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NAVAL POSTGRADUATE SCHOOL

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DUDLEY KNOX LIBRAE

NAVAL POSTGRADUATE SCHOOL

MONTEREY CA 93943-5101

THE EFFECT OF A SINGLE RESONANT EXPANSION

CHAMBER ON THE PROPAGATION OF LONG

WAVES IN A CHANNEL

by

ROBERT TURNER HUDSPETH

A thesis submitted in partial fulfillment
of the requirements for the degree of

MASTER OF SCIENCE

UNIVERSITY OF WASHINGTON
1966

Approved by

Department

Date

\4

I SCHOOL

)40

TABLE OF CONTENTS

Chapter

List of Figures

Acknowledgments

Preface

Page

111

IV

Abstract

VI

Introduction

Wave Reflection by an Abrupt Expansion
The Resonant Expansion Chamber

The Solution

Formulation of the Boundary Value Problem
Using the Boundary Conditions to Obtain a

Solution
Evaluating the Geometric Parameters

Presentation of Wave Data

Recording Wave Data

The Wave Spectrum

The Bretschneider Continuous Spectrum

A Functional Design

An Example of a Functional Design

A Linear Analysis

Wave Data and a Linear Analysis

Discussion

3

7

14

14

20
ZZ

26

27
28
29
30

33

33

34
39

Bibliography

40

LIST OF FIGURES

Figure Page

1. 1 Three conventional methods used to protect

breakwater entrances 2

1. 2 Partial wave reflection by an abrupt expansion

in a channel 4

1. 3 Action of a resonator on the propagation of a

gravity wave (after Valembois, 1953) 9

2. 1 The wave systems considered in evaluating the

effect of the resonator on long waves 15

4. 1 Proposed harbor entrance design 35

4. 2 Determination of the effective frequency

band-width of the resonator 37

4. 3 Determination of the effective frequency

band-width of the resonator 38

ACKNOWLEDGMENTS

For assistance in the preparation of this thesis, I wish to
acknowledge the following:

The Bureau of Yards and Docks, U. S. Navy, who made the
research and study possible by assigning me to the University of
Washington for advanced postgraduate duty during the past two years.

Professor Martin I. Ekse, who served as my advisor during
my graduate study and who assisted me in the selection of the topic
for this thesis. I am grateful to him for his encouragement and
stimulation during the study.

Professor Robert G. Dean, who formulated the problem state-
ment and proposed methods for its solution. My sincere appreciation
to Professor Dean for the copious hours which he contributed by
patiently reviewing my work cannot be adequately or fully expressed
here.

Mr. T. John Conomos for editing the manuscript and for
translating into words many of the results expressed by the equations.

Misses Linda Goodnight and Cherie Cooper for editing and
preparing the manuscript.

PR EF ACE

The study of the effect of resonant expansion systems on the
propagation of long waves is presented in this thesis as a design
problem in coastal hydraulics. Two previous studies which have been
made on similar systems are presented in Chapter I to provide back-
ground material on the subject. The first study presented was made
by Lamb (1916); and it demonstrates that reflection of long waves can
be obtained by an expansion system and that the amount of wave energy
that is reflected is a function of the geometry of the system. The
second study presented was made by Valembois (1953) and is based
on a hydrodynamic impedance theory. An important point to note in
Chapter I is that both of these studies use a scalar equation of pressure
continuity and a vector equation for the conservation of mass to obtain
a solution. A new solution is derived in Chapter 2 by formulating a
boundary value problem which incorporates the same type of boundary
condition equations noted earlier. The problem is formulated in con-
siderable detail to demonstrate the type of problem solution technique
which is required to solve problems arising in coastal hydraulics. The
details given in the problem statement become important later when
the effect of geometrical changes of the resonant system are evaluated.
Chapter 3 discusses the interdependance required between Civil

Engineering and Oceanography to effect a complete solution to a design
problem in coastal hydraulics. Several methods for presenting wave
data are given, and one of these methods is selected for the design
problem being considered. Finally, a linear analysis is employed in
Chapter 4 to evaluate the effect of the resonator on the propagation of
long waves by combining the solution derived in Chapter 2 with the
spectral method of wave data presentation selected in Chapter 3.
The results of the linear analysis are presented in graphical form
as a measure of the amount of wave energy that is reflected and
transmitted by the resonator.

ABSTRACT

The response of a single resonant expansion chamber to the

periodic pressure fluctuations of ocean waves is to act as a rigid

vertical reflecting surface. The effective frequency band-width over

which reflection is significant may be determined for a given harbor

geometry by means of a graphical linear analysis incorporating the

reflection or transmission coefficient and wave data presented by a

Bretschneider Power Sprectrum. The most critical dimension of

the resonator was determined to be the length of the expansion, I ,

R

measured transverse to the centerline of the channel; the effect of
the resonator was maximized when this dimension was either one-
quarter or three-quarters of the design wavelength, L . The influ-
ence of the width of the resonator, b , measured parallel to the

R

centerline of the channel was determined to be maximal when this
dimension was one-half of the design wavelength, la . The effective-
ness of the resonator is also a function of the width of the main

channel, b„ ,, and this parameter was incorporated in the expressions
M

for the coefficients.

Because of its analogy to electrical band- stop filters, the
possibility for improving the effect of the resonator by constructing
several resonators in cascade appears good and offers an opportunity
for extending the results obtained here.

1. INT RODUC TION

As the logistical requirements of nations grow and expand, new
techniques for designing harbors will be required to accomodate the
expansion of littoral ports of communications. The port areas which
are sheltered naturally from the ocean forces and which require little
or no additional protective structures have already been exploited.
Designers, therefore, must address themselves to the problems of
the less protected areas where this expansion must occur. One of
the major concerns in an unsheltered area is the design of breakwaters.

The purpose of breakwaters is to cause a reduction of wave
heights in its lee (Wiegel, 1964). The conventional methods used for
sheltering the entrance of a harbor between breakwaters may be
divided into three general types (Figure 1. 1). All three methods
shown require shipping vessels to maneuver in order to enter or to
leave the harbor. This required maneuvering is both expensive and
inconvenient to the user. In addition, the third method shown requires
a deep shipping channel close to the shore, and may require the addi-
tional expense of periodic dredging. The alternatives which have been
considered to provide a direct access to the shipping channels from
the harbor have included pneumatic and hydraulic devices as well as
resonant expansion chambers (Wiegel, 1964). The development of the
latter is considered here.

(I)

iD

IRECTION OF WAVE PROPAGATION

izzzzzzzzzzzzz&za

\

HARBOR

(2)

I

DIRECTION OF WAVE PROPAGATION

1

HARBOR

DIRECTION OF WAVE PROPAGATION

I

(3)

HARBOR

Figure I.I. Three conventional methods used to protect
breakwater entrances.

Wave Reflection by an Abrupt Expansion

Lamb (1916) investigated the partial reflection of a wave
resulting from an abrupt change in the cross section of a channel.
His results were based on a linear solution to the wave equation.
(A linear solution is presented in Chapter 2).

A right-handed cartesian coordinate system (z-axis positive
upwards) is oriented along the centerline of a channel at the mean-
water surface with the origin located at the discontinuity (Figure 1. 2).
The wave amplitudes and relative velocities to the left of the discon-
tinuity are given by

t,. = F(t--^) + f (t+ -—- ) (1.1)

1 cx c2

r = ■*- • [F(t - — ) - f (t + — )] (i.2)

1 cx Cj Cj

and to the right of the discontinuity by

x

:2

r\z = 0(t - — ) (1.3)

^2 =f~ ' *<t-f-> ; (1-4)

2 2

where F is an arbitrary periodic function representing the incident
wave; f and 0 are arbitrary periodic functions representing the
reflected and transmitted wave, respectively; and c and c are
the wave celerities of the wave systems to the left and to the right of
the discontinuity, respectively. The linear superposition theory is

<ss s ss s s sss s

SsssssS*

s ssssss s\

B,

-oo

F —

4-

^*

)

— ■

— •
•—

->ti

U, —

->

f'ss'^^s's^s'"' sss ^ .

B,

+ oo

PLAN

— oo

^- A

^^

2

■X-L- >

+ 00

ELEVATION

Figure 1.2. Partial wave reflection by an abrupt expansion
in a channel.

valid provided that the small amplitude assumption holds; i. e. ,

1i

« 1 for i = 1, 2. (1.5)

h

The assumption that the waves are propagating in shallow water
requires that

hi

^ ™; (i-6)

L, - 20 '
d

where L is the wavelength in deepwater (Kinsman, 1965). By invoking
d

the shallow water wave assumption given in Eq. (1. 6), the wave
celerities may be approximated by

c. = gh. for i = 1, 2. (1. 7)

There are two boundary conditions which must be satisfied at the
origin (x=y=0). The conservation of mass requires that

Blhiri= B2h2r2; I1"8'

where B , B are the breadths of the channel measured at the mean-
water surface, and h , h are the mean-water depths. The assumption
that the fluid motion will be sensibly uniform along and parallel to the
center line of the channel for small distances (compared to a wajye length) on
either side of the discontinuity requires that there be no sensible
change in the mean-water surface across the discontinuity. This con-
dition requires that the continuity of pressure across the abrupt change
be

rij = v (1,9)

Substituting Eqs. ( 1 . 2) and ( 1 . 4) into ( 1 . 8) gives

Blhl B2h2
g ~ [F (t) - f(t)] = g • 0 (t) at x = 0 ; (1. 10)

1 2

and the substitution of Eqs. (1.1) and (1.3) into (1.9) yields

F (t) + f (t) = <p (t) at x = 0. (l.H)

The ratios of the reflected wave and of the transmitted wave to the
incident wave determine the reflection and transmission coefficients,
respectively; i. e. ,

f - -- _ _Jl

K„ = — and K_ =

R F T F

These coefficients are determined by the simultaneous solution of Eqs.
(1. 10) and (1. 11). By means of Eq. ( 1. 7), the reflection and trans-
mission coefficients, respectively, are

B1C1 " B2C2
rC = ' , J (1.12)

R BlCl +B2c2

and

ZBjC

K_ = . (1.13)

T Vl+B2C2

The energy contained in the reflected and in the transmitted waves is

equal to the energy of the incident wave if the sum of the squares of the

two coefficients is equal to unity; i. e. ,

K^ + K^= 1. (1. 14)

Equations (1. 12) and (1. 13) establish that the effectiveness of the

abrupt expansion for reflecting waves is dependent on the geometry of
the channel.

A similar procedure to the above approximation given by Lamb
( 1 9 1 6 ) could be employed to evaluate the reflection and transmission
resulting from an abrupt contraction of a channel and the two solutions
superimposed in a limiting case. However, the basic assumption that
uniform conditions exist parallel to the centerline of the channel would
now introduce a greater error of approximation, as the dimension of
the length of the expansion measured parallel to the centerline of the
channel approached small multiples of the incident wavelength. These
results obtained by Lamb (1916) indicate that substantial reflection
occurs from an abrupt expansion in a channel and that a further investi-
gation of an expansion system is justified.

The Resonant Expansion Chamber

The resonant expansion chamber (resonator) takes advantage
of the periodic character of the surface disturbances and of the pres-
sure fluctuations produced by a wave (Valembois, 1953). The solution
derived by Valembois includes a wave system transverse to the center-
line of the channel which is induced in the resonator; this wave system
was neglected by Lamb in the preceding example. The derivation again
assumes a linear solution to the wave equation but provides an additional
parameter for evaluating any energy dissipation which may result. The

8
measure of energy dissipation utilizes the Neumann impedance theory
for a hydrodynamic oscillatory system; this theory is analogous to
the impedance theory for electrical oscillatory systems (Defant, 1961).

The hydraulic impedance of the discontinuity (Figure 1.3) is
in accordance with Ohm's law for the analogous electrical oscillatory
system. The effect of the resonator is to alternately withdraw from
and discharge into the main channel a quantity of fluid which is pro-
portional to the surface distortion associated with the incident wave
system. The constant of proportionality is a measure of the hydraulic
impedance of the resonator; i. e. ,

HA

Z=— (1.15)

v

A right-handed cartesian coordinate system (z-axis positive upwards)
is oriented at the orthogonal intersection of the centerlines of the main
channel and of the resonator at the mean-water depth. The linear
shallow water approximation for the wave celerity may be written as
in the preceding example as

c = \Tgh ; (1. 16)

where h is the mean-water depth of the channel. The relative velocity
U(x, t) and superelevation H(x, t) in the section to the left of the dis-
continuity at A are given by the following relationships:

U (x,t) = F (t - — ) - f (t + — ) (1.17)

c c

H(x,t)*-£-= F (t - — ) + f (t + — ) , (1.18)

SS ss SS* s s * r ss sssr Ss* *rs SS*

'S''"'''"''"'S'''''

— OO

^-rr^FA

^f;

+00

ENTRANCE OF

M

q.= Bh

s<-

B
PLAN

k

THE RESONATOR

—00

+00

■yrr

ELEVATION

Figure 1.3. Action of a resonator on the propagation of a
gravity wave, (after Valembois, 1953)

10
and to the right of A by the following:

U'(x,t) = F' (t - — ) (1. 19)

c

H'(x,t)« -S- = F* (t - — ) ; (1. 20)

c c

where F is an arbitrary periodic function representing the incident

wave; and f, F' are arbitrary periodic functions representing the

reflected and transmitted waves, respectively. Since the medium of

propagation is assumed to be at rest, all relative velocities will denote

absolute velocities. The channel is also assumed to be infinite in

extent, both in the upstream and downstream directions; therefore,

no reflection of the reflected wave or of the transmitted wave by a

terminal boundary is considered.

The desired reflection and transmission coefficients will be

determined exactly as before by considering the simultaneous solution

of two boundary conditions at A: the continuity of pressure and the

continuity of flow. The first boundary condition requires that

and the latter requires that

UA = IT. + v*. (1. 22)

A A

Substituting Eqs. (1.18) and ( 1. 20) into ( 1 . 21 ) yields

P. + fA = F"A = -£- • HA , (1. 23)

A A A c A

and substituting Eqs. (1. 17) and (1. 19) into (1. 22) yields

11
fa-£A = f'a+v|- (K24)

Substituting Eq. (1. 15) for v and designating the resulting ratio by

a = -£-• Z , (1. 25)

c

the simultaneous solutions of Eqs. (1. 23) and (1. 24) result in the

following values for the reflection and transmission coefficients,

respectively:

R ' 2 a + 1 ' v '

and

K = z Za . . (1.27)

T 2 a + 1 v

The sum of the squares of the two coefficients is not equal to unity
as in the previous example by Lamb (1916), except when the value of
a (and hence Z) is exactly equal to zero. The requirement that
the value of Z given by Eq. (1.15) be equal to zero corresponds to
no surface elevation in the resonator (e. g. , the resonator discharges
into a large reservoir) or to an infinitely high velocity of discharge.
The hydraulic impedance may be written in a form analogous to the
electrical impedance; i. e. ,

Z = r + j (id L- ) ; (1.28)

Y a

where r is proportional to the energy dissipative forces; and S. , y
are proportional to the inertial and potential (e. g. , gravity) forces.
Requiring Z to vanish is analogous to a tuned resonant circuit in which

12

where (T is the angular frequency of oscillation. The value of Z will
vanish at resonance only in an inviscid fluid (or in an analogous con-
servative system). Equations (1. 26) and (1. 27) may be used for
evaluating the amount of energy dissipation and the corresponding
amount of reflection of waves propagating in a real fluid only when a
value for r is known or may be determined.

The difficulties encountered in the application of the results
of Valembois (1953) lie in the evaluation of the parameter r and in
the criteria for dimensioning a tuned resonator. The values of the
hydraulic resistance and reactances are very small and are usually
determined empirically. Secondly, an important omission in the
theory is a rigorous analytical procedure for dimensioning the geo-
metry of the resonator. The only reference given to a dimension was
the empirically determined value for the length of the lateral expansion
which was equal to one-quarter of a wavelength. Since surface gravity
waves have no natural frequency of oscillation to which the natural
frequency of the resonator may be matched, the electrical analogy
cannot be employed as a means for dimensioning the resonator.

This thesis is addressed to the evaluation of a reflection and
of a transmission coefficient which are independent of any empirical
data required for measuring the small dissipative forces and which
relate any geometric changes in the resonator to its effectiveness for

13

reflecting waves in terms of the characteristic parameters of the
incident wavelength.

2. THE SOLUTION

Formulation of the Boundary Value Problem

The derivation of the desired coefficients will proceed in a
manner similar to that of the examples presented in Chapter 1 by the
formulation of a boundary-value problem. A right-handed cartesian
coordinate system (z-axis positive upwards) is oriented at the orthogo-
nal intersection of the centerlines of the channel and of the resonator
at the mean-water level (Figure 2. 1). The linear second-order homo-
geneous hyperbolic partial differential equation which approximately
describes the instantaneous surface elevation of the fluid in the domain
mutually occupied by the channel and by the resonator is the wave
equation; i. e. ,

(2.1)

where k has a range of values from unity to three. The wave form

2

2

8 T,

= c2 a, 3

2

2

8 t

8 x, 3x,

k k

propagates at a constant celerity of

cr

(2.2)

Neglecting the non-linear effects of the boundary conditions requires

that the following assumptions be made (McLellan, 1965):

(1) The amplitude of the surface disturbance is

small compared both to the wavelength and to
the depth of the fluid.

15

////////////////////////>/////////////////////////

— a>

UM, .

i • >

uMt

• — >

77777

7777 7

UM,

-> X

+ 00

I t

' URi

/

/
/

J I

r

J7T7T7TT7T7TTT7
V

//////////A//////////

PLAN

+ oo

/7777777777777777777777r777777777777777777777777777

ELEVATION

Figure 2.1. The wave systems considered in evaluating the
effect of the resonator on long waves.

16

(2) The channel is of uniform depth.

(3) The fluid is inviscid and irrotational.

(4) The fluid is incompressible and homogeneous.

(5) Coriolis acceleration may be neglected.

(6) Surface tension may be neglected.

(7) The rigid boundaries are smooth and impermeable.

(8) The atmospheric pressure along the water surface
is constant and uniform.

The solution to Eq.(2. 1) for one dimensional space may be

expressed in functional notation by

ti = T,(X, t) . (2. 3)

The most general solution is given by D'Alembert's Theory and is of
the form

r\ (x, t) = F (kx - 0~t) + G (kx + Ct) ; (2. 4)

where F is an arbitrary periodic surface disturbance propagating in
the positive x direction and G is an arbitrary periodic surface dis-
turbance propagating in the negative x direction. One simple-harmonic
function which satisfies Eq. (2. 4) is a cosine function; i. e. ,

n (x , t) = A, cos ( k x. ± 0"t + e, ) ; (2. 5)

k j k j k

where k has a range from unity to five corresponding to the com-
ponents shown in Figure 2. 1 and j has a range from unity to two.

The numerical indices for k are now replaced by alphabetical

17
indices shown in Figure 2. 1 in the following manner;

1 = Mi

2 = Mr

3 = Mt

4 = Ri

5 = Rr;

where the upper case letters M, R correspond to the main channel

and to the resonator components, respectively; and the lower case

letters i, r, t correspond to the incident, reflected, and transmitted

wave components, respectively. The x. components are replaced

by x and y for the coordinate system shown in Figure 2. 1. Equation

(2. 5) is a set of five wave component equations containing ten unknown

constants: viz. , the five values for the wave amplitudes A,,., A,, ,

Mi Mr

A,, ,'A^., A^ and their associated phase angles e,,., e„, , €,, ,
Mt Ri Rr PS Ml» Mr> Mt»

e , e . The set of equations expressed in Eq. (2. 5) are the following:

^a*- = A™- cos ( k x - Ct + c ) (2. 6:1)

Mi Mi Mi

Tlw = A,, c°s ( k x + 0"t + c ) (2.6:2)

Mr Mr Mr

*1 w = Aw cos ( k x - 0~t + e ) (2.6:3)

1 Mt Mt v Mt' v

11 Ri = ARi COs(ky" fft + £Ri} (2' 6: 4)

71 Rr = ARr cos ( k y + (It + 6Rr). (2. 6: 5)

In a well- stated boundard-value problem, the unknowns contained
in Eqs. (2. 6: i) may be evaluated by a given set of boundary and initial

18
conditions (Tikhonov and Samarskii, 1963). In the stating of the con-
ditions required by the above to comply with the criteria for a well-
stated boundary-value problem, the following assumptions are made:

(9) The fluid medium of propagation is at rest

(i. e. , all velocities are absolute).

(10) The terminal boundaries of the centerline of
the main channel are sufficiently removed
from the proximity of the resonator and, there-
fore, are not required boundary conditions.

(11) Steady state conditions exist.

(12) All geometric dimensions considered are
small compared to wavelengths (i. e. , long
waves).

(13) The cross sectional area of the channel and
of the resonator consists of a horizontal
bottom and parallel, vertical sides.

(14) Shallow water conditions exist.

Assumption (11) negates the requirement that the initial conditions
be stated. Assumption (14) permits the use of the shallow water
approximation for the wave celerity; i. e. ,

c2 = gh. (2.7)

The relative velocities shown in Figure 2.1 are related to Eqs. (2. 6: i )
in the following manner:

A c

U„. = — t11 M * cos (kx - (Tt + c .) (2. 8: 1)

Mi h , Mi

M

A c
— Mr M
Uw = ? — ' cos ( k x + 0~t + e w ) (2. 8: 2)

Mr h , Mr

M

UMt

TT

A c

Mt M

M
Ri R

URi "

h

19

cos (k x - 0"t + e ) (2. 8: 3)

cos (k y - 0~t +'€_.) (2. 8: 4)

Ri
R

A c

U = Rr • cos ( k y + Ct + e„ ) . (2. 8: 5)

Rr h v ' Rr7 x '

R

The boundary conditions will be prescribed along the free surface in
terms of Eqs. (2. 6: i ) and along the vertical boundaries in terms of
Eqs. (2. 8: i ).

The free surface boundary condition prescribes the instantaneous
continuity of pressure at the origin for all time:

?lC:+n*fcrr T1Mt= ^Ri^Rr* (2' 9)

The vertical boundary conditions require two statements. The first is

that the continuity of flow into tne domain is expressed by

[Uw+Uw " Uw ] t> h -+ [Ul . + U^ ] bh = 0; (2.10)

L Mi Mr MtJ M M L Ri RrJ R R v '

and the second is that the no flow condition across the rigid terminal

boundary of the resonator is given by

[T? . + U„ ] b^ h^ = 0. (2. 11)

L Ri RrJ R R v '

The above three boundary conditions are evaluated at the following

coordinates:

Eq. (2. 9) at the origin (i. e. , x = y = 0);

Eq. (2.10) atx = ±— , y ="-^- ;

and Eq. (2. 11) at y = -i , x = 0.

R

20
Using the Boundary Conditions to Obtain a Solution

The main channel incident wave may be arbitrarily selected as
the reference wave profile by assumption (11) without any loss in
generality; therefore, its associated phase angle is defined as

<Mi = °- (2-12>

Eight independent boundary equations will result by the substitution of

Eqs. (2. 6: i) and (2. 8: i ) into the three boundary conditions prescribed

in Eqs. (2. 9), (2. 10) and (2. 11) when the expressions obtained after

substitution are expanded by means of the trignometric identities for

the sine and cosine of the sums and differences of angles. The time

dependence of the boundary conditions may be eliminated by equating

the sine and cosine terms which result from the above expansion. The

ten equations which result from the preceding algebra are the following:

A,,. +A,, cose,, =AW cosew (2.13:1)

Mi Mr Mr Mt Mt

-A,, sine,, = A,, sine,, (2.13:2)

Mr Mr Mt Mt

A,,. + A,, cos e,, = A„. cos e- + A_ cos e^ (2. 13: 3)

Mi Mr Mr Ri Ri Rr Rr v

- A, , sin e, , = + A^. sin e_. - A_ sin e_ (2. 1 3: 4)

Mr Mr Ri Ri Rr Rr v

A,, cos e,, = A„. cos e^. + A^ cos e^ (2. 13: 5)

Mt Mt Ri Ri Rr Rr v

AMtSin6Mt = ARi Sin 6Ri - ARr Sin 6Rr (2" ^ b)

ARiCOS(kV£Ri> =ARrC°S(k,E-!Rr» <2'13:7)

21
-ARiSin,WR-eRi) = ARrsLn(1%- ,Rr> ,2.0:8)

b b b

c„bfA,,cos(k— )-A, cos(k-r--£w )-Aw cos(k-r-+ ew )] (2.13:9)
M ML Mi 2 ' Mr v 2 Mr' Mt v 2 Mt/J v '

b bM

= CRbR["AR,COS(k — " «Rl>-+ ARr cos<k — " eRr^

b b b

-^V+AMiSin(kT) + AMrSin(kT-W +AMtSm(kT + W] <2' 13: 10)

b., b

r A #1 M X , A /I M xl

= CRbR [ARi Sln(k— " 6Ri» + ARr s^k~ -«Rr)l ■

By means of Eqs. (2. 13: 7) and (2.13:8), it may be shown that

A„. = A^ (2. 14)

Ri Rr v '

and4 that

+2kl = 6„. + e„ . (2. 15)

R Ri Rr v '

By squaring and adding Eqs. (2. 13: 5) and(2. 13: 6), Eq. (2. 14) may
be rewritten in the form

AMt (2.16)

Ri |2 cos( ki _ )l
R

Equation (2. 7) may be employed to equate the two celerities; i. e. ,

CR = CM- (2-17)

Following the substitution of the known relationships and applying some
additional trignometric identities, the reflection and transmission
coefficients may be obtained by squaring and adding Eqs. (2. 13: 9) and

22

(2. 13:10). The reflection coefficient may be expressed by

b b

R M

M

M

sin (ky) + (— )[sin(k— ) - ItanCkj^)! cos(ky )]
M

K = _
R

b b b b b„

R . R r .M.i .... i ., M.n , , R 2r

bM,2M

l+(^)sin(k^)[sin(ky)-|tan(kiR)|cos(k-p] +(^_)2[sin<kl")'ltan(kVl COs(klp]
M M

f

(2.18)

and the transmission coefficient by

K =

R
cos (k )

bbb b b „ b b , '

R R.r ... M . i . . i . M -, . . R 2r . .. M. i .. . . i .. M, 2

l+(— )sin(k— )[sin(k — )- |tan(k| ) |cos(ky)-]+ (— ) [sin(k— )- | tan(k|R) |cos(k — )]
. M M

(2.19)

The sum of the squares of Eqs. (2. 18) and (2. 19) is equal to
unity and verifies that the energy in the system is conserved.

Evaluating the Geometric Parameters

The criteria for dimensioning the resonator are the maximizing
and minimizing of Eqs. (2. 18) and (2. 19), respectively. This minimizing
and maximizing process requires the evaluation of the arguments of the
trignometric functions involving I , b , and b in terms of the tuned
wave length, L;.., and wave period, TQ, of the resonator.

The dimension which has the most critical influence on the
value of the coefficients is the transverse length of the resonator, H ,
which appears in the argument for the tangent function. The tangent
function becomes discontinuous when its argument approaches a value

23

IT

of (2n -1) — radians, where n is any positive integer. The argu-
ment reaches this value when

*R = (2n - 1) — ; (2.20)

where n must be equal to unity or two by assumption (12). Eq. (2. 20)

agrees with the results obtained by Valembois (1953) when the value

of one-half of the width of the main channel, b, ,, is either a very small

M

per cent of L^ or equal to one-half L . The argument containing

i may be expressed in terms of the angular frequency, 0", by means

of the wave celerity equation; i. e. ,

k*R "f-V (2-21)

L*
Decreasing the dimension | in Eq. (2. 21) to a value less than — - —

increases the tuned resonant frequency, 0~ , by a corresponding

factor; while increasing S to a value greater than — — decreases

the tuned resonant frequency, 0" , also by a corresponding factor; i. e. ,

1_/ 1

Lf i < — — , then 0~> 0~
R 4 o

and if I > — ^— , then 0"< 0" ;
R 4 o

where 0" is the tuned resonant frequency for maximum reflection. It

may be seen from Eq. (2. 16) that the resonator wave amplitude would

become infinite at resonance for any finite value of the transmitted

wave height, A„ , . Since an infinite wave amplitude does not
Mt r

seem to be physically valid, the transmitted wave height, Aw

Mt

24

must be identically zero.

The influence of the width of the resonator, b , on the value

R

of the coefficients is most easily evaluated by means of Eq. (2. 19).

Minimizing the transmission coefficient requires that the value of the

cosine function in the numerator approach zero. The cosine function

approaches zero when its argument approaches (2n -1) — , where n

is any positive integer. Therefore, the value of the width of the

resonator must be

L
bR ^ (2n -1) — y- ; (2. 22)

where n must be unity by assumption (12). If assumption (12) is

valid, the value of b determined by Eq. (2. 22) is a maximum;

R

hence, b may only decrease from this maximum value. The argu-
R

ment containing b may be expressed in terms of the angular frequency

R

as before by

Ir bR °" bR #9 *\\

k — = — — . (^. 23)

Decreasing b from its maximum value given by Eq. (2. 22) increases
R

the tuned resonant frequency 0" .

o

Finally, a critical value for the width of the main channel, b ,

M

is not easily evaluated by means of the equations for the coefficients;
however, because the preceding derivation was based on the assumption
that all -dimensions were small compared to a wavelength, it may be
determined that the width of the main channel must be a small per cent

25
of the tuned wavelength, L .

The evaluation of the geometry of the resonator system has
been discussed by considering the effect of a geometrical change on
the ability of the resonator to reflect an incident wave of a single fre-
quency. Assumption (11) does not allow the application of the coeffi-
cients to a constantly varying input inducing a transitory response.
However, the experimental results obtained by Valembois (1953)
indicate that appreciable reflection does occur over a range of fre-
quencies. In fact, the response curves obtained by Valembois are
similar to the response curves for electrical filter circuits (Olson,
1958). The analogy of these two oscillatory systems introduces the
possibility of determining an effective frequency band-width for which
reflection may be significant.

3. PRESENTATION OF WAVE DATA

The derivation of the reflection and transmission coefficients
in Chapter 2 and their subsequent discussions dealt with an incident
wave of a single frequency and wavelength. An analysis of an incident
wave record reveals that waves in nature are composed of an infinite
and continuous number of frequencies resulting from a complex
phenomenon. Since the resonant structure is static, some means
of determining a single frequency which is a characteristic measure
of the complicated group phenomenon contained in the waves must be
made as well as some means for evaluating the effect of the resonator
when subjected to an input composed of an infinite series of incident
frequencies. Fortunately, the solution to the first problem provides
a means for resolving the second.

The problems encountered in recording and in analyzing surface
waves are currently under research by the discipline of Oceanography.
A complete or even adequate treatment of these complex problems is
beyond the scope of this thesis. The design and construction of the
resonant breakwater serves to illustrate the interdependance required
between Civil Engineering and Oceanography in oceanographic con-
struction. The following discussion is intended only to provide some
background and continuity for the use of the oceanographic data required

11

to effect a complete solution to the design problem posed. The refer-
ences found in the Bibliography will provide a means for initiating a
more detailed study concerning spectral analysis of ocean waves.

Recording Wave Data

The answer to the rhetorical question of how to extract a
characteristic frequency from those occurring in a natural phenomenon
is affected by the methods available for recording the actual physical
profile of the surface disturbance. The various recording devices
currently in use all have the common failure of a lack of a standard
measurement and of distortions inherent in the recording apparatus.
The effect of the recording distortions becomes more pronounced if
small non-linear components must be recorded accurately; but the
presence of these inherent distortions will influence the shape of any
wave profile recorded and will, therefore, be incorporated in the data
for a nonlinear analysis as well. Because the recording process is
quite complicated, the author can only refer to oceanographic literature
and follow Kinsman (1965) by assuming that for the case in question
an adequate and representative wave record is available to the
designer from the oceanographer.

28

The Wave Spectrum

The assumed wave record obtained from the oceanographer
may also provide an answer to the again rhetorical question of how to
extrapolate a characteristic design frequency from the natural wave
phenomenon. The continuous distribution of the frequencies in a wave
train suggest that a statistical technique may be employed to extract
the characteristic frequency. The Sverdrup and Munk Theory, for
example, describes the sea surface by a single sinusoidal component
called the "significant wave" (Kinsman, 1965). The next level of
complexity involves a Fourier analysis.

In a Fourier or harmonic analysis, the wave record is mechani-
cally transformed into a spectrogram (McLellan, 1965). If the charac-
teristic solutions to Eq. (2. 1) are orthogonal with respect to the
weighting function (which arises from the Sturm -Liouville problem
generated by the separation of variables) and are piecewise differenti-
ate in the recording interval T (Hildebrand, 1962), the solutions
may be represented by a product series of the form

00

Mt) = ) n (t). (3. 1)

L-i n

n = 1

The solution determined in Chapter 2 was a cosine function. Since
this function is symmetrical with respect to the z-axis, it may be
expressed in a Fourier series for an even function; i. e. ,

29

00

V H

n (t) = ) —2— cos ( a t - € ) . (3. 2)

n l_j 2 n n

ri= 1

The coefficients H are determined by multiplying each side of Eq.

th
(3. 2) by the weighting function and by the k solution, n , given

in Eq. 3. 1, and then integrating over the truncated recording interval,

T.

H

oo
_2
T

ou

\ ri(t) • cos (d t- € ). (3. 3)

*J „ n n

0

The value given by Eq. (3. 3) is equal to twice the average
value of the product n (t) * cos ( 0" t - e ) over the recording interval
T. The wave energy spectrum may also be derived from Eq. (3. 2)
by squaring both sides of the equation and integrating over the recording
interval T. The harmonic analysis of the wave record does not pro-
vide any information regarding the shape of the surface disturbance
when

| t| >T (Kinsman, 1965).. (3.4)

The next logical progression from this discrete spectrum is a repre-
sentation of the wave record by a continuous spectrum.

The Bretschneider Continuous Spectrum

The purpose of this discussion on the methods of presentation
of wave data is to enable the designer either to .request wave data in
a particular form or to transform a wave record available from the

30
oceanographer. One such method of wave data presentation currently
in use is the Bretschneider Power Spectrum. This spectrum requires
that the average height of the highest one-third waves, H / , as
determined from a Rayleigh probability distribution of wave heights,
be known (Dean, 1966). From the value of the significant wave height,
H / , a parameter which is equal to twice the total energy contained in
the waves is defined by

Hl/3 = 2.83 nTE£ . (3.5)

Bretschneider determined that the spectral distribution of the wave
energy, S £, as a function of frequency (7 could be represented by

S„2(<r)= ^A-'IT'5 e"'-25'^'4 (3-6)

o

The parameter 0" is the frequency at which the peak energy occurs
and is defined in terms of the significant wave period, T /„, by

0" = , ,Z, T m . (3.7)

o 1. 17 • T . v '

Integrating Eq. (3. 6) from zero to infinity yields twice the energy
contained by the waves E .

A Functional Design

The design of the resonant breakwater considered in this thesis
is an example of a functional design discussed by Wiegel (1964);

31
however, only part of the protection function (which is but one aspect
of the over-all function of the breakwater) is developed. The functional
criteria also affect the method required for presenting the wave data.
The purpose for which the harbor is being constructed dictates both the
functional requirements of the resonator and the required method of
wave data presentation.

As discussed previously, the wave height and wave period
parameters normally available from the oceanographer are those of
the significant wave, H , and T / . The design criteria will
normally require protection from the effects of the highest waves
present at the structure. The significant wave height, H , , has
been determined to be an accurate index to structural damage by
waves (Wiegel, 1964).

Harbor oscillations or seiching will also normally be found
in the design criteria for a harbor. Seiching is a complicated response
of the harbor to a periodic force. This response is a function both
of the geometry of the harbor and of the period of the force. Wave
data for designing protection from seiching, therefore, must be a
function of frequency.

One solution to the problem posed at the beginning of this
chapter of how to determine a single design frequency from the con-
tinuous spectrum of frequencies occurring in the natural phenomenon
appears to lie with the statistical significant wave period, T / .

ss

32 -

The Bretschneider Power Spectrum derived from the significant wave
height, H /_, will then provide a means of evaluating the effectivenei
of the resonator to an input composed of an infinite and continuous
spectrum of frequencies. By means of this power spectrum, the effect
of a resonator on a wave system occurring in nature may now be
evaluated.

4. AN EXAMPLE OF A FUNCTIONAL DESIGN

The Bretschneider Power Spectrum offers a solution to the
design problems posed at the beginning of Chapter 3. Evaluating the
effectiveness of the resonator over a spectrum of frequencies by com-
bining this power spectrum with the reflection and transmission coef-
ficients represents a linear analysis. The analysis is linear since
both the power spectrum and the expressions for the coefficients were
derived from linear solutions.

A Linear Analysis

In a linear process, the output, 0, is a linear function of the
input, I ; i.e.,

0 = R (d) • I . (4. 1)

R(0~) is the linear transfer function and is a function of frequency, 0".
The essential features of a linear process are the following (Dean, 1966):

1. Changing the amplitude of the input function,

I, changes the amplitude of the output function,
0, proportionally.

2. The constant of proportionality is the linear
transfer function and is a function of frequency,
0", only (i. e. , the transfer function is inde-
pendent of the amplitude).

3. Input functions may be superimposed.

The effectiveness of the resonator at a given frequency, therefore,

34

may be determined from the product of the square of Eq. (2. 18) or

(2. 19) and the value of the ordinate S ? from Eq. (3. 6) for the

H
Mi

same corresponding frequency; i. e. ,

or

S 2 (0- ) = KJ: ((I ) • S 2 (0" = > (4.2)

H l K l H l

Mr Mi

! 2 (GT ) = K" ( (T ) ' S 2 (0\ ). (4.3)

H^ i T l H <- 1

Mt Mi

Wave Data and a Linear Analysis

As an illustrative example of a linear analysis, it is assumed
that a previously undeveloped coastal area has been selected for the
construction of a new harbor facility. The harbor has been designed
to accomodate large commercial vessels (Hennes and Ekse, 1955). The
proposed entrance to the harbor between the breakwaters is to have
a depth of fifty (50) feet and a width at the mean-water level of two-
hundred (200) feet (Figure 4. 1). The effectiveness of the construction
of a single resonator on the local wave conditions is to be measured.

The preliminary engineering report includes the collection of
extensive oceanographic data on the proposed location. From this
report, the significant wave height and period were determined to be
the following:

35

q:
O
on
cc
<

o

ID

It

ii 002

c
o>
'55

Q>
"D

0)
O

c
o

w

c

o

n

\-
o

(0

o
a
o

a>

o

LU

0_

Q_

I
CO

<1

I
o

36
H / = 12 feet
T , = 12.7 sec

The total energy contained in the waves is determined from Eq. (3. 5)

to be

E = 18 sq. ft.

The significant wave period, T , , determines the value of 0" in

Eq. (3. 7) from which the spectral distribution of the incident wave

energy, S 2 » may be determined by means of Eq. (3. 6) (Figures

HMi
4. 2 and 4. 3).

By satisfying the requirement of Eq. (1. 6), the shallow water

approximation for the wave celerity is determined to be forty (40) feet

per second through the breakwater entrance. By means of Eq. (2. 2)

and the value of the wave celerity, the significant wave length, L , is

found to be six-hundred (600) feet. Utilizing the design parameters

from Eq. s (2. 20) and (2. 22), the dimensions of the resonator are as

follows:

I =150 feet

b = 300 feet

The values of the reflection and transmission coefficients may now
be determined as a function of frequency from Eqs. (2. 18) and (2. 19),
respectively. The frequency dependence of the square of these
coefficients is also shown in Figures 4. 2 and 4. 3.

1.0

37

T (sec)
20 17 15 13 12

cvi oc

0.5

0.0

1R= 150ft
bR=300ft
bM = 200ft

TTTT

0.2

0"0 0.6

CJ (rod /sec)

SQUARE OF THE REFLECTION COEFFICIENT

75

CVI

I 50[-n

I

25

Ef=l8fr
H,/, = 12 ft

Tl/3=l2.7ft

0.2 Q0 0.6

0" (rod /sec)

MAIN CHANNEL INCIDENT WAVE ENERGY SPECTRUM

CM

CVI 2

I
(/)

0~0 0.6

0" (rod /sec)

MAIN CHANNEL REFLECTED WAVE ENERGY SPECTRUM
Figure 4.2. Determination of the effective frequency band -width of the resonator.

1.0

38

T(sec)
20 17 15 13 12

0.5-

0.0

iR= 150ft

bR = 300ft
bM= 200 ft

0.2

0" (rod /sec)

SQUARE OF THE TRANSMISSION COEFFICIENT

0.2 0*o 0.6

(J (rod /sec)

MAIN CHANNEL INCIDENT WAVE ENERGY SPECTRUM

3

M 2

X

0.2

0.6

Co
0" (rad/sec)

MAIN CHANNEL TRANSMITTED WAVE ENERGY SPECTRUM
Figure 4.3. Determination of the effective frequency band-width of the resonator.

39
The reflective and transmissive effectiveness of the resonator
at a given frequency is determined from the product of the ordinate of
the incident wave spectrum and the ordinate of the transfer coefficient.
The resulting spectral distributions of these two linear analyses are
shown on Figures 4. 2 and 4. 3.

Discussion

The resonator appears to have a frequency band- width ranging

from 13 to 17 seconds in which reflection is approximately 100%. The

effectiveness of the resonator is determined from the amount of energy

which it reflects and transmits. This amount of energy is measured

from the area under the power spectrum curves for the reflected and

transmitted waves, S £ an<^ S £ » shown in Figures 4. 2 and

H Mr H Mt

4.3, respectively. The narrow spike occurring between the periods

of 13 and 14 seconds prevents the resonator from being greater than

85% effective in reflecting the incident wave energy. The proposed

design for the given wave conditions is approximately 75% effective

for reflecting the energy contained in the incident wave spectrum

between the periods from 10 to 17 seconds.

BIBLIOGRAPHY

Dean, Robert G. Unpublished Lecture Notes, Coastal Hydraulics
CE 544. Seattle, Washington: University of Washington, May,
1966.

Defant, Albert. Physical Oceanography, Vol II. New York:
MacMillan Co. , 1961. p. 179.

Hennes, Robert G. and Ekse, Martin I. Fundamentals of Transportation
Engineering. New York; McGraw-Hill Book Company, Inc. ,
1955. pp. 393-394.

Hildebrand, Francis B. Advanced Calculus for Applications. Englewood
Cliffs, New Jersey: Prentice-Hall, Inc., 1962. p. 217.

Kinsman, Blair. Wind Waves. Englewood Cliffs, New Jersey:
Prentice-Hall, Inc., 1965. 676pp.

Lamb, Horace. Hydrodynamics. Third Edition. Cambridge, England:
University Press, 1916. p. 253.

McLellan, Hugh J. Elements of Physical Oceanography. London:
Pergamon Press, 1965. pp. 95-102.

Olson, Harry F. Dynamical Analogies. Second Edition. Princeton,
New Jersey: D. Van Nostrand, Inc. , 1958.

Tikhonov, A. N. and Samarskii, A. A. Equations of Mathematical
Physics. New York: The MacMillan Co. , 1963. 765 pp.

Valembois, J. Study of the Effect of Resonant Structures on Wave
Propagation. Translation No. 57.6, U.S. Army, Corps of
Engineers, Waterways Experiment Station, Vicksburg, Mississippi.

Wiegel, Robert L. Oceanographic Engineering. Englewood Cliffs,
New Jersey: Prentice-Hall, Inc. , 1964. 532 pp.

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