Introduction. The potential theory of sur-face waves has no provision for energy dissipa-tion. However, in a real fluid viscosity will cause energy dissipation. This subject has been stud-ied recently, both theoretically and experi-mentally, with respect to shallow water waves. The theoretical studies have included the solu-tion of the Navier-Stokes equasolu-tion for a linear-ized one-dimensional case of a shallow water wave [Biesel, 1949] and a nonlinear solution of a similar problem [Grosch, 1962]. The experi-mental work has included direct propagation studies [Spies, 1958; Grosch et al., 1960; Wat-son and Martin, 1962; Lukasik and Grosch,
1963] and- bottom shear stress measurements [Lukasik and Zepko, 1961; Eagleson, 1962].
Natural shallow water waves can occur as ocean swell. Therefore, the above studies have a direct relation to the problem of the energy dissipation of ocean waves as a result of their interaction with the bottom in nearshore re-gions. Biesel's linearized solution of the Navier-Stokes equation for a shallow water wave yields for the fluid velocity
u = u.[sin (kx cut)
e"-" sin (kx wt + my)] (1)
where
Lab.
v. Scheepsbouwkunde
Technische Hogeschool
5689
uo = cold
c =- wave phase velocity = (gd) a = wave amplitude
d = water depth k = wave number
= angular frequency v = fluid kinematic viscosity y = height above the bottom and
m = (co/201/2 (2)
The nondimensional velocity u/u,, is plotted as a function of kx cot in Figure 1. A physical picture involving two regions of space clearly emerges. According to potential theory, the ve-locity profile is u u0 and is independent of depth. However, owing to the no-slip viscous boundary condition, there must be a thin bound-ary layer region so that the sum of the poten-tial and the 'viscous' contributions vanishes at the bottom. Above the boundary layer region, in the potential flow, the surface elevation, the bottom pressure, and the particle velocity are in
phase, and viscosity is unimportant. In the
boundary layer region the viscous terms in the equation of motion dominate, and the in-phase relation between bottom pressure and particle velocity no longer exists.
JOURNAL OF GEOPHYSICAL RESEARCH VOL. 68, No. 20
Diftfr
15, 1963Pressure-Velocity Correlations in Ocean Swell
S. J. LIIKASIK AND C. E. GROSCH Davidson Laboratory
Stevens Institute of Technology, Hoboken, New Jersey
Abstract. Thirty-minute simultaneous records of bottom pressure and bottom fluid ve-locity have been made in 12-m water depths off Block Island, R I Small glass-enclosed bead thermistors were used in the velocity measurements. Four runs were made with the thermistor at various heights above the sand bottom, ranging from zero to 38 cm. These data have been digitized at 1-sec intervals and analyzed to extract power spectrums and cross spectrums. Both the pressure and the velocity measurements made at a height of 38 cm above the bottom show a typical double-peak spectrum having maximums at wave periods of 8 and 13 sec. In the peak regions of the spectrum the coherency of the tsVo variables is of the order of 70 to 80 per cent. Quantitatively, the pressure and the velocity power spectrums agree in magnitude to within 30 per cent over the major part of the frequency range. The pressure and the ve-locity are in phase to within 10° over the major part of the frequency range. Theoretical cal-culations suggest that at the bottom. there is a viscous boundary layer about 0.5 cm thick. Since the thermistor bead is approximately 012 cm in diameter, it is capable of resolving a boundary layer of this thickness. Measurements made with the therraistor on the bottom do in fact reveal the presence of such a boundary layer. The operation of the velocity sensor has been checked independently by means Of convolution -Calculations based on the measured bottom pressure.
5690 7 -kx-alt. 5° 6 5 2 15° 30° 45° 60° 900 0.2 0.4 U/uo
Fig. 1. Velocity profiles at various instants of time predicted by linearized theory [Biesel, 1949]. The quantity 1/m has the dimension of length, and from an examination of Figure 1 it is clear that some multiple of 1/m, say 3/m, can be
considered as a boundary layer thickness. The insertion of numerical values typical of ocean swell, e.g. a 10-sec wave period, into (2) yields
a boundary layer thickness of the order of
0.5 cm.
As a first step in linking the cited theory and laboratory experiments with oceanographic ob-servations, we would like to find (a) a region above the bottom where viscosity is unimpor-tant and pressure and particle velocity are re-lated by potential theory and (b) a region at the bottom where a viscous boundary layer
exists.
The work reported here consists of field meas-urements of bottom pressure and fluid velocity. Simultaneous measurements were made in shal-low water south of Block Island, R. I., at a depth of 12 m and at a position about 400 in offshore. The position of the measuring station relative to Block Island and the bottom topog-raphy is shown in Figure 2. The bottom was
0.8 hard sand, flat except for 2-cm-high wave
rip-ples, and free of rocks and vegetation. The measurements were made under conditions of a very calm sea. The tidal current, as further dis-cussion will show, was negligibly small in com-parison with the wave-induced particle velocities. The instrumentation consisted of a pressure -1.0 r f).118.4",.. ...*:rg12... -jc.--'''. ...C8. no -"'ii9,... *-9rky- ....J.7.-ii 24 hrd ..a.-rgxicr-k...-- 27 24 30 30 111 ,.. ---4_.,d 26 . 19 21- . '23 34.- - -
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..--- s... 57 57 58..."..e° 64 62s,
.. 86...
-.N 48
".. 2-- .65 671
47 54 39 50 rky .50 40 30 20-
10'
50 710-36' 35 34Fig. 2. South shore of Block Island, R. I., and adjacent the position of the measuring station are shown. Soundings are in nautical miles. 42 37 rky 5I
5155
39 57 rky 56 /66 57 - 1/2 64 59 waters. Bo a 41°-09' 34 40 5lr6
41° oe' .,..17.7. I.. ...a...,as/41-116. AiiLlitial..--"IIMMI. Ira,,...:4-' 4... :,:. 1
egiirar....pass -.,-.411'..7" '''-' 5....,...,12.--ir---.:14 _IL,.§,-.--ji.
u.IT., 6 ..nrife'' 26 26 24 rky 26 19 28 21 rky 33 34
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5691 least squares. This is the line shown in Figure 4. Over the range of velocities of interest, from 0 to 6 cm/sec, it was found that the thermistor sensitivity was temperature-dependent by about 5 per cent/°C. Calibrating the thermistor at the ambient ocean temperature of 18°C caused errors arising from this effect to be negligible.
The velocity sensor calibration shown in
Fig-)
tire 4 was performed in a constant velocity flow,. while the field measurements were performed in. a time-varying flow. Therefore, the high-fre-quency response of the sensor is important, and the results must be corrected for the finite re-sponse time of the instrument. The velocity
dicated by a thermistor subjected to an
in-stantaneous change in velocity is u 1
where T is the time constant of the thermist,or [Rasmussen, 1962]. For a sinusoidal variation Fig. 3. Thermistor velocity sensor. Scale is
indi-cated by inch rule.
gage buried flush with the ocean bottom and a small thermistor velocity sensor capable of trav-ersing vertically. The instruments were placed
on the bottom and adjusted by two Scuba
divers. Four 30-minute pressure-velocity records were made at various heights above the bottom. Data were recorded in analog form by means Of a Sanborn recorder on paper tape.
Equipment. The pressure sensor consisted of a 0- to 50-1b/in.' absolute pressure gage mounted in a water-tight box. An absolute calibration of the pressure gage was made in the field with a manometer. The signal resulting from the static head of 12 m of water was suppressed electri-cally, and the differential signal was written with a Sanborn recorder. The high-frequency response of the pressure-measuring system is limited by the response of the pen recorder.
The therraistor velocity sensor is shown in Figure 3. It consists of a small bead 0.04 cm in diameter and 0.12 cm high, encased in a thin glass tube. The thickness of the glass at the tip is about 0.03 cm. The therraistor was used in a constant-temperature bridge circuit analogous to that used in conventional hot-wire anemom-etry [Larsen, 1960; Hollenberg, 1962; Rasmus-sen, 1962]. The output signal from this bridge was recorded on the Sanborn recorder. This sensor was calibrated in the laboratory by tow-ing it in a small tank. A typical calibration of the pen deflection as a function of velocity is shown in Figure 4. To reduce data from the field measurements, the velocities were sepa-rated into two ranges corresponding to deflec-tions less than or greater than 0.5 cm. Separate quadratic expressions were fitted to the calibra-tion data in the two ranges by the method of
2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0
00
8 2.0 4.0 6.0 8.0 10.0 VELOCITY (CM/SEC)Fig. 4. Calibration curve of velocity sensor in terms of recorder deflection. Water temperature is I8°F.
5692 A I i i
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4 8 12 16 20 24 11 40 32 24 (1) 16(0EG)Fig. 5. Amplification factor and phase shift of thermistor velocity sensor as a function of the frequency parameter h.
in velocity at a point in space given by u = U sin wt, the velocity indicated by a thermistor
is u = U A(co)
sin [cot (co)], whereA(w) = (1 ± oirYila and cA(co) = .arctan
cuT. For the thermistor used in the measure-ments reported here the time constant T is de-termined essentially by the thermal conductiv-ity and the thickness of the glass coating. The thermistor material is at a roughly uniform temperature because its thermal conductivity is greater than that of the glass and the
constant-temperature circuit responds almost instantane-ously. We estimate that the time constant is 0.3 sec. For convenience in comparing the thermistor frequency response with the spectral estimates given later, A and ch are shown in Figure 5 for T = 0.3 sec plotted in terms of a parameter h that is related to co by h = 50 to/71-.
After being calibrated, the velocity sensor was tested by placing it in a wave channel in the laboratory. A train of shallow water waves was generated, and the surface elevation as well
as the signal from the velocity sensor was
recorded. The velocity of the fluid was calcu-lated from the curve shown in Figure 4. This was compared with the fluid velocity calculated from potential theory and the measured wave elevation and period. The two were found to agree to within 3 per cent.
One further characteristic of the thermistor that was clearly displayed in this laboratory ex-periment is that the thermistor acts as a recti-fying device. That is, it is insensitive to the fore-aft sense of the velocity. This is illustrated in Figure 6. The upper trace shows the surface elevation as a function of time for a shallow water wave. The lower trace shows the output of a thermistor velocity sensor. Because of the
Fig. 6. Response of velocity sensor to a regular wave train. The top trace shows the surface elevation and the bottom trace shows the thermistor output. One-second timing marks are at the bottom. The ordinates are in arbitrary units.
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rectifying nature of the therrnistor, the maxi-mums and minimaxi-mums shown in the surface ele-vation are not found. The negative peaks are now positive, resulting in a series of maximums at twice the frequency. The zeros of the sur-face elevation correspond, for a shallow water wave, to zeros in the velocity. These appear as cusps in the velocity record. One-second timing marks are shown at the bottom.
-Analysis of data. The four 30-minute runs were digitized at 1-sec intervals and punched
directly onto cards. From the absolute calibra-tion of the pressure gage in the field, the actual pressure amplitudes are known. From the cali-bration of the velocity sensor made in the
lab-oratory, absolute velocities can be obtained from the record. However, in view of the recti-fying property of the velocity sensor, it is neces-sary to examine the cusp structure of the ocean velocity data in order to restore the fore-aft sense of the velocity. Alternate peaks between axis crossing must be reversed in sign and the negative signs punched onto the raw data cards. Although this judgment adds a certain degree of subjectivity to the data, we think that the result is the actual fluid velocity in the ocean at the time of the measurement.
It is important to note that we are treating a vector velocity field by means of a magnitude and a sign, the magnitude being derived from the thermistor record and the sign supplied from inspecting the cusp structure of the thermistor record. This can be done only for a
one-dimen-sional velocity fieldthat for a long-crested
shallow water wave, for example. The presence of another velocity component noncollinear with the wave particle velocity, such as a tidal cur-rent or a cross-sea, would greatly complicate the analysis. It would also require the use of a directional velocity sensor.
Because of the distribution of frequencies present in the random swell, a simple single-fre-quency model of the processes taking place will be inadequate. Instead, a statistical analysis is required in order to extract numbers that can be compared with simple linearized sinusoidal wave solutions such as that in (1). Therefore, the analysis has consisted of a power-spectrum and cross-spectrum analysis of the velocity and pressure data. The spectral estimates have been corrected for the thermistor frequency response using the results shown in Figure 5.
0 200 80 2 40 2000 1000 50 16.7 10.0 7.2 PRESSURE 12 16 20 24 (10c) go) 25 12.5 8.3 6.3 5.0 4.2 CO-SPECTRUM 300 120 60 0 UADRATURE spEcrks;
Fig 7. Smoothed pressure and velocity power spectrums and cross spectrums. the velocity sen-sor was 38 cm above the ocean bottom.
When the thermistor is at a height of 38 cm above the ocean bottom, it is expected, in view of the numerical result based on (1) and (2), to be in the potential flow region. The potential pressure is transmitted through the boundary
layer, and so it can be measured by a gage
mounted, flush with the bottom. In this region the pressure p and the particle velocity u for a pure progressive wave are in phase and are related as
p/u = pX/r
(3)where p is the fluid density, X is the wavelength, and is the period. Figure 7 shows smoothed pressure and velocity power spectiums and cross spectrums for a velocity sensor at a height of 38 cm above the bottom. The horizontal axis is a linear frequency scale in terms of a quantity
h which is related to the period by h = (2n
At)/T, where n = number of lags (50, here) and At = sampling interval (1 sec). A wave period scale is also shown, for convenience. We see from both the pressure and the velocity record a typical double-peaked ocean wave spectrum having dominant periods of about 8 and 13 sec. The pressure spectrum shows a very-low-frequency spike that is believed to be attributable to a long-period drift. This drift, which may be either a tidal or a temperature effect, is clearly observable on the pressure rec-ord. The velocity spectrum is uniform at low
4.5 3.9 T (SEC) 5.6
5694 40 20 0 340 IS 320 wo 300 280 260 240 220
Fig. 8. Coherency and phase Of the data shown in Figure 7.
frequencies. It is believed that this is spurious and is due to low-frequency variations in the sensor zero level. The co-spectrum. is large and the quadrature spectrum is practically zero, in-dicating that the pressure and velocity are re-lated and are in phase. This is what one expects in the potential flow region. This information is presented in Figure 8 as the coherency and phase of the two variables. The peak coherency is about 75 per cent, and the phase angle is nearly zero over the region of the spectrum con-taining most of the energy. There is an anomaly
280 240 co 200 120 2 BO 40 100 LAGS SO LAGS 12 16 20 24 h (100 SEC')
Fig. 9. Comparison of smoothed pressure power spectrums for 50- and 100-lag analysis.
160 VELOCITY FROM PRESSURE
12 16 20 24
h (100 SEC-I)
Fig. 10. Comparison of the directly measured velocity power spectrum with the velocity trum calculated from the measured pressure spec-trum (equation 3). the velocity sensor was 38 cm above the ocean bottom.
in the phase angle between pressure and velocity for waves having periods of about 25 sec. The wavelength for such waves hi this depth of water is about 300 m; it is about the same as the distance from the measuring station to the shoreline. We believe that this anomaly is the
result of the reflection of the swell by the
island, since in a standing wave the pressure and velocity differ in phase by 900.
The adequacy of the spectral analysis can be judged from Figure 9, where the 50-lag pressure spectrum is compared with the result from a 100-lag analysis. The changes are completely explained by the effects of the change in resolu-tion and the increased sampling variability. There is a corresponding small increase in the peak coherency from 75 to 83 per cent (not shown) that would be expected according to the results of Pierson and Dalzell [1960].
The pressure and particle velocity are related quantitatively in Figure 10. This figure com-pares the directly measured velocity spectrum with the velocity spectrum obtained from the pressure spectrums by means of (3). For each frequency in the observed swell spectrum the corresponding wavelength for a water depth equal to that at the measuring station was cal-culated.. For the shorter wave periods there is a small but significant departure from shallow water conditions. Thus, the velocity spectrum calculated from the measured pressure spectrum rests on the assumption that the ocean bottom is at least locally flat; that is, the change in water
depth should be small over linear distances of the order of a wavelength. Nevertheless, we see that there is a very close agreement between the two different velocity spectrums. This agree-ment indicates that the magnitudes of the pres-sure and velocity are those expected on the basis of potential theory. For the spectral analy-sis that has been performed, the number of degrees of freedom is approximately 72. Accord-ing to Blackman and Tukey [1958], for this number of degrees of freedom there is a 90 per cent probability that a band ±-25 per cent wide enclosing these spectral estimates will contain the true spectral value. Over the region of the spectrum containing most of the energy, the two velocity spectral estimates are found to be within about 30 per cent of each other.
Similar results are shown for the measure-ments taken with a thermistor mounted in con-tact with the bottom. Owing to the size of the thermistor, it actually averaged the velocity
over a total height of about 0.12 cm. This
amounts to about one-third of the boundary layer thickness according to linear theory. Fig-ure 11 shows smoothed pressFig-ure and velocity power spectrums and cross spectrums. The pres-sure spectrum is the same as for the previous case, but now the amplitudes of the velocities
a 40 0 600 500 400 300 200 100-100 PRESSURE 12 16 20 24 h ((00 SEC-') I 1 I 1 I 1 I I I--I- I I CO-SPECTRUM ClUADRATURE SPECTRUM 300 240 In 180 a a 120 -.°) 60
Fig. 11. Smoothed pressure and velocity power spectrums and cross spectrums. The velocity sen-sor was in contact with the ocean bottom.
20 w 0 cc co 340 ° 320 300 40 CALCULATED PHASE 280
Fig. 12. Coherency and phase of the data shown in Figure 11. Shown also is the phase cal-culated from equation 6.
are very much smaller. This is, of course, what we expect in a boundary layer. Again the co-. spectrum is large, indicating that the two vari-ables are correlated, and the quadrature spec-trum is almost zero, indicating that the two variables are in phase. This information is
pre-sented again in Figure 12 in terms of the
coherency and the phase. The peak coherency is again about 75 per cent, but it occurs only for the higher-frequency peak of the spectrum. The coherency for the lower-frequency peak is now less. The phase angle is not as uniform as before, and the low-frequency anomaly in the phase, though still present, is slightly modified. The linearized solution of the Navier-Stokes equation (1) can be used to predict the velocity in the boundary layer. With a probe whose bead is in contact with the bottom and which extends to a height L in the boundary layer, the average velocity (u) to which this probe responds is
(u) = (1/L) fo' u(x, y,
t) dy. Substituting u(x, y, t) from (1), we obtain for the average velocity(u) = icon)) sin [kx we ± 0(w)]
( )
where
F(co) = [f2(co) g2(co)11/2 (5) 0(co) = arctan [g(co)/f(co)] (6) 5695
5696
TABLE 1. F(0)) and 0(0)) as Functions of the Frequency Parameter h, for L = 0.12 cm
1
Aco) = 1 -
2mL (e--112mL)(sin rnL - cos mL) 1 g(W) = 2mL (e-'z'/2m1)(sin mL ± cos rnL) With L equal to an effective thermistor height of 0.12 cm, we calculated F(w) and 0(w) as a function of h. They are given in Table 1.To calculate the spectral estimate of the ve-locity in the boundary layer using linearized viscous flow theory, the potential velocity u was first calculated from the pressure as indicated by (3). Then, from (4), the magnitude of the velocity in the boundary layer is
(u)! = F(cOuo (7)
Figure 13 shows the spectral estimate of the potential velocity calculated from the pressure and the spectral estimate of the velocity in the boundary layer from (7). Also shown is the measured velocity spectrum. It can be seen that the agreement between the measured velocity
BOUNDARY LAYER VELOCITY PREDICTION u 0.8 6.4 POTENTIAL THEORY VELOCITY PREDICTION DIRECT MEASUREMENT OF VELOCITY M M 20 24 h (100 SEC-I)
Fig. 13. Comparison of the directly measured
velocity power spectrum with the velocity spec-trum calculated from equation 7 on the basis of
linearized viscous flow theory. Also shown is the velocity spectrum calculated from potential theory. and the result based on (7) is excellent. How-ever, this is partly due to the use of an 'effec-tive thermistor height' of 0.12 cm instead of the actual height of 0.15 cm (a 0.12-cm bead with 0.03 cm of glass coating on the bottom).
The phase shift, 0(w), between the potential velocity and the velocity in the boundary layer was also calculated from the equations given
above. It is listed in Table 1 and plotted in Fig-ure 12. FigFig-ure 12 shows that the linearized the-ory predicts the correct dependence of phase on frequency but overestimates the magnitude of
the phase shift by about 20°.
With reference to the frequency dependence of the phase shift, it should be noted that the velocity measurements taken with the thermistor touching the bottom also give a measure of the bottom shear stress. The effective velocity that the thermistor responds to is the velocity at a
height from the bottom of L/2. For a wave
period of 10 sec (the peaks in the spectrum oc-cur at 8 and 13 sec), the quantity rri.L/2 is0.35.
From Figure 1 it can be seen that between
my = 0.35 and zero the velocity profiles are very nearly linear. Thus in this range of y, the
bottom shear stress To being given by To =
(aWay),.., the velocity is approximately u = (y/p)7,,, where p. is the viscosity. Therefore, to this order of approximation, the velocity at
y = L/2 and the bottom shear stress are
in phase. From (1) it can easily be shown that the potential velocity and hence the bottom pres-sure lags the bottom shear stress by 450. This phase shift decreases with height and is zero at the top of the boundary layer. The approach to 45° of the calculated phase shift in Figure 12 as h 0 is understandable, since m hv2 and7, sec F(w) 0(w), rad 3 3333 0.2270 0.6719 4 25.00 0.2572 0.6556 5 20.00 0.2829 0L6415 6 16.67 0L3054 0.6289 7 14.29 0.3254 0.6176 8 12.50 0.3435 0.6072 9 11.11 0.3601 0.5976 10 10.60 0.3753 0.5886 11 9.09 0.3895 05801 12 8.33 0.4027 0.6722 13 7.69 0_4151 0.5646 14 7.14 0.4267 0.5574. 15 6.67 0.4377 0.5506 16 6.25 0.4482 0.5440 17 5.88 0.4581 0.5377 18 5.56 0.4675 0.5317 19 5.26 0.4766 0.5259 20 5.00 0.4852 0.5202 21
476
0.4934 0.5148 22 4.54 0.5013 0.5096 23 4.35 0.5090 0.5045 24 4.17 0.5163 0:4996 25 4_00 0.5233. 0.4948 1.6 w L2200 cow 160 re w 120 t5 80 2 40 700 600 500 400 too 200 100 100 200 PRESSURE 4 8 12 16 20 24 h ((00 SEC') CO-SPECTRUM 300 OUADRATURE SPECTRUM 240 I., 180. 120 60
Fig. 14. Smoothed pressure and velocity power spectrums and cross spectrums. The velocity sen-sor was 2.50 cm above the ocean bottom. as h ---> 0, the boundary layer. thickness 1/m approaches infinity. In this limit the finite size of the thermistor is negligible, and the thermistor
indicates the true bottom shear stress. It
isclear, however, that at any nonzero frequency the finite size of the thermistor results in aver-aging phases that are less than 45°, thus yield-ing values of ((o) that are less than 45° and that decrease with increasing frequency.
Regarding the measured phase shift between the bottom pressure and the averaged boundary layer velocity, Figure 12 shows that, instead of leading, the velocity has been retarded and is more nearly in phase with the pressure. The calculation of the average velocity, given by (4), may be over-simplified but no definite rea-son can be given for the discrepancy.
In addition to the previously discussed meas-urements with the velocity sensor at 38 cm above the bottom and with the velocity sensor on the bottom, two additional runs were taken at heights of 125 and 2.50 cm above the bot-tom. Figure 14 shows the results of the spectral analysis for the 2.50-cm height. The velocity spectrum no longer displays double maximums like that shown at the bottom and in the poten-tial flow region. Also, there is a 'much larger quadrature spectrum than was observed in the
40 20 gn 0 La re 340 1,1 o 320 PHASE 300 280
Fig. 15. Coherency and phase of the data shown in Figure 14.
previous runs (see Figures 7 and 11). The dif-ference between this run and the two previous runs is shown most markedly, however, by Fig-ure 15, which presents the coherency and phase of the two variables as a function of frequency. Comparing Figure 15 with Figures 8 and 12, we see that the peak coherency is very small, of the order of 30 per cent or less, and the phase shows large fluctuations. These reiults are also evident, but to a lesser extent when the velocity sensor was at a height of 125 cm (Figures 16 and 17). Here the velocity spectrum resembles that at the bottom, but the peak coherency is
200 160 cc u, 120 2 4 ao 1000 800 600 400. 200 0 PRESSURE 12 16 20 24 h '(100 SEC') 300 240 0 120 2 60 200
Fig. 16. Smoothed pressure and velocity power spectrums and cross spectrums. The velocity sen-sor was 125 cm above the ocean bottom.
5698 Velocity Sensor Height, cm 40 20 m 0 cc 340 sz 320 300 280
Fig. 17. Coherency and phase of the data shown in Figure 16.
only 55 per cent and the phase again shows large fluctuations.
All four runs are compared in Table 2. In addition to the peak coherency, the ratio of the spectral estimate of the directly measured ve-locity to the veve-locity obtained from the pres-sure spectrum averaged over the frequency re-gion containing significant energy (6 < h < 20) is presented. In the potential flow region (height = 38 cm), the velocity measured is 94 per cent of the potential velocity and the maximum co-herency is 78 per cent. On the bottom (height = 0.06 cm), the measured velocity is only 38 per cent of the potential velocity and the maxi-mum coherency is still high (76 per cent) ; this indicates the presence of a boundary layer. At the two intermediate heights (125 and 2.50 cm) the average velocity lies between the two limits
TABLE 2. Comparison of Pressure-Velocity Correlation at Various Heights
/Measured Velocity)
Vel. from Pressure CoherencyMaximum
of 38 and 94 per cent, but the coherency be-tween pressure and velocity is very low.
We think that the results at the two inter-mediate heights can be explained by the pres-ence of the 2-cm-high sand fipples that were observed by the divers. The presence of these ripples, even though the bottom was smoothed locally by the divers, *ill contribute to an inter-ference and shielding of the velocity sensor, and extraneous flows introduced by the ripples, such as eddy shedding or turbulence, can contribute to the loss of coherency.
Discussion. In order to assess the subjectiv-ity introduced into the analysis when the lost sense information is manually restored to the velocity data, a section of the pressure record was selected and used to predict the fluid ve-locity by means of potential flow theory and conventional convolution techniques. That is, if
we rewrite (3) as u(t) = COO p(t), where
(1)((o) = 71/43A, then p(t) can be considered an input function and (13(w) a transfer function. The velocity u(t) is
u(t)
f
K(t r)p(r) dr (8)where the kernel function K(t 7-)
is the
Fourier transform of c13((0) [Solodovnikov,
1960]. The result for a typical section of the record is shown in Figure 18. It is seen that the treatment of the velocity data has yielded results that are close to what is expected on the basis of the measured pressure. Therefore, we believe that this method of handling the data, while partially subjective, has nevertheless not done violence to the data. The discrepancies between the two appear to be due to
instru-e 6 0 24 28 32 36 4 TIME (SEC)
Fig. 18. Pressure record and corresponding ve-locity record calculated on the basis of equation 8. Values of the directly measured velocity are
indicated by circles on the lower trace.
38 0.94 0.78
2..59 0.83 0.30
1.25 0.64 0.55
5699 mental effects rather than to the data handling
process. For example, the part of the record between 5 and 15 sec suggests that there has been a zero shift in the velocity sensor, since the total excursion on both the calculated and measured velocities are identical.
On the basis of this work, we can conclude that the amplitude and phase relations between pressure and velocity in ocean swell are cor-rectly described by the potential theory of shal-low water waves. Furthermore, the existence of a viscous boundary layer on the bottom has been directly demonstrated. This boundary layer is described qualitatively by a linearized laminar flow solution of the Navier-Stokes equation. The interpretation of the velocity measurements in the boundary layer is not without difficulty; nevertheless there appear to be quantitative dis-crepancies between the theory and observations. Any detailed calculation based on an assumed model of the flow condition near the ocean bot-tom must therefore be approached cautiously. It is instructive, however, to calculate an en-ergy loss rate due to viscous dissipation in the bottom boundary layer. Using the results of Lukasik and Grosch [1963], we find that the relative energy loss per unit time for a shallow water wave is
1 dE
=
E dt
If we insert numerical values typical of ocean swell in shallow water, a 10-sec period wave in a 12-m ocean depth, for example, we get a
value for dE/Edt of 5 x 10
see. This
en-ergy loss rate can also be expressed in terms of a wave amplitude decay with propagation tance. For the same wave parameters, the dis-tame necessary for the wave amplitude to de-crease by a factor of 1/e is 4450 wavelengths, or about 500 km. Although this distance is greater than the continental shelf width (and the ocean depth on the continental shelf is also
greater than 12 m), it is clear that at least
under some conditions the energy loss due to viscosity can be appreciable.
Acknowledgments. We wish to acknowledge the many helpful discussions we have had with Professor Willard J. Pierson, Jr., concerning the work reported here. The thermistor instrumenta-tion was developed by Dr. Karl Larsen; the ve-locity sensor was designed and calibrated by Mr. Joel Hollenberg. Mr. Robert Canary and Mr.
Al-bert Stover assisted in the collection of the data as Scuba divers. The spectral analysis and con-volution programs for the IBM 1620 computer were written by Mr. Richard Clapp. Data proc-essing on the 1620 was performed with the assist-ance of Mr. George Zepko.
These calculations were done at the Computer Center of Stevens Institute of Technology, which is supported in part by the National Science Foundation. The 100-lag spectral analysis was car-ried out on the New York University IBM 7090 through the courtesy of Professor Pierson and Mr. Emmanuel Mehr. Finally, the support of the Office of Naval Research is most gratefully
ac-knowledged.
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