NEL/REPORT 1432 Walon CEpl LYOddu/ 14N DISTRIBUTION OF THIS DOCUMENT IS UNLIMITED 0 0301 0040540 3 ONO L AOA PROBLEM Investigate and report on the nature of internal waves of 2-to-20-minute periods at the U. S. Navy Electronics Laboratory (NEL) Oceanographic Research Tower. Specifically, study their relation to surface waves. RESULTS Il Resonant interaction between amplitude-modulated swell and internal waves may create internal waves of 2-to-20-minute periods. Internal waves with ampli- tudes of 1-3 meters can be produced within an interaction time of 15 minutes if the stratification of the water is such that — +~—- 2 9 Ry — k| (oy — Wo)” is an eigenvalue of the internal wave equation (Ey, Eo, @ 1, and w2 being wave numbers and frequencies of the modulated swell). De Internal waves due to amplitude-modulated swell have the same characteris- tics as the modulation. Specifically, they have the same wavelength and period and travel in the same direction as the amplitude modulation. Bo The resonance process is most efficient in the case of a modulation which travels in the same direction as the carrier wave (the main constituent) of a swell. The creation of internal waves of this type is strongly dependent on the stratification of the water. 4. Preliminary evaluations of temperature and swell records from the NEL Tower show good agreement between periods of swell modulation and internal wave periods. 5. It is likely that internal waves are created on the entire continental shelf off Southern California during times of favorable stratification. This depends on tides and wind. RECOMMENDATIONS IL. Verify, with additional measurements at the NEL Tower, the theory that internal waves are produced by swell. De Acquire, for complete analysis, long records of temperature fluctuations and surface waves in order to compute spectra with high resolution and statistical confidence. 3. Establish, by measurements of surface waves at two positions, and slick observations, that internal waves travel in the direction of the swell modulation. ADMINISTRATIVE INFORMATION Work was performed under SR 104 03 01, Task 0594 (NEL L41171). The report. covers work from August 1966 to December 1966, and was approved for publication 30 January 1967. Financial arrangements for travel and salary were furnished by the Office of Naval Research (ONR) and administered by Scripps Institution of Oceanography. The author is grateful for the assistance of J. L. Cairns, D. E. Good, J. R. Olson, and C. L. Barker in collecting data; of Mrs. R. P. Brown in programming and computer analysis; of C. V. Hart in computations and plotting; and E. C. LaFond, O. S. Lee, and J. L. Cairns for discussions. CONTENTS LIST OF SYMBOLS .. . page 5 INTRODUCTION .. . 7 ENERGY TRANSFER FROM SURFACE WAVES TO INTERNAL WAVES BY RESONANCE .. . 9 EQUATIONS AND BOUNDARY CONDITIONS . . . 10 THE PERTURBATION EQUATIONS . . . 11 THE EQUATION FOR w(x, zt)... 13 THE PRIMARY WAVE FIELD AND THE FORCES fj and fo .. . 15 THE SECONDARY WAVE FIELD IN THE RESONANCE CASE . . . 17 Ft (2) Gt_(0), and At, IN CASE OF AN EXPONENTIALLY ii, > nia) mnr STRATIFIED SEA... 19 THE SECONDARY RESONANCE WAVE FIELD DUE TO SWELL IN CASE OF AN EXPONENTIALLY STRATIFIED SEA . . . 23 MEASUREMENTS ON SWELL AND INTERNAL WAVES . . . 24 CONCLUSIONS . . . 26 RECOMMENDATIONS . . . 27 REFERENCES . . . 27 ILLUSTRATIONS 1 Temperature recorded by thermistors 3 to 21 at the NEL Tower (4 October 1966, 1900-2000) . . . page & 2 Amplitude (cm) of internal waves due to modulated swell after an interac- tion time of 103 seconds (16.6 minutes) as a function of the angle between the two primary waves... . 23 3 Swell spectra of wave height sensors 1 and 2 at positions near the NEL Tower, and phase difference between both records (4 October 1966, 1230-1330, sampling rate 1 second); the arrow indicates the 95-percent confidence limit... 25 4 Histograms of periods of swell modulations (left) and of internal waves (right) at the NEL Tower. . . 26 REVERSE SIDE BLANK 3 fi i D ‘a 0] cy" be 4 rs Posies a ig ] F ¥ ‘i aT via a Md: i a «| 7 it q J i { i ar 3 $y LIST OF SYMBOLS If not defined in the article, the symbols have the following meaning: A, Amplitude of W Acian Expansion coefficients, see equations (58) and (62) On, Os See equations (63) and (64) ee See equation (57) fifo See equations (39) and (40) + . Sav (IRE Ge See equation (52) g Acceleration of gravity H Bottom depth h Horizontal components, used as an index R= (Kk 47) Wave number vector p Pressure, p mean pressure w (u, v, w) orbital current velocity of the waves, with components u, U, W in direction x, y, Z W (2) Eigenfunction of nth mode, according to equations (46) to (48) xX (x, y), horizontal plane z Vertical coordinate, pointing downward 1 dp 1 (z) — —; V9 = const p dz ; G Amplitude of a wave p Density, p(2) mean density distribution co) Gravitational potential @ Angular frequency @ @ @ V =, =, =} del operator Ox Oy az Ve Diver ivergence v2 Laplace operator REVERSE SIDE BLANK INTRODUCTION It was shown by E. C. LaFond!-3 and O. S. Lee4 that internal waves of 2-to-20-minute periods are a dominant feature in the thermal structure of the sea around the NEL Oceanographic Research Tower. From extensive measurements, E. C. LaFond derived that 50 percent of all waves had periods greater than 7.3 minutes and 50 percent of waves had heights of more than 5.6 feet (170 centi- meters). Wave heights of more than 20 feet (6 meters) can sometimes be observed in water only 60 feet (18 meters) deep. These short-period oscillations are only one part of the entire internal wave spectrum. They are superimposed on longer fluctuations of the mean ther- mocline, with changes mainly due to internal tides and wind.5 Nevertheless, these waves in the 2-to-20-minute range are the most striking fluctuations besides the internal tides. Their occurrence seems to depend on several factors. There is neither a close relationship to the surface tides nor to the internal tides, but the changing stratification due to the tides seems to be of importance. Most likely, these waves are long-crested, progressive waves traveling toward shore with a velocity of 20-to-40 feet per minute (10-to-20 centimeters per second). The measurements by O. S. Lee4 indicate a beamwidth of only +15 degrees. This agrees with former measurements by C. W. Ufford,® G. Ewing,? and E. C.LaFond.1 C. §. Cox® got similar results. They are supported by observations of sea surface slicks, which are often closely related to internal waves. According to O. S. Lee,4 the mean speed is 27 feet per minute (13.7 centimeters per second) and the direction 85 degrees. Figure 1 gives an example of these waves measured on 4 October 1966, 1900-2000. The temperature fluctuations are shown for thermistors 3 to 21. The distance between the thermistors is 2.5 feet, thermistor 3 being 5 feet above the bottom and thermistor 21 about 20 feet below sea surface. The water depth is 60 feet. After a calm period of several hours the waves start to occur at about 1900. The isotherm depth decreases during the next half hour and during that time high- amplitude internal waves are present. The period is not quite independent of depth. Shorter periods are generally observed near the surface than in deeper layers, but all waves are of first mode. This seems typical for the area. The origin of these waves is rather obscure. There are no obvious mete- orological or tidal forces that could produce the regular wave trains. If there were a constant coupling between the tidal phase and the occurrence of these waves, one would be inclined to interpret them as an adaptation of the changing mean stratification. But this is not possible. The only reason for their creation therefore seems to be surface waves. F. K. Ball9 has shown that, in the case of a two-layered model, resonance is possible for second order interactions between surface and internal boundary waves. S. A. Thorpe!0 extended the theory to wave interactions in a contin- uously stratified fluid. He showed that a transfer of energy from surface to internal waves may occur, and an internal wave generation mechanism will exist. The theory has been applied to situations which might be realized in the labora- tory, but an application to natural conditions has not been attempted. For tank experiments (under somewhat extreme conditions) he found that the internal wave amplitude will be equal to that of the surface waves after an interaction time of only 28 seconds. eae NO y, is JX BS 2S, iO Ue LS Ney, \ — 0 14 ee ae = / 2 \ < SS ‘ YEN ag 12 sg Ss WwW he a oO \ — i 10 — Kt 8 6 4 1900 iron 1930 tinea aaa "ies 2000 Figure 1. Temperature recorded by thermistors 3 to 21 at the NEL Tower (4 Octo- ber 1966, 1900-2000). (Each curve represents the variation of the temperature from the mean for that numbered thermistor.) K. E. Kenyon!! extended the theory to a description of the entire internal wave spectrum which may result from the interaction of surface waves in the ocean. He gave a computation of scattering of swell energy into internal wave energy, using observations from the NEL Tower, from which he concluded that this process may be of minor importance for the creation of internal waves. The following computations are similar to his, but we do not compute the energy spectra, and derive only the equations for the amplitudes of the waves. We come to different results, the reason for this being that Kenyon considered surface wave spectra only for the case that the directionality of the spectrum is given by cos4a. This means that all surface waves are contained within an angle of about +30 degrees. From this narrow directional swell spectrum, energy is fed to internal waves traveling mainly perpendicular to the swell. This, obviously, does not agree with the observations mentioned above. Both swell and internal waves travel mainly toward shore in the sea off Southern California. Kenyon’s conclusion that surface waves with a narrow directional spectrum do not feed considerable energy into the internal wave spectrum is supported by K. Hasselmann.!2 This agrees with our results, but it will be shown here that the process is about 100 times more effective for amplitude-modulated swell, which may be described by two surface waves traveling in directions which in- clude an angle between 90 and 180 degrees. This still leads to a progressive wave, if the two amplitudes differ considerably in magnitude. If these two waves are given by CG) =A, cos hy: x—0,t) , Ge, =A, cos Ry-x— oot) (A) with A, > A,, the resulting wave can be described by C(t) =6,(8, 1) + 6%.) = AG, t) cos (hex — at — Hx, ) (2) where the amplitude A(x, t) and the phase A(x. t) are given according to Ales) =|A_ 22 A 22 OA. eos, Plo" wl @ Me ila oe ay) Silay Soa rreeerar or she 19) ; and = A. sin Be Bayon = ¢ Bae | f(x, t) = arctan —2 ese sole? (4) A, fi Ay cos [" Tee: ~ (@;—@o) | thus representing a swell which travels in a direction given by the wave vector k, with an amplitude changing between A, + Ag and a variable phase. Wave measurements at the NEL Tower indicate the reality of such a swell. ENERGY TRANSFER FROM SURFACE WAVES TO INTERNAL WAVES BY RESONANCE The process of resonant interaction is easy to comprehend, but the detailed analysis involves a great deal of algebra. A short outline of the main ideas may help in understanding the following computations. The hydrodynamic equations are nonlinear. They include terms of the form wUeVu, u being the velocity vector. In internal wave theory, these equations are linearized by perturbation methods. The result is a sum of mutually independent waves, each one conserving its own energy if friction is neglected. Many fea- tures are adequately described by this linear theory, but there exist circumstances under which the nonlinear terms can give rise to significant energy transfer be- tween these primary wave solutions. Surface waves are zero mode solutions of the internal wave equation. In- ternal waves may be of any mode larger than zero. The computations in the following sections are based on this idea. Consi- der the simple nonlinear wave equation d2¢ at? + @*6— $7 =0 (5) 10 Applying perturbation methods, ¢ = 6\!) + 6‘2) + - - -, the first-order equation reads with the solution AO) » ae en" (7) n These are mutually independent waves. The second-order equation is 9 2g'2) 2 i Lal?) Spr » aac! m +@,Jt (8) ™m nN at The secondary waves ¢‘2) are therefore forced waves with frequencies w,, + w,.- As long as these combination frequencies w,, + @n are not the same as a natural frequency of the system, the amplitudes of the secondary waves will remain small. However, when one of the combination frequencies w,, + @, is the same as one of the natural frequencies w of the system, this wave never gets out of phase with the forcing wave, and a continuous energy transfer, only limited by the available energy in the primary waves, is possible. If this occurs, the solution of (8) is of the type 4{2)«t, that is, 62) increases linearly with time. This is the resonance case. Its physical meaning is that energy from frequencies w, and w, is continuously fed into the frequency range w,, + @,. We will study the problem of whether energy from surface waves in the frequency bands w,, and @, can be transferred into internal waves in the frequency band @,+@,. Obviously, if am and w, are swell frequencies, a transfer to the frequency band @, — wy iS pos- sible only because internal waves have a cutoff at the Vaisala frequency, which iS considerably lower than the swell frequencies. Additionally, the frequencies ®m and w, must be very close together in order to feed energy into a frequency band corresponding to periods 2 to 20 minutes. EQUATIONS AND BOUNDARY CONDITIONS The hydrodynamic equations of a nonviscous, incompressible, stably stratified fluid in a Cartesian frame of reference with z pointing downward may be written as i pene p ai + pueVu = —Vp — pV (9) 4 HOV =O (10) at + U p Vou = @ (11) The symbols have the following meaning: w= (u,v, w) is the fluid velocity, p the pressure, ¢ the gravitational potential, and p the density. The effect of the earth’s rotation is neglected. The boundary conditions are OG as = W = -(= + Tove) at the sea surface, z = —C (x,t) (12) t p= const at the sea surface, z = —C(x, t) (13) w = (0 at the bottom, z = H = const (14) In order to apply the boundary conditions (12) and (13) at the undisturbed sea surface z = 0, we expand them about z = 0. The Taylor series of (12) is Om 2 ern ac le uC tu ) iL acy Hee b pa se hoy Ah SS poe lo A = LE 7) Sa 5 om ay NG Cae te amo VGcjat z=0 (15) equation (13) yields dp 2 a2 p— Ss SEE RE anette (16) ~ @z 2D @z THE PERTURBATION EQUATIONS Suppose the variables in (9) to (11) can be written as perturbation series ars ; ar Ras oo CMe ON ee: (17) with p‘) — p (z) and pod = p(z), then, entering the equations (9) to (11) and (15) to (16), we arrive at the following equations, omitting terms of higher than second order. Hal Oth Order Equations dp - Es (18) dz oe with p = const at z = 0. (19) Ist Order Equations _ ae 8 vp')) +p) yd= 0 (20) Eyal) dG MOL (21) ot dz Veu'l) 0 (22) with 9c) TOUS ate —0 (23) ot dp oe) a) s =(Oatz=0 (24) dz We = Ovat 2 = (25) 2nd Order Equations _ Gh) Ep oa ee pews Ve) + p'2yd=—( \)) ee (26) 42) = op of 2 =i) yo) (27) at dz V:u2 =0 (28) with Qe? aw? BP 22 2 CO SVM a z= 0 (29) t Oz 7 dp ap\)) Gow d2p (DQ (SA) = = (OW sae —= fs 2 = 0 (30) x ; B : dz DQ dz® Wo? =0 ai z= /4l (31) The Oth order equations characterize the mean stable conditions; the 1st order ones describe linear internal and surface waves in that fluid; and the 2nd order.equations give nonlinear effects of these waves, including energy transfers to other frequency bands. It is this set of equations which leads to energy flux from surface waves to internal waves. THE EQUATIONS FOR wx, z, t] We intend to eliminate wu, v, p, and p in the left-hand sides of the 1st and Ind order equations in order to get equations for w. The procedure is well- known.13 For the Ist order equations we get a2)? a4w) aew')) alee +e. wi), a fe ae = 0) (32) where - ae Mie) = = = a tz Combining the boundary conditions (23) and (24) together with the horizontal components of (20) and the continuity equation (22), we find LBV p,pw™ =) a 2=0 (33) and (25) remains the same w 0 atz=H (34) Internal and surface waves of the first order are governed by equation (32) with boundary conditions (33) and (34). The problem has been solved analytically 14 for many density distributions p(z) and may be handled numerically for any dis- tribution p(z). The result is a surface wave wo!) and a set of internal waves co ,\1’, each one being mutually independent and traveling with a phase w ell velocity given by the eigenvalues of the problem. The same procedure is used for deriving an equation for w‘2) from (26) to (28): The operator Vj,* applied to (26) together with (28) yields 9 42) ; ap) a 5 a cf Vp-p = jp a a 2 5 DW) (35) Zz Differentiating with respect to z and t we have = | 43,,(2) 44, (2) 2,(2) 2 (1) oo Oe = Te V2 oP a9 (0 ana) Bp Pee "aPag® aaa AB; Combining the vertical component of (26) with (27) gives (1) : = fa) Ow cy 2) = gu!) 7p) ee: tae abet gga) 2 a2ut2) a2p?) dp paronnet ff — = (Cte dzat dz at ot Pp which after applying Nine and substituting p‘2) in both equations above yields 2 (2 44. (2 23 (2 5 d7w'?) (2) Jw! ? dw! ! 1 Dl Gay i) Vir oO glw + =e FSH == Vp gu "Vp dt dt~0z ot ~dz D 2 a (36) d aw) iene) ang, OD) x ( =(1) 1) 7 off Guy) Sela 2 aU sgae CO) + ae 1) 5 a, pu Vw a Asan Vp Pp en + pu Vup, Combining the boundary conditions (29) and (23) together with (35) and eliminating ay (iL) in the inhomogeneous part by means of the vertical component of (26), we 43, (2) GQ) gt QD gy) 0° w i Cu eh Ou es ae fa) g ow \ ae + EN, wl? = = vn(é ~ 4 TVG) oye (@ ss “(1) ot 2dz B dt 5 Ob Js ped en a + V2 es FO .9,6M Jat z <0 (37) a Zz and (31) remains unaltered w'?) = at 2 = A (38) There are two driving forces responsible for second order internal waves: (i) The body force f; in the fluid due to the nonlinear terms in the equations of motion and the incompressibility condition. It is given by the inhomogeneous part of (36) <= 1 alls d aw'!? (x,z,t) = — — V,24 eu Vo + Af + pur fy 2 z h \é ? ain at p 1 @ SI | esa om += g Wes (© seal + pu). Vit, p dzot dt (ii) The surface force fo due to the deviations of the L.Viy6 )) (39) sea surface from its mean level, given by the inhomogeneous part of (37) ot te d OO) Fry Quine dis fy Gz = 0,1) = oo eel +O). VR) te 7 oat eS _) ae Sa eS Pp oh A1)2 5 (1) gl i ow Js a ° | v2 =—- BP-7,6) (40) Comparing these equations with the simple example given at the beginning, the problem (9) to (14) corresponds to equation (5), and (32) to (34) corresponds to (6), and (36) to (38) corresponds to (8). THE PRIMARY WAVE FIELD AND THE FORCES f, AND f, Suppose that the primary waves are progressive waves traveling in a direc- tion given by the wave number vector ? = (x, 7). We will use the notation k =|hl-yk2 + 72. A set of solutions of (32) to (34) is U,)) (%2,t) = S A Ea sin (k.-x —w,t) (41) } ap) ar ae y) hy a a n he. dz n Tl e= il 22 > w (Xz, t) = s AW cos (Rx = wb) (42) p= il po?) (2,0) = — » SEV itn cee — a) (43) ‘e Op io= il 15 > e2 @ dW > => Diet) =p » fh, =e 2 atin (gk ef) (44) k dz p= il CONG) = » A,—= —"| sin (h_-x— o,) (45) g R, Oe || 9 Me n=1 eo where W_(z) is governed by d2W dW k 2 2 ] aD a TEs + (gC — w,,*) ae V0 (46) with dW, ek? =z W=0 at z=0 (47) dz O, and WK =O a Zee (48) and represents either surface or internal waves. From (41) to (45) we can derive the driving functions f, and fg according to (39) and (40). This includes a great deal of algebra. The final result is ee) ee) fy (x, z,t) =— 5 » |e « Beets (z) sin (Ro R)X AOm +o,)t| + a = (z) sin | eB) — Cody ~op)t| | (49) f (60,1) = ay Slee (0) sin|(k_ +h,)-* —(o,,+0 an m= n= £C> ManlG =P Je —@, = ont (50) where Fie (z) and Ge (0) are given by Bobb. dW dW, + ieee) = [= as [er 20) ((@)_ 2@),, mee = |e tw, (Om, a0 i) | om c 1 dp (oto) a). 6 6 aoe ote | WW hee bal oO, p digas Om aWi. FR awe. dW d2W By of G2 @)= | <2 py 2 22 ek ki oll: a (om * o,) lane m m y) i Om dz Re dz dz d R / ) el —@ ap i I @ i dW aN On dW ; & Te 2 i W TU 2 I im ip (Om a Wy) @ DB ip - dz apy ® dz m m gk. _->_l js a Do 1 diy Wy |e > 2 & ey gel ee | eT 52 ar 5 OO, * Om k 2 an ie i ip a) Res *m fs n Ana THE SECONDARY WAVE FIELD IN THE RESONANCE CASE The secondary wave field is given by (36) to (38), which with (49) and (50) read a2 w! iD a) 462) a3w') Wie eee iain elw! ot eS MO ED |’ ——— = ae d Pe ae (z) sin at dz“odt * ot Az =(@- = ot | (53) > => b +b (k tR,,) RH 2 1 is ae +g V 2 wi?) = 5 » Ges (QO) sin [é,, tr By Nose == (Om = + o,)t| (54) n m n at z=0 2)_Qatz=H (55) where either the plus or minus sign holds. If eigenvalues of (53) exist, which fulfill the condition > => k m + al v= zi (56) (o, as @,) if 18 the solution of (53) to (55) is + oat @, 3 @.)AS We@) om [@, SS \(2) m iO Dupre BE do fh ose — fe t u y || ra are cos| (k,+ h,)-x— (om ta,) mnr 4b W(z) sin [@,, + hk )-x— (oa, to,)t | (57) + : : P where F’- (z) has been expanded in an eigenfunction series according to R= (2) fee his ae) = » Ane We (z) (58) r and the functions W=(z), we (z), and W= (z) are governed by the following equa- tions 2 wr it 21h eee ali d We ap awe x lero, + On) Rm tk, Wt = 0 dz2 dz @,, 20)" B + Bo 2 dW, & [Pmt R| Woe hates 0 (59) dz (@, Ba JY W-=Oatz=H wwe 7+ 2 5 2 Ce ce [et (on # on?) [Fn #2, | es dz” dz (oa, + w 2 i nat + = Gi ~ (@. 3 a) GE \ r ~~ - Seg at z= 0 (60) dz 4a, t @,) i 4g |k,, + F, Tr iF Oy ° > > > > ay, = aie [eP — fon # o,)?) |2,, +, | 2 ce 2 |h th, |eT as dz? Ge (Om toe i (@ 2a.) i + hy ay (2 a a ae g|k,, +h,,| uses eres ls NOY (61) dz (Gn Beyt Ol ee ae The factor (a, ta») Antar Vide IP, Len + +2) in (57) is dimensionless; W (z) and we (z) are velocities and W*(z) is an acceleration. Krom (57) to (61) we Ronclude that the primary waves create new waves of wave number Roy =p and eCeNey @m + my With amplitudes depending on the driving forces Fz, (z) and Gx, (0). These new waves grow linearly with time if the resonance condition (56) is met. The amplitude factor Az, is given according to (58) by H Si).— ADO 8 (a) = f i lp F* (2) W*(z) dz Sy en eS ee eee (62) mnr H D) fle — (ep, £0,)?|pWe(2) dz 0 If more than one internal mode is to be considered, G,=,(0) has to be expanded into eigenfunctions, too, jand evaluated at z = 0. Formulas similar to (58) and (62) will hold then for @= (0). mn FY (z), G*,(0) AND A*_ IN CASE OF AN moar EXPONENTIALLY STRATIFIED SEA In order to evaluate solution (57) of the secondary resonance waves, we have to know the eigenfunctions of W(z), which are given by (59). They depend on the mean stratification p(z) and determine the amplitude factor A,,7,,, given by (62). Equations (59) can be solved numerically for any stable stratified fluid, and it is known that the eigenvalues may vary considerably due to density changes. However, solving (59) for an exponentially stratified sea, where I‘p= const, may be sufficient at the present time where only the magnitude of the effects is of interest. Pes Using (2) = poe 0* as density distribution, W,, 1S given as Solution of (46) to (48) SUG DZ W, (2) = A e sin a,(z — H) (63) where 19 and similar equations hold for W_(z). Entering equation (51) and (52), we arrive at > ni) a —I0z [Pm a Ald 2 2 Tae Brn = AWA’ v | z h D) (sV5 to, (Om to,))(Fa,k —a_k Rk) On” m a hk. 2 Bi ie a ay n Nie fe ay) a, _=9 zon V5 +m s ad, 2 R oO 2; k ue n we a | lo? i) iw) Se SS | D. i=} = fe =) v * te acit 5, at ae @ (@—== @.,)) RR ‘i k RB eo (0) n m n mn m oy | 20 2 See le es SoS (6,20) 2 ( Op, @:, m n = .. Dy=- On, [ Dy 2 Om @ 1 > > n 0 , 6/2 a, (@— = ) cos (a, —@,,) (2-H) ion li, ph k y) ; elo= (Op + a _@n{Om* “| Yee (ee 7 m n k a On On On n Rk. S [ (Gieconetam es 2 He +a,)(2—-H) aT m 3 + 1 Giseen (0) =e B\_ ah ((@) @,) M Vo IN 20 PIsi JH . a, % (a, a, S (1+ a,) sin (@_ ©, Vp V5 S + —a M a (a,,—a,,)N Tor (1+ a,)P} sin (a,, + a,JH a iM Ir Tp AS nn i i M + (+, + ron, (1,2 tt) | cos (a, —ar) iT / r.2 el Dee \ + E M+(a,a, ; \x (4,2 Be) P|cos (a,, +a,,)H | (66) with = Vio ON ele, \\ By =O O. ese 2 = 4 >) iF >) > +k rs h ae apy 2 m n m n m Car > => —_> _> k_-k k_-k “k Nat—m™ [q+ mirriaetr in is i 2 ako Omen fe ine k 2 k 2 m n 2 gk 2p Dy) h 2 ©n m n = orTagT on 1 + WwW m) }, db RO (@,, = @.) > => kk P= lez“ k2 Ate follows then from (62) and (65) m -(10/2 i Aa A Ane (LO )H TED sin a,H cos a,H rit eae) a ‘ . In/: .T q , Litovaya/ La — +e a cos (a, == aH + (a7—a —a) sin (a,,—a, -—a) H 4 r Me r + (10/2)H @ |b. 0 (<2cos (a =@7_ i @ al + (@ 9) m n If ™ a =@ = @) Bim (C_=o. + @ u)| 2 n r m n T D) Vo 2 + (C=. + a,) ou fie On Riya) et R |- aC) 5 698 (G_, + a,—a)H + (a, + @,—a,) sin (a, + aay) a 2 a7 TY . om (a+ a, a,) T Ir \ ge 0 (To/2)H{*0 : R L ae tt e i cos (a+ a, + aJH + (a,,+ a, + a) sin (G+ @, + a JH) are = mee o a 2 ae (a,,+ at a,) Ip/: I 2, (6 0) al 0 a sin (a,—a,, + aH = (@_—@_ + a_) cos (a.—a-, + 2) 2 — + (a,—a,, + a,) It a + (0) (10/2)H(*0 «; % Ne: S 5a G4 C—O. +e ie sin (a, 4 a —a-)i = (a, +a_—a_) cos (a, + a,, —4,JH) eit ae BONES (a, + a. =.) Ir re Ir pe — (0 (10/2)H Oe: ap =a a —a, a7 +e (2 sin (a,—a,,—a,)H — (a_—a,,—a,) cos (a,—4,,~4,)H) - 5)) ome (C.=0,. 30)" Tr m n IP ye 1 Bs 2) (10/2)H/40 _- oli a a@+a-+ a +e (0 sin (a.+ a, + a JH — (a, +O 3s a,,) cos (a_+a_ + a,JH Re (CR GES TOM) T m n (67) with kk |2 ms + Q = ae k “5 540 tow,(o £0,!|[e0 Ra? = anFe®s| ) ~“m — be) 9 ) >_ y 2 = “hk & Tr k-k 1S nr ne It on 2 (@, + @,) Oe m aod = === z (en0, eer ) 2 [eo De Oe ae R 4 k tre Ss 5 9 7; z [ary t0,(@n* ooy)| [Fak ate Anh nh SO” m Gena 2 Oe ana 2 aa Sah 3 abt (@,.. = @)_) Bend yen Foy Fats (a,,a, 0") 2 era ee tO vo umon kare a st IF ale Pema gf) to, (o,,+ Oran gl’ _ @n!@m to,) Me is pee 20, ®,, Om men + Rie kp Qn 1 | + (a —a") 1+ _ (@ = o,,)|F h (G_ = @_) + @., toca = or m oe oN Te Naa a aR. lle. Bey) fe = [5 Te\7 x1+— = 0 at ay 0 m n m On On Dm rae es Me te (a a + Die + ob ah 2 (a + a_)+ ea i) (68) bo bo THE SECONDARY RESONANCE WAVE FIELD DUE TO SWELL IN CASE OF AN EXPONENTIALLY STRATIFIED SEA With reference to observations described in the following section, we eval- uated the resonance part of solution (52), using (59), (60), (66), (67), and (68). The results are based on the following numerical values: The swell frequencies of the two primary waves are @1 = 3.808 - 10-1 see-l and @» = 3.900 - 10°! sec -1. The corresponding wave numbers for a water depth of H= 18 m are k, = 3.00 - 10-4 em™!, ky = 3.08 - 10-4 cm-!. The density strati- fication is given by Ig = 5.811 - 10-7 cm-!. The corresponding periods and wave- lengths of the primary waves are 7; = 16.50 seconds, 79 = 16.11 seconds, )j = 209.4 m, A» = 204.0 m. Furthermore, for surface waves we have a, = 1|k,|, An = 0 . The product between the amplitudes A, and A,, of the vertical com- ponents of the velocity is supposed to be A; A» = 400 cm? sec-2. The corresponding amplitude product of the two swell waves is € ¢5 = 889.76 em2, which may be based on ¢, ~ 50 cm and G5 = 17.8 em. Because of these swell frequencies we expect an internal wave of frequency @9 — w1 = 0.0092 sec"! or period 75_; = 11.38 minutes. The result is shown in figure 2, which contains the amplitude of this internal wave after an interaction Rm OS te aanlageaaae lan als paula al hayes 102 CM 10 0° 30° 60° 90° 120° 150° 180° Figure 2. Amplitude (cm) of internal waves due to modulated swell after an interaction time of 103 seconds (16.6 minutes) as a function of the angle between the two primary waves. time of 103 sec (~ 16.7 minutes). The amplitude is given in a logarithmic scale as a function of the angle @ between the two primary swell waves. Amplitudes of about 13 centimeters are obtained if the primary waves are traveling in direc- tions containing an angle of about 60 degrees. The amplitude increases to more than 3 meters for angles of more than 150 degrees, and is only a few centimeters for angles less than 30 degrees. This seems to support the results obtained by K. E. Kenyon!! and K. Hasselmann!2 that a narrow directional spectrum of surface waves does not lead to significant internal waves. On the other hand, the figure demonstrates that surface swell which can be described by two surface waves traveling in different directions may create internal waves very rapidly. A swell of that type would be interpreted according to equations (1) to (4) as amplitude-modulated and traveling in a direction given by k,, because the ampli- tude A, is much larger than Ay and therefore would govern the swell. MEASUREMENTS ON SWELL AND INTERNAL WAVES In order to test the theory, simultaneous measurements on surface swell and internal waves were made in October 1966 at the NEL Tower, running from3 October, 1130 to 6 October, 1215. Two wave-height sensors were used to derive the directional spectrum of the swell — one fixed at the NEL Tower, the second at a distance of 39.3 meters toward the west. The voltage output was recorded on punch tape. The temperature fluctuations were measured by means of two vertical thermistor arrays. The spacing between the thermistors was 75 centi- meters, reaching from the bottom up to 2.75 meters below mean sea surface. The two arrays were located in positions along a line from southwest to northeast. The distance between the arrays was 284.4 meters. The temperature fluctuations were recorded on an analog recorder and on punch tape. Simultaneous records of swell and temperature fluctuations were obtained. The complete analysis of these data will be published later. Preliminary results support the theory given above. Figure 3 shows the power spectra of the two wave-height sensors for data from 4 October, 1230-1330. The voltage output of the two sensors has been used directly for these calculations. This output is different for both, and this is the reason for spectrum 2 showing higher intensities than spectrum 1. Other- wise, both spectra coincide quite well. The phase difference changes linearly with frequency as is to be expected. But there occurs a remarkable hole at frequencies of about 0.061 cycle per second, corresponding tow =0.38 sec7! or 7 =~ 16.4 sec. The energy decreases in both spectra from about 12 to 1.2 (arbitrary units), and the phase differences change from 35° to —70°, which indi- cates that the swell in this spectral range consists of two waves, one with w= 3.910! sec"! traveling in direction 30° and a second one with w, = 3.808x 10-! see-! traveling in a direction of about 270°. The angle between both is about 120°. From figure 2 it follows that within a quarter of an hour these two swell waves would create an internal wave of 7 = 11.4 minutes with an amplitude of 1.5 meters The swell in the vicinity of the NEL Tower therefore seems to be modulated as indicated by figure 3. If the stratification is favorable, that is, if |k,, — Rk, |2. (om — @y)2 iS an eigenvalue, a very intensive energy transfer from the swell frequencies «,, and,,, to internal waves of the first mode occurs, producing internal waves of amplitudes of several meters. These internal waves have the same characteristics as the swell modulation. Their period coincides with that of the modulation, and the wavelength is the same. Additionally, they travel in the same direction as the modulation. The agreement between internal wave periods and the periods of the swell modulations was tested. Data from the wave-height sensors for the time from 4 October 1966, 1900 to 5 October 1966, 0145, were used to determine the periods of the modulations. The internal wave periods were taken from the analog records for the time 3 October, 1500 to 5 October, 0300. Histograms of both SECONDS Ie Ie 14 12 10 8 ENERGY 0.06 0.08 0.10 0.12 0.14 €/SEC Figure 3. Swell spectra of wave height sensors | and 2 at positions near the NEL Tower, and phase difference between both records (4 October 1966, 1230-1330; sampling rate, 1 second); the arrow indicates the 95-percent confidence limit. periods are shown in figure 4. They show the number of occurrences of periods between 2 and 22 minutes in percent. The number of waves contained in each histogram is about 120. The figure is in good agreement with the theory given above, both distributions are peaked in the period range of 4-to-10 minutes and decrease toward periods larger than 20 minutes. Final conclusions, however, are possible only after the spectral analysis of the entire data. A better agreement between these histograms cannot be expected because the internal waves are strongly dependent on the changing local mean stratification,® whereas the swell is governed by quite different sources. 30+ SWELL i IL INTERNAL WAVES MODULAT ION kK ii OC) 20 \- al In jaa Lu oO I@ i= 0 5 10 15 20 0 MINUTES Figure 4. Histograms of periods of swell modulations (left) and of internal waves (right) at the NEL Tower. CONCLUSIONS 1. Resonant interaction between amplitude-modulated swell and internal waves may create internal waves of 2-to-20-minute periods. Internal waves with amplitudes of 1-to-3 meters can be produced within an interaction time of 15 minutes if the stratification of the water is such that —_ > 2y 2} lf, ip k| N@, — 9) is an eigenvalue of the internal wave equation (Py, ‘ss @ 1, and a» being wave numbers and frequencies of the modulated swell). 2. Internal waves due to amplitude-modulated swell have the same charac- teristics as the modulation. Specifically, they have the same wavelength and period and travel in the same direction as the amplitude modulation. 3. The resonance process is most efficient in the case of a modulation which travels in the same direction as the carrier wave (the main constituent) of a swell. The creation of internal waves of this type is strongly dependent on the stratification of the water. 4. Preliminary evaluations of temperature and swell records from the NEL Tower show good agreement between periods of swell modulation and internal wave periods. 5. It is likely that internal waves are created on the entire continental shelf off Southern California during times of favorable stratification. This de- pends on tide and wind. RECOMMENDATIONS 1. Verify, with additional measurements at the NEL Tower, the theory that internal waves are produced by swell. 2. Acquire, for complete analysis, long records of temperature fluctuations and surface waves in order to compute spectra with high resolution and statistical confidence. 3. Check, by measurements of surface waves at two positions, and slick observations, the result that internal waves travel in the direction of the swell modulation. REFERENCES 1. Navy Electronics Laboratory Report 937, Slicks and Temperature Structure in the Sea, by E. C. LaFond, 2 November 1959. Dp LaF ond, E.C., ‘‘Internal Waves,’’ p.731-751 in M.N. Hill, The Sea, v.1, Interscience, 1962. Bo Navy Electronics Laboratory Report 1342, The U.S. Navy Electronics Laboratory’s Oceanographic Research Tower; Its Development _and Utilization, by E.C. LaFond, 22 December 1965 4. Lee, 0.S., ‘‘Observations on Internal Waves in Shallow Water,’’ Limnology and Oceanography, v.6, p.312-321, July 1961 Seo t 5. Cairns, J.L. and Lafond, E.C., ‘‘Periodic Motions of the Seasonal Thermo- cline Along the Southern California Coast,”’ Journal of Geophysical Research, v.71, p.3903-3915, 15 August 1966 6. Ufford, C.W., ‘‘Internal Waves Measured at Three Stations,’’ American Geophysical Union. Transactions, v.28, p.87-95, February 1947 Ue Ewing, G., ‘‘Slicks, Surface Films and Internal Waves,’’ Journal of Marine Research, v.9, p.161-187, 1950 28 Cox, G.S., ‘Internal Waves, Part 2,’ p.752-763 in M.N. Hill, The Sea, v.1. Interscience, 1962 ae cd Ball, F.K., ‘‘Energy Transfer Between External and Internal Gravity Waves,”’ Journal of Fluid Mechanics, v.19, p.465-478, July 1964 Thorpe, S.A., ‘‘On Wave Interactions in a Stratified Fluid,’’ Journal of Fluid Mechanics, v.24, p.737-751, April 1966 Kenyon, K.E., Wave-Wave Scattering for Gravity Waves and Rossby Waves, (Ph.D. Thesis, University of California at San Diego), 1966 Hasselmann, K., ‘‘Feynman Diagrams and Interaction Rules of Wave-Wave Seattering Processes,’’ Reviews of Geophysics, v.4, p.1-32, February 1966 nar ce ea Krauss, W., Interne Wellen, v.2; Methoden und Ergebnisse der Theoretischen Ozeanographie, Gebrtider Borntraeger, 1966 UNCLASSIFIED Security Classification DOCUMENT CONTROL DATA-R&D (Security classification of title, body of abstract and indexing annotation must be entered when the overall report is classified) 1. ORIGINATING ACTIVITY (Corporate author) 2a. REPORT SECURITY CLASSIFICATION Navy Electronics Laboratory UNCLASSIFIED San Diego, California 92152 3. REPORT TITLE INTERACTION BETWEEN SURFACE AND INTERNAL WAVES IN SHALLOW WATER 4. DESCRIPTIVE NOTES (Type of report and inclusive dates) Research and Development Report, August 1966 - December 1966 5. AUTHOR(S) (First name, middle initial, last name) W. Krauss 6. REPORT DATE » TOTAL NO. OF PAGES 7b. NO. OF REFS 30 January 1967 28 13 8a. CONTRACT OR GRANT NO 9a. ORIGINATOR'S REPORT NUMBER(S) b. PRosEcTNO.OR 104 03 O1 1432 Task 0594 (NEL ILA LAL) - OTHER REPORT NO(S) (Any other numbers that may be assigned this report) 10. DISTRIBUTION STATEMENT Distribution of this document is unlimited. 11, SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY Naval Ship Systems Command Department of the Navy 13. ABSTRACT The relationship of internal waves, with 2-to-20-minute periods, to surface waves was investigated. It was found that (1) the interaction between amplitude- modulated swell and internal waves may create internal waves of 2-to-20-minute periods; (2) internal waves due to amplitude-modulated swell have the same char- acteristics as the modulation; and (3) the resonance process is most efficient in the case of a modulation traveling in the same direction as the carrier wave of a swell. DD R47 Ss asc UNCLASSIFIED S/N 0101- 807-6801 Security Classification UNCLASSIFIED Security Classification KEY WORDS Internal Waves - Shallow Water Ocean Waves - Analysis DD 12"..1473 (sack) UNCLASSIFIED (PAGE 2) Security Classification t Ge G3ISISSVIONN S$! paed styy (TZT171 T4N) p60 ¥SEL ‘TO €0 VOT US "w ‘ssnesy | sisAjeuy - Sane ueaQ “Zz Ja}eM MO|JEUS - SAAR [PULAU] *T GAlSISSVIONN S! p4ed sIyy (TZT171 14N) y6S0 YSE1 ‘TO €0 VOT US “M ‘ssneiy *| SisKjeuy - SaAeN\ UeIIQ “2 JayeM MOJJEUS - SAARM [eUJa}U] “T "{]@MS & JO AAEM Jald4ed a4} Se UOIPAJIP awWes aU} U! 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(2) :Spolsad 3\NUIW-QZ-0}-2 JO SAAEM JEUJa}U! ayeasd AeW SaAeM |eU4A} -U| pue jams payejnpow-apny!|dwe Uaamjag UOl}Ie4Aa,U! 94} (T) Jeu} puNoj sem }| “payeBlsanul sem Sanem adejins 0} ‘Spoliad JINUIW-QZ-0}-2 Y}IM ‘SAaAeM jeUJA\U! JO dIYSUO!e|a4 ay) GaldISSWIONN "19 uer gg ‘'d gz ‘ssnedy) “MAG YaLVM MOTIVHS NI S3AWM IWNUSLNI ONY JOVINNS N3aML9d NOILOVYALNI Z2€p1 4oday ‘yyjeg ‘oaig ues ‘ qe] $9!U01}99)9 Aven "{J@MS B JO ARM Ja144ed aU} SE UO!}AIIP awWes au} U! HuljaAes} UOIE|NPOW e Jo ased a4} Ul JUA!DIJJa JSOW S| SSad04d adUeUOSAaL AU} (€) PUB ‘UO|NPowW a4} Se Sd1}Si4a}IeVeY awWes ayy aAey }|ams payejnpow-apnyiidwe 0} anp sanem jeusayu! (2) ‘spolsad a}NUIW-QZ-0}-2 JO SAAEM |BUJa}U! a}ea19 ACW SaAeM |eU1a} -UI PUB ||aMS payejnpowW-apnyi|dwe Uaamjaqg UO!}IeJA}U! AU} (T) y2U} PUNO} Sem }| ‘payeblsanul Sem Sanem adeJANS 0} ‘spoliad J\NUIW-0Z-0}-Z UYIM ‘SAAeM jeUJA}U! Jo dIySuo!eja4 ayy daldISSVIONA "19 uer og ‘*d gz ‘ssmedy “M AQ YaLWM MOTIVHS NI S3AVM IWNYSLNI ONY JOV4UNS NIIMLIA NOILOVYILNI ZEp1 j4oday ‘yyeg ‘oBaiq ues ‘ qe] $2|U01}99)9 Aven CHIEF OF NAVAL MATERIAL MAT 0331 COMMANDER, NAVAL SHIP SYSTEMS COMMAND SHIPS 1610 SHIPS 1631 SHIPS 2021 SHIPS 204113 COMMANDER, NAVAL AIR SYSTEMS COMMAND AIR 5330 AIR 5401 AIR 604 COMMANDER, ORD 03C ORD 0322 ORD 9132 COMMANDER, COMMAND FAC 42310 COMMANDER, NAVAL ELECTRONIC SYSTEMS COMMAND TECHNICAL LIBRARY COMMANDER, NAVAL SHIP ENGINEERING CENTER CODE 6120 CODE 61798 CODE 6179C03 CODE 6360 CHIEF OF NAVAL PERSONNEL PERS 11B CHIEF OF NAVAL OPERATIONS OP-03EG OP-311 OP-312F OP-322C op-07T OP-702C op-71 op-716 oPp-09B85 OP-922Y4C1 CHIEF OF NAVAL RESEARCH CODE 416 CODE 418 CODE 427 CODE 466 CODE 468 COMMANDER IN CHIEF US PACIFIC FLEET CODE 93 US ATLANTIC FLEET COMMANDER OPERATIONAL TEST AND EVALUATION FORCE KEY WEST TEST AND EVALUATION DETACHMENT DEPUTY COMMANDER OPERATIONAL TEST AND EVALUATION FORCE, PACIFIC COMMANDER SUBMARINE FORCE US PACIFIC FLEET CODE 21 US ATLANTIC FLEET COMMANDER ANTISUBMARINE WARFARE FORCE US PACIFIC FLEET COMMANDER FIRST FLEET COMMANDER SECOND FLEET COMMANDER TRAINING COMMAND US ATLANTIC FLEET OFFICE OF THE OCEANOGRAPHER OF THE NAVY COMMANDER OCEANOGRAPHIC SYSTEM PACIFIC COMMANDER SUBMARINE DEVELOPMENT GROUP TWO COMMANDER, DESTROYER DEVELOPMENT GROUP, PACIFIC COMMANDER FLEET AIR WINGS, ATLANTIC FLEET NAVAL AIR DEVELOPMENT CENTER LIBRARY NAVAL MISSILE CENTER TECHNICAL LIBRARY PACIFIC MISSILE RANGE CODE 3250 NAVAL ORDNANCE TEST STATION CHINA LAKE CODE 753 PASADENA ANNEX LIBRARY NAVAL WEAPONS LABORATORY KXL LIBRARY PEARL HARBOR NAVAL SHIPYARD CODE 246P PORTSMOUTH NAVAL SHIPYARD CODE 242L PUGET SOUND NAVAL SHIPYARD CODE 246 SAN FRANCISCO NAVAL SHIPYARD HUNTERS POINT DIVISION NAVAL RADIOLOGICAL DEFENSE LABORATORY (2) NAVAL ORDNANCE SYSTEMS COMMAND NAVAL FACILITIES ENGINEERING CODE 222A DAVID TAYLOR MODEL BASIN LIGRARY NAVY MINE DEFENSE LABORATORY CODE 716 NAVAL TRAINING DEVICE CENTER TECHNICAL LIBRARY INITIAL DISTRIBUTION LIST NAVY UNDERWATER SOUND LABORATORY BUREAU OF COMMERCIAL FISHERIES LIBRARY LA JOLLA, CALIF. 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DIVISION 221 DIVISION 730 NAVY UNDERWATER SOUND REFERENCE LABRATORY LIBRARY FLEET ASW SCHOOL TACTICAL LIBRARY FLEET SONAR SCHOOL NAVAL UNDERWATER WEAPONS RESEARCH AND DOCUMENT SECTION ARMY ELECTRONICS RESEARCH AND DEVELOPMENT LABORATORY ARMY ELECTRONICS COMMAND MANAGEMENT & ADMINISTRATIVE SERVICES DEPT AMSEL-RD=MAT COASTAL ENGINEERING RESEARCH CENTER ARMY CORPS OF ENGINEERS AIR FORCE HEADQUARTERS DIRECTOR OF SCIENCE AND TECHNOLOGY ENGINEERING STATION AFRSTA LIBRARY AIR UNIVERSITY LIBRARY OFFICE OF NAVAL RESEARCH BRANCH OFFICE AUL3T-5028 PASADENA AIR FORCE EASTERN TEST RANGE CHIEF SCIENTIST AFMTC TECHNICAL LIBRARY - MU-135 BOSTON AIR PROVING GROUND CENTER CHICAGO PGBPS-12 SAN FRANCISCO HEADQUARTERS AIR WEATHER SERVICE LONDON AWSSS/SIPD NAVAL SHIP MISSILE SYSTEMS ENGINEERING WRIGHT-PATTERSON AIR FORCE BASE (1) STATION SYSTEMS ENGINEERING GROUP CRTD) CODE 903 SEPIR CHIEF OF NAVAL AIR TRAINING TRAINING RESEARCH DEPARTMENT NAVY WEATHER RESEARCH FACILITY NAVAL OCEANOGRAPHIC OFFICE UNIVERSITY OF MICHIGAN OFFICE OF RESEARCH ADMINISTRATION NORTH CAMPUS COOLEY ELECTRONICS LABORATORY CODE 1640 UNIVERSITY OF CALIFORNIA-SAN DIEGO SUPERVISOR OF SHIPBUILDING, US NAVY MARINE PHYSICAL LABORATORY GROTON, CONN. SCRIPPS INSTITUTION OF OCEANOGRAPHY CODE 249 LIBRARY NAVAL POSTGRADUATE SCHOOL DEPT. OF ENVIRONMENTAL SCIENCES LIBRARY FLEET NUMERICAL WEATHER FACILITY NAVAL APPLIED SCIENCE LABORATORY CODE 9200 CODE 9832 NAVAL ACADEMY ASSISTANT SECRETARY OF THE NAVY CRESEARCH AND DEVELOPMENT) NAVAL SECURITY GROUP UNIVERSITY OF MIAMI THE MARINE LABORATORY LIBRARY MICHIGAN STATE UNIVERSITY LIBRARY-DOCUMENTS DEPARTMENT COLUMBIA UNIVERSITY LAMONT GEOLOGICAL OBSERVATORY DARTMOUTH COLLEGE RADIOPHYSICS LABORATORY CALIFORNIA INSTITUTE OF TECHNOLOGY JET PROPULSION LABORATORY HARVARD COLLEGE OBSERVATORY G43 HARVARD UNIVERSITY AIR DEVELOPMENT SQUADRON ONE GORDON MCKAY LIBRARY VX=1 LYMAN LABORATORY OREGON STATE UNIVERSITY DEPARTMENT OF OCEANOGRAPHY UNIVERSITY OF WASHINGTON ra DEPARTMENT OF OCEANOGRAPHY FISHERIES-OCEANOGRAPHY LIBRARY APPLIED PHYSICS LABORATORY NEW YORK UNIVERSITY DEPARTMENT OF METEOROLOGY AND OCEANOGRAPHY UNIVERSITY OF ALASKA GEOPHYSICAL INSTITUTE UNIVERSITY OF RHODE ISLAND NARRAGANSETT MARINE LABORATORY LIBRARY YALE UNIVERSITY BINGHAM OCEANOGRAPHIC LABORATORY FLORIDA STATE UNIVERSITY OCEANOGRAPHIC INSTITUTE UNIVERSITY OF HAWAII HAWAII INSTITUTE OF GEOPHYSICS ELECTRICAL ENGINEERING DEPARTMENT A&M COLLEGE OF TEXAS DEPARTMENT OF OCEANOGRAPHY SUBMARINE FLOTILLA ONE, US PACIFIC FLEET DEFENSE DOCUMENTATION CENTER (20) DEPARTMENT OF DEFENSE RESEARCH AND ENGINEERING WEAPONS SYSTEMS EVALUATION GROUP DEFENSE ATOMIC SUPPORT AGENCY DOCUMENT LIBRARY SECTION NATIONAL OCEANOGRAPHIC DATA CENTER CODE 2400 COAST GUARD OCEANOGRAPHIC UNIT NATIONAL ACADEMY OF SCIENCES/ NATIONAL RESEARCH COUNCIL COMMITTEE ON UNDERSEA WARFARE COAST GUARD HEADQUARTERS OSR-2 ARCTIC RESEARCH LABORATORY WOODS HOLE OCEANOGRAPHIC INSTITUTION DOCUMENT LIBRARY LO-206 ENVIRONMENTAL SCIENCE SERVICE ADM. COAST AND GEODETIC SURVEY ROCKVILLE, MD. WASHINGTON SCIENCE CENTER 23 WASHINGTON, D. C. THE UNIVERSITY OF TEXAS US WEATHER BUREAU DEFENSE RESEARCH LABORATORY DIRECTOR, METEOROLOGICAL RESEARCH ELECTRICAL ENGINEERING RESEARCH LABORATORY PENNSYLVANIA STATE UNIVERSITY ORDNANCE RESEARCH LABORATORY STANFORD RESEARCH INSTITUTE NAVAL WARFARE RESEARCH CENTER MASSACHUSETTS INSTITUTE OF TECHNOLOGY ENGINEERING LIBRARY LINCOLN LABORATORY RADIO PHYSICS DIVISION FLORIDA ATLANTIC UNIVERSITY DEPARTMENT OF OCEAN ENGINEERING THE JOHNS HOPKINS UNIVERSITY APPLIED PHYSICS LABORATORY DOCUMENT LIBRARY INSTITUTE FOR DEFENSE ANALYSES DOCUMENT LIBRARY LIBRARY GEOLOGICAL SURVEY LIBRARY DENVER SECTION ESSA/INSTITUTE FOR TELECOMMUNI CATION SCIENCES AND AERONOMY BOULDER, COLO. FEDERAL COMMUNICATIONS COMMISSION RESEARCH DIVISION NATIONAL SEVERE STORMS LABORATORY CENTRAL INTELLIGENCY AGENCY OCR/DD-STANDARD DISTRIBUTION NATIONAL BUREAU OF STANDARDS BOULDER, COLO.