chart divisions.
The standard deviation of the record was found from one hundred
points picked at random. An arbitrary zero was chosen as a line
well below the record and the square root of the second moment about
the computed mean of the sample as measured from this arbitrary mean
was found. By some strange accident, the mean of the sample fell
right on one of the scale lines within a few thousandths of a unit,
and thus the estimated mean of the record falls, within the accuracy
of the measurements, on one of the chart lines.
Now suppose that the mean and standard deviation of the sample
which was taken are close to the true mean and standard deviation
of the record. Then another sample of one hundred other points
chosen at random would have nearly the same mean and standard de-
viation. In fact, an infinite number of different samples of points
could be taken from the record and if the points were far enough
apart, each sample would have essentially the same mean and standard
deviation. More technically the means should be normally distributed
with a mean near the true mean, etc. The only thing that could not
be done would be to take a sample of one hundred points from, say,
a portion of the record one second long such that the points were
only one one hundredth of a second apart. In this case, Since the
80
points are so strongly autocorrelated the distribution would not be
Gaussian.
Also, all of the different samples could be combined into one
big sample, and that sample would again have an approximately Gaussian
distribution. And also if points were chosen, say, one one hundredth
of a second apart throughout the total record length, then the
150,000 points so obtained would have an approximately Gaussian
distribution.
Finally the distribution of every point on the whole record
would be approximately Gaussian, and, since the record is continuous,
this permits a computation as to how long a time out of the total
twenty-five minutes the record will occupy a given range of pressure
values. From equation (7.33) modified by a substitution of P(t,) for
n (t,) and EpHmax Lor Bax? it is possible to compute the probability
that a point will exceed a certain value. If the probability that
the record will exceed the value Py is p(I) and if the probability
that the record will exceed the value Pir is p(II), and if the value
of Py is greater than the value of Prt then the probability that
the value will lie between Py and P,, is (p(II) - p(I)). Also
(p(II) - p(I)) multiplied by the length of the record (25 min), then
gives that fraction of the total time of the record that the pressure ©
value will be between Py and Prt
For the record under study, one scale division was equal to
0.855 standard deviations. Therefore the probability that the re-
cord would lie between the scale line for the mean of the record and
the scale line one unit above was equal to 0.3034, and theoretically
for 7.58 minutes out of the total 25 minutes, the graph of the wave
81
record should have been between these two scale lines. As actually
measured it was between the two scale lines for 8.03 minutes. This
is a discrepancy of about 6% between the theoretical and observed
values.
Table 18 shows the other values as computed from the theory and
as checked by measurement. The greatest error in minutes is 0.45
minutes between the predicted and observed values. Thus the error
in prediction is only about 2% of the total pressure record length.
For the greater departures from the mean, the percentage errors are
larger, but the whole table shows remarkable agreement between pre-
dicted and observed values. The last row, for example, predicts that
the record will be more than three positive scale divisions from the
mean for about eight seconds out of twenty-five minutes and that the
record will be more than three negative scale divisions from the mean
for another eight seconds. Actually the record never went below
three scale divisions and it was above three scale divisions for
ten seconds.
What has just been done should be reemphasized. Points were
taken at random from a pressure record. The standard deviation of
these points in terms of scale units on the paper was then computed.
Then the total time that the record would occupy a certain range of
values was computed on the basis of the fact that the record was
Gaussian. The predicted and observed values were found to agree
remarkably well out to 2.5 standard deviations of the distribution.
Usually statisticians are well pleased if an observed distribution
fits a normal curve two standard deviations away even crudely, and
in this case the agreement is even good 2.5 standard deviations away.
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83
Note that the average values give very good agreement. They seem
to remove the remaining non-linear effects in the pressure record
explainable by a tendency toward a trochoidal forn.
A very simple statistic therefore describes many of the features
of the record. Were it actually a wave record, this statistic could
have been forecasted by forecasting the power spectrum and integrat-
ing the power spectrum of the wave record over # and © to find Bax’
Consequently, without even mentioning the significant height and
period, important information can be obtained about the forecasted
waves.
If the above record had been a wave record, it would be possible
to predict, for example, that a given spark plug on a recorder like
the one developed by the Beach Erosion Board (Caldwell [1948]) would
be submerged for 4.90 minutes during the next twenty-five minutes,
and the actual observed time would have been 4.63 minutes. if the
waves were passing, Say, an oil drilling rig in the Gulf of Mexico,
(and if the rig could be located at a point compared to the dimensions
of the waves), then the length of time the water would cover any given
mark on the rig could be predicted. These considerations would not
be valid for a free floating object like a life raft because it moves
horizontally with the waves, but it is not too difficult to visualize
extensions which would yield information on the motion of the raft
above and below mean sea level also. A ship with a length comparable
to the wave lengths associated with the spectral periods involved
in the power spectrum would have a different up and down motion, but
again the Gaussian character of the motion would have to be true and
statements similar to the above could be made about the ship's motion.
84
a SS LE
—
The autocorrelation function was not determined by the proced-
ures given in Chapter 10 and equations (10.29) and (10.30). Such a
computation is laborious, and instead the record was mechanically
autocorrelated by the machine described by Seiwell [1950a]. Eighty
lags of two seconds each were evaluated and the values were corrected
so that they essentially correspond to the Q, of equation (10.30)
apart from a constant factor. Reduction of % to unit value then
yields the normalized autocorrelation function, and multiplication
of each value by E which is known from direct computation of the
PHmax
standard deviation of the Gaussian distribution would then yield the
non-normalized autocorrelation function.
Since the record was taken in 30.5 feet of water, two second
lags were used with the assurance that aliasing would be negligible.
From considerations in Chapter 10, only about two per cent of the
height of a 4 second elemental component would show up in the pressure
record.
The autocorrelation function given at the top of figure 38 shows
several interesting features. It dies out in a few oscillations to
low values after about 18 lags (or after 36 seconds). Then it re-
combines to rise again to values near 0.15 after 26 lags (or after 52
seconds). After 56 lags, the autocorrelation function dies down to
small values and from then on it never amounts to anything substantial
again rarely assuming values near 0.10.
If there had been one pure sine wave component (or cyclic com-
ponent) present in the record of an amplitude containing great enough
85
power to contribute an important part to the total power, then the
autocorrelation function would not have died down completely and
there would have been a cosine wave present out at the far end of
the autocorrelation function.
If there had been several pure sinusoidal waves present in
the record, it is possible that by accident they would be in phase
cancellation at the end of the number of lags shown. Under these
conditions more lags might show that the autocorrelation function
would rise to more substantial amplitudes.
Thus it is proved that this record does not contain one pure sine
wave of appreciable amplitude. No finite number of lags can prove
the absence of several discrete sine waves (several can be 3, 5 or
50). A finite number of lags only makes it more and more unlikely
that there are some given number of pure sine waves present. With
more lags, one is more sure that a small number of discrete components
are not present.
Although it is possible for there to be several pure sine conm-
ponents of appreciable amplitude in this record, the autocorrelation
function seems to contradict the possibility of just a few, say one,
two, or three. Also the fact that the record is Gaussian, seems to
suggest that the record is of the form of equation (12.19) although
again a few pure sine waves of low amplitude plus a superimposed
Gaussian disturbance would yield an autocorrelation function quite
Similar to the one obtained, and the sampling procedures of Table 18
above might not detect any difference. The presence or absence of
"cyclic" or purely periodic discrete components in wave records in
general will be discussed in detail later in this chapter.
86
80
90 100 No
FROM 2258 TO 2323 E.S.T.
OFF LONG BRANCH, NJ.
: rn ENS re
22 re) 0
=a
ee
8
— 1,0
° 10 20 30 40 50 60 70
NON- NORMALIZED AUTO-CORRELATION FUNCTION
REDUCED TO UNIT AMPLITUDE BY DIVISION BY Ep max
FOR PRESSURE RECORD OF /0-18-2I
Al A (DEPTH OF 32.5 FEET MS:b
i Ane,
5
210
=
25 26 NA
fe) i) 2 3 4 -) 6 7 8 9 10 Ww 12 3 14 15 16 17 1@ 19 20 21 22 23 24
am) 7 T T T Tv T Ww T WT =: WT 7 WT Tr Tr us us Tr T Tr Tr Tr 7 T T us
20° 60 30 20 15 i2 10 6675 67 6 55 5 4643 4 38 35 3332 3 29 27 26252423
ad @ i120 60 40 30 24 20 I7) 15 13.3 12 109 10 92 866 8 75 71 67 63 6 57 55 5.2 5 48 46 T seconos
BEST ESTIMATE OF THE PRESSURE POWER SPECTRUM
OF ABOVE AUTO-CORRELATION FUNCTION IN TERMS OF
FRACTION OF TOTAL POWER PER UNIT BAND OF THE
yw AXIS. SUM OF VALUES AT: w= 222, h-0,1,2,...30 IS EQUAL TO 1.018
Lp
220
a5
20
205
°
h oO ' 2 3 4 5 6 7 8 9 1o Wt 2 #1 4 18 16 17 #18 t9 20 2) 22 23 24 25 26 27 28 29 30h
"RAW" OR UNFILTERED POWER SPECTRUM OF ABOVE
Fig 38 The Analysis of a Pressure Record
—s87—
120
30
140 150
LAG IN SECONDS
160
The "raw" pressure power spectrum
The next step in the analysis under discussion is to apply equa-
tion (10.31) to the normalized autocorrelation function given on the
top of figure 38. The value of m was chosen to be equal to 30 in
order to retain a sufficient number of degrees of freedom for each
band. The use of the entire function would more than treble the labor
involved and the results would be very unreliable (see Table 16 and
equation (10.39)). The result of the computation is shown on the
bottom of figure 38. The "raw" estimate is irregular, and were it
to represent a power spectrum there might be reason to suspect that
great difficulty would be encountered in attempting to forecast ocean
Wavese
The "filtered" pressure power spectrum
However, as has been shown, the "raw" estimates must be smoothed
by equation (10.32) and upon smoothing the beautifully regular esti-
mate is obtained which is shown in the center of figure 38. From
Table 16 for 50 degrees of freedom, the true value in each band will
be between 1.45 and 0.74 times the value indicated by the solid curve.
These bounds are shown by the dashed lines on the figure. The sum of
the values given on the solid curve is very nearly one, and this is
both to be expected and to be considered a good check of the accuracy
of the computations since the normalized autocorrelation function was
employed. These results, upon the proper choice of scale, will yield
the estimate of the true pressure power spectrum.
Quantitative interpretation of the filtered pressure power spectrum
So far for reasons of convenience, all of the computations have
88
employed the normalized autocorrelation function and in figure 38
the power under the spectrum is essentially one (a 2% error due to
rounding seems to have occurred). The total power under the power
spectrum is known from the results of Chapter % and it is now a simple
and straightforward procedure to modify the ordinate scale of figure
38 in order to obtain the complete representation of the power spect-
rum given by the top part of figure 39. The scale on the left is in
units of em?-sec and ranges from zero to slightly above 2000 units.
Suppose that the peak is at 1700 cm’ sec. Then the power from 2m 23/240
to 29 25/240 (or from 10.43 to 9.60 seconds) is given by 1700 times
27/120 or by 88.8 em*. This is equivalent in power to a sinusoidal
component 9.41 cm high.
Many interesting things can be deduced about the original wave
record from the pressure power spectrum. Important amounts of power
are contributed to the pressure record over the entire band, and all
values of # from 27/15 to 27/6 are important.
A finite net such as those described in Chapter 7 would thus
require at least 12 sine components to approximate the record. All
components would be of the same order of magnitude in amplitude.
Even if no autocorrelative function were available, the power spectrum
would show that a pure sine wave component with, say, 3/4 of the total
power in the record is not present because the power spectrum would
be markedly different from what it actually is.
There is reason to believe that "white noise" (Tukey and Hamming,
[1949]) has been introduced into the data by the process of analysis
since the original values could be read accurately to only about
three significant figures. If so, then the small amount of power
(about 10% of the total) indicated below 27/15 is not really present.
89
4
Ee) ex
fox AREA UNDER SOLID LINE
EQUALS 601.5 CM.?
IN CM2- SEC.
if
612 13) 14 156? BS 20 at 22 0623 624 25 26 h
Sree Te et Te ee ee i, a =
625 24 23 + IN SEC
= 4846 T SEC.
r( 8 9 10
ub uk fue SB)
3.8 3.5
™ 7 a
12
AUTOCORRELATION FUNCTION PLOTTED IN TERMS OF [Apy(#)]? Vs.
TRUE SPECTRUM LIES BETWEEN DASHED BOUNORIES 90% OF THE TIME.
4000 [4 1°
fe) a
IN CM.2—SEC. os
| a
3444 ! \
| \
i] \
i ‘
3000 | \ AREA UNDER SOLID LINE
i a EQUALS 1263 CM2
I
IN SEC.’
46 43 4
e 60 30 20 15
2 10 86 75 6.7
POWER SPECTRUM OF THE FREE SURFACE, [Atz2]?
Fig 39. Quantitative Power Spectra of the Pressure Record and the Free Surface
The points determined by the circles represent the average
value of [Apy(#)]° over the band which straddles the point. The
curve joining the points is simply an aid to the eye since any
curve can be drawn over each band under study just as long as the
area under it equals the value which has been determined. Thus
the true power spectrum can be an extremely irregular function with
very rapid (even if continuous) fluctuations. Even worse than that
the power spectrum could have been of the form discussed in Chapter
10 and the same graph would have been obtained in figure 39.
To discover if really rapid fluctuations in the power spectrum
are present, it would be necessary to increase m and the length of
the record. Thus a 50 minute record and twice as many lags would
give 60 bands of the w axis instead of 30 with the same reliability.
A 100 minute record with 120 lags would give four times as many
values. Would the 120 values (instead of 30) thus determined follow
the same general curve as shown by the solid line? The question can-
not be answered until the work is done, (and it is not planned to do
it), but it is very difficult to think of any physical mechanism which
would cause the power spectrum to be irregular within any conceivable
limits of resolution.
The above process of narrowing the band width and increasing
the length of the record would also detect any purely sinusoidal com-
ponent in the record. Thus with greater resolution, a discrete com-
ponent would produce a sharp narrow spike rising out of the general
function. The spike could be made as high as desired and as narrow
as desired, and in the limit it would become infinitely high and in-
finitesimally wide such that the product of the height and the width
gL
would be equal to the square of the amplitude of the discrete com-
ponent.
Thus, to within the resolving power of the analysis which has
been carried out, there is no proof of the presence of any discrete
components, nor is there any proof that they are not present. A
little thought shows that one can never prove either the presence
or absence of very small power discrete components by taking one fin-
ite section of a time series since there is always the possibility
that the function being studied is represented by a sum over a finite
net such as in Chapter 7 with many more terms than could possibly
be resolved by the choice of m and N in the numerical analysis.
The analysis of the pressure record given above has yielded the
power spectrum of the pressure record. The time has now come to put
back the high frequency waves (low period) filtered out by the effects
of depth. The power spectrum of the free surface will be the result.
The filtering process is not completely reversible because the waves
with periods below four seconds have been irretrievably lost. Since
the water is essentially infinitely deep for these low periods, a
modified application of the results of Chapter 11 could estimate the
amount of power left out completely.
It will be assumed that the pressure recorder responds to the
actual pressure fluctuations at its indicated depth. This statement
is equivalent to stating that purely sinusoidal pressure fluctuations
at the depth of the instrument and of equal amplitude but different
periods are recorded with the same amplitude.
The procedures are then straightforward and the results of
92
Chapter 12 apply. Each spectral band must be multiplied by a dif-
ferent correction factor as given by equation (12.21). The power
spectrum on the bottom of figure 39 is then the power spectrum of
the free surface. The true power in each band will lie between the
dashed lines 90% of the time and the solid curve is the best esti-
mate of the power spectrum.
Figure 40 is a comparison of the pressure power spectrum with
the free surface power spectrum. It shows that the low period end
of the power spectrum has to be amplified very much more than the
high period end. The minor wiggle in the pressure spectrum at a
period of 5 seconds may even be an important secondary peak in the
free surface record. The free surface record will be more irregular
and choppy then the pressure record. The spectra also show that
the "significant" (or "characteristic") period of the free surface
wave record will be lower than the "significant" (or "characteristic" )
period of the pressure record.
It is now possible to see where the 10% to 25% error described
by Snodgrass [1951] comes from when the "significant" (or "character-
istic") period is used along with the "significant" height to zo from
a pressure record to the waves at the free surface. The "significant"
height is crudely proportional to the square root of ee and the
"significant" height of the pressure record is crudely proportional
to the square root of Epmax’ rhe "significant" period method of
pressure record analysis multiplies (Bomex) by a constant
amplification factor [cosh (1 °H/e) I(+,,H)], for a fixed py
which depends on the choice of the "significant period. This choice
varies from analyst to analyst on the same record.
93
WINMIAAS 4OMOd BaIDJANS
Bad4 AYf YsIM WNMIAUS JAMO BANSSAd AYf JO UOSIIDAWOD ‘Op BI4
SLNINOAIWOD AININOFIYS MOT FHL FO NOMLVIIAITdWY HILV IYI
HINW FHL FLON “WNYLIFdS YIMOd FJIvVIYNS FIA
HLIM WNYLIIdS HYIMOd FYNSSIYd JO NOSIYVdWOD
JHL
ve S@ 9242 G2 ¢€ ze ce ¢
ian fe a aE ea a ue ae ae
ooo!
0002
Jha
HOIH WIC'’GE FAVM JINIS
JHYNd V OL LNITIWAINOZ
z2WIE9a] = “YNZ
‘XUW Hd
2W9 G/109 = e|
ooo¢€
Consider then a range of possible "significant" periods (de-
pending upon the analyst) and the multiplication of the "significant"
height of the pressure record by the possible amplification factors.
Then the quantity
2
ea 4
1/2
= ]
2
[(E, 4 /[cosh( I(w4,H))) E
Pmax
is a ratio which represents roughly the value obtained by the correct
method divided by the value obtained by the erroneous method. If
the ratio were one then the error would not be apparent; if it is
sreater than one then the part after the decimal point represents the
percentage error referred to by Snodgrass [1951]. Table 19 gives
some of the ratios which can result from the assumption of various
significant periods.
Thus for this depth, which is quite shallow compared to most
depths at which pressure recorders have been installed, if the pres-
sure record were given any "significant" period greater than &.6 sec-
onds, then there would be a considerable error in the computation of
the "significant" height of the free surface. At greater depths and
for differently shaped pressure power spectra the errors would be
different and there is no hope of consistency in the old methods of
analysis. Note that the power lost above 4.8 seconds would serve
only to increase the error if it were included. Also note that the
filtering nature of the pressure recording method always tends (given
a widely variable power spectrum) to give too large a sicnificant
period to the free surface record and too small an amplification
factor by the old methods.
oO
Table 19. Ratio of correct significant height
to value obtained by erroneous
extrapolation of the pressure record
upwards
Significant Amplification Ratio
period (sec) factor
24.0 1.071 1.332
20.0 1.107 1.311
ce 1.138 1.293
15.0 1.197 1.260
13.4 16257 1.230
12.0 1 349 1.192
sO 1.433 26151
10.0 1.548 1107
9.2 1.662 1.069
8.6 1.812 1.023
8.0 2,025 0.967
7.6 2.223 0.925
7.0 2.512 0.869
6.6 2.843 0.817
6.4 3.342 0.753
Significant height and period
The remarks so far in this paper have been in many cases directed
against the concept of the "significant" (or characteristic) height
and period method of wave analysis. There is really nothing wrong
fundamentally with these concepts. The thing that is wrong is the
way that the concepts have been misapplied.
The physical meaning of the average height of the one third
highest waves, for example, can possibly be deduced from the power
integrals and the autocorrelation function and the fact that the
records are Gaussian. Such a number depends in a very complicated
96
way on the set of points in the record which determine the suc-
cessive relative maxima and minima of the record. The probability
distribution of this set of points may depend on the power spectrum
in addition to the fact that the record as a whole is Gaussian. It
is not too difficult to believe that the various ratios, 1/10 high-
est waves to the 1/3 highest waves, etc; such as summarized by Snod-
grass [1951] are all consequences of the fact that the records are
Gaussian. The trouble with these methods of analysis and of attempts
to extend them such as those described by Putz [1950, 1951] is that
the features of the wave record are obscured by concentrating attention
too sharply on the waves. Paraphrasing an old saying: "such methods
of analysis cannot see the wave record on account of the waves."
Similarly, the "characteristic" or "significant" period is a
number determined from the time interval between successive relative
maxima of the record if the relative maxima exceed a certain value.
Given a high crest, the autocorrelation function says that the next
crest is also likely to be high and that the next crest is most likely
to occur at a time given by the first relative maximum after lag
zero of the autocorrelation function. For a "swell" record the first
maximum of the autocorrelation function has an amplitude which comes
quite close to the original peak value and thus the "significant"
period would have a useful meaning if the band width of the swell
could be given. For a "sea" record the first relative maximum can
be.quite low, which means that the "significant" period is not a
very useful number at all.
If the autocorrelation function in figure 38 is used to obtain
the significant period of the pressure record studied at the start
97
of this chapter, then the value turns out to be about 9.2 seconds.
Then from Table 19, the best estimate of the percentage error which
would result from an incorrect upward extrapolation of the pressure
record to the free surface is 6.9 per cent.
Seiwell's results
Publications by Seiwell [1949, 1950] and Seiwell and Wadsworth
[1949] have claimed that a purely cyclic (or sinusoidal) component
is present in wave records. Later the original interpretation was
modified to include the presence of two or three cyclic components.
The autocorrelation method is quite laborious, and the earlier con-
clusions were based on one second lags for the first complete “oscil-
lation" of the autocorrelation function followed by skipping some
arbitrary number of lags and then finding another "cycle." For ex-
ample, if the autocorrelation record shown in figure 38 were given
for only the first 10 seconds followed by no data from 12 seconds to
40 seconds and then by another cycle from 42 seconds to 52 seconds
it might be very easy to conclude that one "cyclic" component was pre-
sent. This conclusion is of course shown to be incorrect by the rest
of the data. Once one cyclic component is found, then a little more
detail in the autocorrelation leads to the hypothesis that several
"cyclic" components are present.
ED ES TE ES ES
Tukey and Hamming [1949] have analyzed Seiwell's data, and al-
though the autocorrelation function employed was normalized in a way
which makes the values somewhat different from the correct procedure
given in equation (10.30), the results are of interest here. The
following paragraphs are quoted from Tukey and Hamming and figure
98
41 is a copy of the figure referred to in the quotation.
"The next two examples, provided through the kindness of
Dr. He. R. Seiwell of the Woods Hole Oceanographic Institution,
are based on pressure recordings taken off Cuttyhunk Island,
Massachusetts in 1946. They represent the pressure at a depth
of 75 feet and reflect wave heights. The basic data are:
Station 53-W 537X
Date 15 Sept. 46 15 Sept. 46
Time 0500 hours plus 0650 hours plus
270 to 600 seconds 325 to 636 seconds
Serial correlations 0O(1) 20 seconds 0(1) 16 seconds
for lags of
Length of run 331 seconds 301 seconds
This type of data has been subjected to a few-constant fitting
procedure based in part on quadratic autoregressive residuals
as reported by Seiwell and Wadsworth --- and by Seiwell ---.
"In this case also, the serial correlations have been
analyzed as if they were serial products......The we values
obtained by a simple equating method, show substantial negative
values. Since true negative values are impossible this makes
such equating methods entirely useless on such data. tie to."
values, on the other hand, show a very reasonable behavior
and, in particular are never negative by more than 0.004, which
presumably results from accumulated errors and the use of ry
instead of Qp° -
"The upper frequency limit is 0.5 cycles/second for each
record, since there is 1 sample/second. Thus for record 53-W
we have a power density estimate every 0.025 cycles and for
record 53-X every 0.03125 cycles. The results are plotted
in [the] figure ..... We see that the general character of
the results is the same, namely an unresolved peak near 0.075
cycle/second and essentially no energy beyond 0.15 cycle/second.
The peak frequency may have increased in record 53-X as compared
with 53-W.
"In order to study the nature of the peak near 0.075
cycles/second, it would be natural to repeat the analysis so
that the upper frequency limit would be at, say 0.125 cycles/
sec, which would be obtained by analyzing the record at 4
second intervals and using lags of 0, 4, 8, ....., 80 seconds.
Unfortunately this would lead to widely fluctuating results
since there would then be only 82 points in the longer record,
and there would be only
82- + (20)
ae eee
20/2
a7
degrees of freedom for each U,* if m = 20 were again used.
Thus any attempt to put the peak under too powerful a micro-
scope is doomed to failure unless a longer stretch of obser-
vation is available. The length of the record, the spacing
of the observations, and the lags used are ideally suited to
show that there is essentially no power above 0.12 to 0.15
cycles/second (at periods less than 8.3 to 6.6 seconds), but
is‘not well suited to the detailed investigation of the structure
of the peak. The 53-X record has been analyzed by Seiwell and
Wadsworth in terms of a combination of
(1) a single frequency, and
(2) an auto regressive scheme as proposed by Kendall....
The latter scheme would involve a finite amount of power in the
region 0.12 to 0.50 cycle/second now seen to contain at most
a negligible amount. Almost any analysis containing simple
auto-regressive components will similarly fail to fit the
observed facts."
The above analysis shows that the one second lags chosen and
the number of lags made were quite inadequate to describe the power
spectrum. At depths of the order of 78 feet, faith in hydrodynamic
theory would tell us that all periods less than about 6 second would
not be recorded by the pressure recorder and the spectra shown sure-
ly confirm this fact since essentially 2/3 of the values obtained
are zero. Note that for a lag of three seconds and for the same
amount of work on a record three times as long, considerable val-
uable information would have been obtained.
Noise versus signal
The problem of proving that a wave record contains one or
several pure sine waves is analagous to a problem treated originally
by Wiener [1949] in his famous book on communication theory. Con-
sider an A.M. radio receiver a great distance from the transmitter.
Let the detected signal, say, one of the notes in the chimes of
100
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N.B.C., be so weak that it is drowned out audibly by static and
tube noise. Also suppose that a long record of the voltage graphed
as a function of time is available. The noise can be described by
an integral similar to, say, equation (7.1). The chime would be
the fundamental and harmonics of a pure sine wave. An autocorre-
lation of the record would cancel out the noise, and eventually the
oscillation due to the sine waves would be all that is left. The
discrete components correspond to jumps in the cumulative power
density such as in the first examples in Chapter 7, and the noise
yields a continuous increase between the jumps.
If the signal is very weak, many autocorrelations must be made
and the weak oscillation cannot be detected until the autocorrelation
of the noise has gone nearly to zero. If the signal is strong, not
so many lags must be taken in order that it become visible as an
oscillation in the autocorrelation function.
The cumulative power distribution functions for the case with
eee
cyclic components
Figure 42 shows two cumulative power distribution functions
which illustrate the problems connected with the analysis of wave
records. The first contains an easily recognizable cyclic component.
The second contains many small cyclic components.
The upper one shows a discrete jump in E(p# ) at 20/74. Let
Y(p) at ps 2r/T, be 1/4. The jump has about half of the power of
the total record. Given this form for E(#), then equation (7.1)
would consist in part of a limiting form like equation (7.7) plus
jump in the record). With such a pure sine wave present, the distri-
102
E(x)
as aa
E(p)
Emax
en en Qa Qa Qn ———
Figure 42. Some Cumulative Power Density Functions for time series
with "cyclic" components present.
== | HO} ae
bution would be recognizably non-Gaussian. After a sufficient
number of lags, the autocorrelation function would settle down to
the form of a pure coSine wave with an amplitude equal to one half
of the original power. The autocorrelation function could not
possibly -become small like the one shown in figure 38.
The lower cumulative power distribution function shows five
small but still discrete jumps in E(w). Again there would be a
term of the form of equation (7.7), but now in addition there would
be five pure sine waves present at 20/T,, 2n/T., 2n/T 35 2r/T, and
2n/T x. (Let the phases be fixed by defining ~ (yu) at these points.)
It would be quite difficult to detect these five pure sine waves by
autocorrelating the record. However after enough lags, they would
be all that remains of the record. If the record were truly station-
ary, in fact, the discrete components would still show up upon cor-
relation of a record, say, 30 minutes long, with another record,
say 30 minutes long, taken several hours later.
Final conclusions of the autocorrelation function
Thus by analogy to the above comments, the autocorrelation
function of the record studied in figure 38 proves that there is
not one pure sine wave present with an amplitude squared equal to
25% to 50 % (or greater) of the total average square of the record.
It is not proved that there is no pure sine wave present with, say,
an amplitude squared equal to 1% or .1% of the total average square
of the record.
In the derivation of the theory of previous chapters, it has
been assumed that wave records are essentially pure noise. The most
powerful argument in favor of this assumption lies in computed power
104
spectra which show appreciable power in bands throughout the entire
analysis. A second powerful argument lies in the spectra obtained
by Barber and Ursell [1948] and Deacon [1949] which show a gradual
essentially continuous shift as the power spectrum of a swell follows
the theories derived herein. One is forced to conclude that discrete
sine waves of appreciable amplitude have not been proved to be pre-
sent in wave records, and that the best interpretation of a wave re-
cord is that it is just colored noise.
The free surface power spectrum given in tigure 39 is a function
of # alone and nothing can be said about the short crestedness of
the free surface. All power in the power spectrum for periods less
than four seconds has been lost due to the filtering effect. Extra-
polation of the high end of the spectrum suggests that the power
lost above # equal to 27/4 is not too great.
If it is assumed that most of the wave energy flux is in one
direction and if this direction is assumed to be very nearly direct-
ly toward the shore since the winds were almost directly on shore,
then the flux of energy toward the shore can be computed from equation
(12550) 5
The top part of figure 43 is a graph of the integrand of the
integral given in equation (12.50) for the particular power spectrum
under study with pe /4 absorbed in the scale on the left. For the
depth under consideration (30.5 feet), values of » near 27/4 seconds
yield essentially the form (pg/2)+(A(,))°-(g/2u ) which means that
the energy flows forward with the group velocity of "deep" water waves,
(g/2). For low values of » , the energy is essentially moving
105
1.250x107
11.200x107
iz
1O.130x10
9570x107
FLUX OF ENERGY TOWARD SHORE = 4.5x107ergs~
i ém.sec.
9.000x10
8 ee 2
7.660x16"
7320x107
6750x107
6.180x10’F
ae
5,060 xi0"
4490x107
3.935x107
3.378x10"
2.815x10"
2.252x10"
1689x107
1.126x10"
0.563xi0"
i i ee a Viet neal 1 ll a SSE EE EE ss
i a a a i a a nk in aa a a ae ae ee i a a Ae ee a a ce cine one Aumann
2 30 20 I5 12 10 86 75 67 6 545 5 46 43 4 3.75 35 33 314 3 286 274 26 25 24 2.3 222 214 206 2
B
2 ee 2p oe ae ae a, Oe ie ek leak Ar an NG AE at
46 43 4 3.75 35 33 3.14 3 286 274 26 25 24 2.3 222 214 206 2
g
—_>— (wave speed)
uI(nH)
a
a
q
4
5
Ply
= = Bes
60 30 20 15 12 10 86 75 67 6 545
Bla
als
nla
ols
Sis
Pa
Bla
bf
60 30
Figure 43. Graphs of the functions needed in the computation of the energy flux toward shore,
and the integrand of equation (12.50) in the power spectrum given on figure 39.
OSs
forward with the speed, (gn) V2, i.e. the group velocity of shallow
water waves.
The various terms involved in the computation of the top part
of figure 43 are shown below the graph of the integrand of (12.50)
for the case under study. The term in the square bracket, namely,
_ 2H KTH H)
g sinn[2H—L(H aH) By
is graphed first. It ranges from the value of two to the value of
one and is equal to two at # equal to zero and asymptotically equal
to one as # approaches infinity. For practical purposes, it is
equal to one at # equal to 27/4. With a one half from out in front
of the integral the graph is simply the classical expression, Gie52).,
graphed as a function of » , i.e., (2r/T), over the range of interest.
The other term, namely g/u I(p,H), is the wave crest speed (a
g is needed from out in front of the integral). At# equal to zero,
it equals “gH and tor large # it approaches zero values (since capil-
larity is neglected). The value at # equal to zero is 985 cm/sec
since the depth is 991 cm.*
The bottom graph is the group velocity of the various spectral
components. It equals 985 cm/sec for low values of # and falls to
half this value at 27/5.4. This graph times the energy in the wave
record per band of the » axis given by (pe/2)[ Ay (py eA then gives
the flux of energy toward shore.
Finally, a numerical integration of the top of figure 43 yields
the result that the energy flux toward the shore is equal to
4.58 x 107 ergs/sec per centimeter of length along the wave crest.
* The mean low water value was corrected to mean sea level, and a
possible two foot tidal amplitude was neglected.
107
I et
This is equivalent to 4.58 watts/cm, or along one kilometer offshore
there are 458 kw of wave power fiowing toward the shore in the
vicinity of the point of observation. This amount of power is rather
puny compared to values which can result from the action of high
waves, but at least it is an accurate theoretical value based upon
a sound analysis of the original pressure record.
Table 20 shows the numbers which are appropriate to the com-
plete determination of the energy flux toward the shore as has been
given above for the example being studied in detail. The first
column is the number, h. The second column is the spectral frequency.
The third column shows the values of the pressure spectrum in re-
duced units as it is shown in the center of figure 38. The fourth
column shows the amplification factors for the pressure power spect-
rum. The fifth column shows numbers related to the group velocity.
The product of the last three numbers across each row would yield a
value for each spectral frequency and the sum of all of the values
for each spectral frequency would be a number which, apart froma
constant, would yield the energy flux toward shore.
The power per unit band in the pressure power spectrum varies
over a factor of fifty from the greatest to the least. The ampli-
fication factor varies over a factor of ten and the group velocity
factor varies from 2.01 to 0.85. Some of the values in the function
to be integrated, which result from the product of these numbers,
are thirty-eight times greater than other values. In the significant
height and period method, one value for the significant height of the
pressure record and one value for the significant period would result
in an extremely inaccurate estimate of the energy flux toward shore.
108
Numbers relevant to the computation
of the flux of energy toward the shore
Table 20.
be Normalized Amplification Group
Bernd to U Teen pec
ON ed
FOOD MDINHANKHEPWHEHO
pressure factor velocity
power factor
spectrum
6) -0178 1.00 2.01
21/120.0 -0166 1.004 2.00
21/60.0 -0119 1.008 1.99
27/40.0 .0141 1.026 1.98
217/30.0 -0161 1,049 1.97
217/24.0 Pron ly ak 1.071 1.95
21/20.0 -0163 1,107 1.928
21/172 0109 1.138 1.915
217/15.0 0161 ee OF Ube key ss%
217/13.4 0433 1.257 1.805
21/12.0 0685 1.339 1.745
21/11.0 1146 1.433 1.686
217/10.0 °1501 1.548 1.630
217/92 01357 1.662 1.580
217/826 21199 1.812 1.524
21/8.0 0861 2.025 1.493
21/726 0718 2.223 1.408
21/7.0 -0593 2.512 1.340
27/6 .6 -0400 2,843 1272
217/64 0282 36342 16217
217/6.0 0214 3.787 1.149
217/528 20119 4.435 1.078
21/564 0048 5.480 1.012
21/562 20029 6.807 0.950
217/520 20075 8.225 0.918
27/4.8 -0033 10.336 0.854
217/4.6 ~ 0 ~ ~
109
The number which finally resulted in the above computations is
an important number for beach erosion problems. The result is
valuable, but it is still a long way from the data which are actually
needed. The wave direction is unknown, and the form of the breakers
and the angle they make with the coast upon breaking cannot be deter-
mined from one pressure recorder and from the theories presented
herein.
What percentage of the wave power moves sand at the beach, what
percentage might have been surf beat actually flowing outward, what
percentage is dissipated by friction when the waves finally break,
and what percentage goes into the kinetic energy of a littoral cur-
rent (if the waves are at a slight angle to the beach) are all
questions for future theoretical investigation.
Wave record analyzers
Wave record spectrum analyzers have been reported in the liter-
ature by Barber and Ursell [1948] and Klebba [1946]. Wave record
autocorrelators have been described by Seiwell [1950a] and Rudnick
[1951]. The spectrum analyzers yield some function which is supposed
to be some sort of spectrum of the record. They have no scale for
the amplitude of the spectrum, and they have not been adequately
calibrated.” Until the work of Wiener [1949] and Tukey and Hamming
[1949] there was no way to interpret such analyses and there was
considerable confusion on how the machines were to be constructed
and on the design of the electronic circuits needed.
*As far as is known as of the date of this paper.
Aro
Compare the irregularity of these results and the lack of
quantitative values with the numerical analysis which has just
been presented. The spectrum was quite regular and the results were
precise in a statistical sense. The accuracy could have been in-
creased by taking a longer record and the results would be precisely
defined.
A record of a given length, with a fixed degree of resolution,
has a certain inherent statistical inaccuracy, due to the size of
the sample and the band width of the analysis, which cannot be re-
duced; and Tukey and Hamming have described this inaccuracy and
given the precise procedures for stating the results in a statis-
tical sense.
The wave analyzers mentioned above have the same inherent errors
(except possibly aliasing) as the results of the numerical methods
plus others due to design characteristics. The analyzers can be re-
designed so as to approximate the numerical method of analysis em-
ployed above, and, moreover, they can be calibrated against a numer-
ical analysis in order to check their response.
The numerical wave record spectrum analysis presented above re-
quired many months of work and effort. It would be impossible to
analyze an adequate supply of wave records by the same slow computing
techniques. One nice thing about the overall problem of torecasting
ocean waves is that huge quantities of these records can be made avail-
able and much larger quantities will be becoming available from deep
water observations. Thus it is important that a speedy and accurate
means be provided for the quantitative analysis of a large number of
pia
records. If the wave record spectrum analyzers mentioned above could
be modified so that they will give reliable results, then instead
of months per analysis it would require only five or ten minutes to
analyze a twenty minute record. It is therefore advisable to analyze
a number of records such as the one treated above numerically and
then to compare the results with the electronic analysis in order
to calibrate the analysis.
Design features of wave record analyzers
The design features of an electronic analyzer will be described
in general in order to show what is needed in such an instrument.
Plans are being made to modify the instrument devised by Klebba
[1946], and a Kay Electric Company sonograph is being studied in
order to convert it to a wave analyzer. The above instruments will
be modified and interpreted in the light of these considerations.
Wave analyzers should have the following features as suggested
by the numerical analysis given above.
1) The length of the record to be analyzed should be of the order
of 20 to 35 minutes. Provision for the analysis of variable length
records over a range of from 10 to 45 minutes would be advisable but
not essential.
2) The band pass filter should be square shouldered and it should
have a Ap proportional to the same value employed in the numerical
analysis above. Too wide a band pass would result in poor resolution
of swell spectra and too narrow a band pass would result in an ex-
tremely erratic analysis. The shape of the filter is very important
and the typical tuned circuit response curve is not very good for
this application (see Tukey and Hamming [1949] for further details).
-112
Provision for different width filters would be advisable.
3) The band pass filter should not tune through the record too
rapidly; that is, the entire record should pass through the filter
before it has been tuned through, say, one tenth of its band width.
4) The rectification time constants which provide the output
voltage to portray the spectrum should be long enough to average
effectively over the entire record.
5) A square law detector would be best so that the graph of
the spectrum would be that of a power spectrun.
6) Variable controls should be eliminated, and a choice of four
or five calibrated set switch positions provided.
Present results of wave record analyzers
Figure 44 is a collection of examples of electronic analyses
as taken from the literature. Various spectra are shown as analyzed
by the machines described by Klebba [1946, 1949] and Barber and Ur-
sell [1948]. An autocorrelation as performed by Rudnick's device
[1951] is also shown. Some of the spectra have been modified by add-
ing some dashed and dash-dot curves in order to illustrate some
points in the forthcoming discussion. &
Spectrum number one as shown on the upper left of figure 44 is
taken from a paper by Seiwell [1949a]. It is an analysis on Klebba's
machine of a pressure record taken in 120 feet of water off Bermuda
on 25 Uctober 1946 at 1405 for 350 seconds. Fér periods less than
about 7 seconds the amplitudes are negligible due to the filtering
effect of depth.
The dashed curve drawn by eye through the ‘irregular curve of the
figure is a smoothed interpretation of what the spectrum might just
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as well have been given the length of the record and the statistical
reliability of the analysis. Stated another way, to prove that the
spectrum is actually as irregular as shown a very much longer
stationary record would be needed.
The upper and lower dash-dot curves might represent the 90% con-
fidence limits of the analysis, and if the record is related to the
Square root of the power spectrum then rough computations suggest
that the number of degrees of freedom of the analysis lies between
4 and 20 and that it is most likely about 9. Thus the individual
peaks and troughs are extremely unreliable.
Nevertheless the analysis shows that there are important contri-
butions to the entire spectrum from » equal to 27/20 to # equal to
or/7. The record is undoubtedly that of a pressure filtered "sea"
record, and the sea surface would best be represented by a spectrum
covering a wide band with possibly important contributions even for
periods below seven seconds.
Spectrum number two is from the paper by Klebba [1949]. It was
taken in 78 feet of water on 15 September 1946 at 0650 EST off Cutty-
hunk. Spectral components with a six second period would begin to
show in the spectrum if they were present and certainly important ten
second components would be evident. They are not present; the highest
important value of » is at 27/11 and the lowest is at 27/15. Since
the dashed curve could represent the spectrum just as accurately as
the one shown, (and since the dash-dot curves again suggest the degrees
of freedom of the analysis), it would appear that this record is a
clear cut example of a power spectrum such as those predicted in
Chapters 7 and 10. The record must have been a "swell" record with a
aa
well defined band width, and the waves must have come from a distant
source. Local chop below periods of 6 seconds would be undetectable.
This second spectrum as shown on the upper right of figure 44
is the electronically analyzed spectrum of the classical wave record
53-X. The electronic analysis was first given by Klebba [1949],
Seiwell [1949b] gave the same electronic analysis and stated that
the analysis “does not permit a reliable interpretation of the physi-
cal properties" [of the record].
Seiwell [1949b] then proceeded to interpret the record in terms
of a cyclic component of 12.25 seconds and a superimposed series of
random fluctuations. His results were debated by Deacon [1951] at
the National Bureau of Standards Symposium on Gravity Waves.
Tukey and Hamming analyzed Seiwell's autocorrelation data and
the results of the analysis were quoted a few pages back. The power
spectrum analysis of the autocorrelation data from record 53-X is
given in figure 41. The quotation from Tukey and Hamming and the
theoretical results contained in this paper effectively refute the
claim of a cyclic component.
Tukey and Hamming were limited at the very start by inadequate
data since the original record was too short, the lags were too close
together, and there were not enough lags. Their results consequently
yielded a spectrum which has practically no resolution over the band
of frequencies of importance. From their analysis and from figure
44, it is not too difficult to see how Seiwell might have reached his
erroneous conclusions since the swell did have a rather narrow band
width. However, the important point is that the electronic analysis
in this particular case, when properly interpreted, yields the most
nearly correct qualitative spectrum.
116
Spectrum number three is from a paper by Rudnick [1951].* The
record was taken offshore from Guam and additional information on
the record can be found in a paper by Miller [1949]. The spectral
analysis was made on Klebba's machine. If again the dash and dash-dot
curves can be interpreted as before, this record strongly suggests
the simultaneous presence of a local "sea" and a "swell" from a dis-
tance. The contribution from the "swell" rises significantly above
the level of the "sea" record at the same frequencies. As would be
expected the correlogram of the record is quite irregular, and it would
be difficult to detect the simultaneous sea and swell conditions on
the basis of it alone.
The three small spectra on the lower right were taken from the
paper by Barber and Ursell [1948]. They were made at Pendeen England
on 14 March 1945 at 2100 and on 15 March 1945 at 1700 and 1900. The
third spectrum is from swell and the first two spectra are from the
same storm after it had moved closer to the coast of England and
intensified.
According to Barber and Ursell, the analyzer responds only to
certain frequencies which have an integral number of cycles around
the wheel on which the record is placed. Barber and Ursell [1948]
make the following statement:
"The record is fastened around the circumference of a wheel
which rotates about a horizontal axis carrying the record past
an optical system which throws the record a horizontal line
of light. The reflected light illuminates light-sensitive
cells whose electrical output is, therefore, a continued repe-
tition of the curve on the record. This electrical output is
*In this very interesting paper, Rudnick reports that wave records
are Gaussian. This important discovery was thus first published
by him in 1951. His paper was not known to the author when Part
One was published.
117
amplified and made to drive a vibration galvanometer. It is
clear that if there is a component in the record having N
complete cycles in the peripheral length of the wheel, this
will produce a resonance of the galvanometer at its natural
freouency of p cyc./sec. when the wheel is rotating at a
speed of p/N rev./sec. The wheel is made to revolve at a
speed which gradually decreases from a high value and the
vibration galvanometer performs a series of transient
resonances, one for each periodicity in the record. The
resonances of the vibration galvanometer are converted to
an electrical signal which drives a pen recorder, and the
curve drawn by this pen is a series of peaks which constitute
a Fourier amplitude spectrum on the curve on the record. ...."
The envelope of the individual spikes in the record would seem
to be related to the power spectrum of the record. The width and
shape of the spike would therefore be related to the band pass filter
of the analysis and the figure suggests that the resonant galvano-
meter is very sharply peaked and responds to an extremely narrow band
of the power in the wave record. Note how the amplitude of the record
falis down to very low values on each side of each peak.
Now note how extremely irregular the envelope of the peak appears
to be. From 1700 to 1900 in the first two spectra marked gaps appear
inside of the range of # where one would expect only minor variations
from the theories contained in this paper. If the irregularities
were to reflect actual physical changes in the record, this would
be most disconcerting, but they really do not.
The irregularities from record to record and from point to point
ir tse same record are simply due to too great a resolution for too
small a record length. The wave records were 20 minutes long and
there are about 15 spikes between 27/15 and 27/12 in the spectra
shown. This suggests a band width of the analysis given byApw equal
to 27/4-15-15. From equation (10.39), and since 20 minutes times
118
60 equals 1200 seconds which in turn must equal NAt, it follows
from equation (10.39) that the analysis has approximately five de-
grees of freedom.
Table 16 then shows that adjacent peaks can vary by a factor of
four above the true value and by a factor of one half below the true
value in a power spectrum determined by these conditions.
The spectra shown must probably be squared value for value to
get a shape like a power spectrum, and if this is done the variation
just described actually occurs.
The resolution employed is very much greater than is needed, and
replacement of the galvanometer by a square shouldered band pass cir-
cuit about five times as wide as the one employed would be the first
step in obtaining quantitative results from this instrument. This
would result in twenty-five desrees of freedom and the shape of the
spectra obtained would be much more regular. High resolution such as
that employed in the above analyzer would require a record five times
longer than the one given and very careful design considerations, es-
pecially with reference to integration time constants, to yield reliable
results.
It would also be interesting for the reader to return to the
Appendix to Part One and study the various spectra shown there in
the light of these considerations. All the spectra shown, both in
the Appendix and in the last figure, show important observational and
theoretical properties of the sea surface, but they are not quanti-
tative. They must be made quantitative to provide reliable and useful
numerical results.
119
Conclusions
Power spectra can be computed or determined electronically ina
reliable statistical way which will yield valuable information on
ocean waves. Two dimensional power spectra are also badly needed,
but the one dimensional spectra, such as have been shown, have veri-
fied many of the theoretical properties of the sea surface, which were
derived in previous chapters. In particular, sea and swell records
appear as predicted, and a quantitative spectrum of a pressure record
yields correct values for the computation of the properties of the
free surface and of the energy flux toward shore.
120
Acknowledgements
The author again wishes to express his sincere thanks to
the many people who have helped in the preparation of this work.
The continued help, cooperation, and interest of all of the
people and of the organizations mentioned in Part One is deeply
appreciated. The interest with which Part One has been received
is gratefully acknowledged.
Thanks are also due to Mr. Dean F. Bumpus for the aerial
photographs used in Chapter 12. They illustrate many important
properties of wave refraction.
July 1, 1952 Willard J. Pierson, Jr.
Department of Meteorology
New York University
Ted:
Fig.
31.
326
40.
43.
44,
Continued Index to the Figures
Part II Page
Graph of the Itcoth as a function of H #“/g and
Obner related Tunctions. < . sce 6 « « e608 «© © % «eos
Definition of terms for wave refraction theory... 43
Aerial photograph over Oracoke ......+.+e-+-+ 7
Aerial photograph over Swash Inlet .......+ee Zl
Enlarecement over Oracoke « 3 « 6 « « «© « 2 s+ 6 « « « 72
Bnlarzement over Oracoke =. 2 at's. « «6 <:3 « © ss se
Enlargement over Swash Inlet ......e«-«-e-e-«- P74
The analysis of a pressure record .....+.e+.-e+e 87
Quantitative power spectra of the pressure record
and the free surface sn6 sos. seers «° 6. ens «els « SOO
Comparison of the pressure power spectrum with
the free surface power spectrum ..... «+e. 94
Power spectra computed from Seiwell's data (after
Dukey-eand’ Hamming) . «296.60 sa ess See) ee) eel
The cumulative power distribution functions for
time series with cyclic components present .... . 103
Graphs of the functions needed in the computation
of the energy flux toward shore and of the integrand
of equation (12.50) for the power spectrum given
On PreurastOn . ele Weare chee ee | oisl am Geant ae eee OG
Graphs of various spectra and autocorrelation
functions obtained by electronic methods ..... . 114
122
Continued Index to the Plates
Part ITI Page
Plate LVII Additional properties of a short crested Gaussian
sea surface in infinitely deep water. Equations
Ga br Ba to (11.6) e ° e e ° e e e e e e e e e e e e e 2
Plate LVIII Additional properties of a short crested Gaussian
sea surface in infinitely deep water. Equations
CH?) GO (31.14) ° e ° e e ° ° e e ° e e e e ° e e 5
Plate LIX Additional properties of a short crested Gaussian
sea surface in infinitely deep water. Equations
Clelee 15) to Clie) e e e e e ° e e e e e e e e e e e 8
Plate LX Additional properties of a short crested Gaussian
sea surface in infinitely deep water. Equations
CR 22) to (11.29) e e e es e e e e e e e e e e e e e 14
Plate LXI Waves in water of constant depth. Ecuations
CBI CEO (CLAS) enn aller tanc'co ie) dis cents: el geluienerees eae
Plate LXII Waves in water of constant depth. Equations
(12.14) to (le.t7) e e e e e ° e e e e e e e e e e e 34
Plate LXIII Pressure records in water of constant depth.
Bavations. (12.18) to. Gi2.22) cy pte wie os Se ee 0
Plate LXIV The transition zone. Equations (12.23) to (12.31) . 42
Plate LXV The transition zone. Equations (12.32) to (12.38) . 49
Plate LXVI The transition zone. Equations (12.39) to (12.43) . 52
Plate LXVII The transition zone. Equations (12.44) to (12.48) . 62
Plate LXVIII The transition zone. Equations (12.49) to (12.52) . 64
Plate LXIX Additivity of power spectra. Equations (12.53)
to (12.59) e e e ° e ° e ° e ue eo e e ) ° ° e ° e ° 67
Plate LXX The breaker zone e e e e e ° e e e e e e e e e e e e 77
123
Continued Index to the Tables
Part II Page
Table 17. Computation of the Itcoth by iteration ...... 31
Table 18. Predicted and observed time during which pressure
record occupies a portion of the graph of the 25
Minute. record © sc... @ “sc ve) Ss) eee ce ene wo. ce ae ie 83
Table 19. Ratio of correct significant height to the value
obtained by erroneous extrapolation of the
pressure record upward... .« «+e «ee ce ee 96
Table 20. Numbers relevant to the computation of the flux
energy toward the shore .....-.e«-«-e+e¢ceee 109
124
Supplementary List of References
Arthur, R. S., [1946]: Refraction of water waves by islands and
shoals with circular bottom contours. Trans. A. G. U., ve 27,
no. ll.
Davies, T. V., [1951]: The theory of symmetrical gravity waves of
Soaks amplitude. I Proceedings of the Royal Society, A,
Vie 200, 195K.
Deacon, G. E. R., [1951]: Analysis of sea waves. Symposium on
Gravity Waves, National Bureau of Standards.
memes SS ee ee
Lee, Y. W., [1949]: Communication applications of correlation
analyses. Symposium on Applications of Autocorrelation An: Lyses
to Physical Problems, Woods Hole, Mass., 13-14 June 1939.
ONR Dept. of Navy, Washington, D. C.
Luneberg, R. M., [1944]: Mathematical theory of optics. Brown
University, summer 1944. (Notes no longer available.)
» [1947]: Propagation of electromagnetic waves. Lecture
notes, New York University.
Mason, M. A., [1951]: The transformation of waves in shallow water.
Coastal Engineering Council on Wave Research. The Engineering
Foundation, (pp. 22-32).
Miller, R. L., [1949]: Wave and weather correlation at Apra Harbor,
Guam, M. I., from 18 March to 31 May 1949. Wave report from
Scripps Institution of Oceanography, No. 90.
Pocinki, L. S., [1950]: The application of conformal transformations
to ocean wave refraction problems. Trans. A. G. U., v. 31, no. 6;
Rudnick, P., [1951]: Correlograms for Pacific Ocean waves. Proc.
of the Second Berkeley Symposium on Mathematical Statistics
and Probability. University of California Press, pp. 627-638.
Snodgrass, F. E., [1951]: Wave recorders. Coastal Engineering,
published by Council on Wave Research, The Engineering
Foundation, pp. 69-81.
125
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