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Interaction of waves and a shear flow Hughes, Blith Alvin 1960

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INTERACTION OF WAVES AND A SHEAR FLOW by BLYTH ALVIN HUGHES B.A. , University of Br i t ish Columbia, 1956 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in the Department of PHYSICS We accept this thesis as conforming to the required standard: THE UNIVERSITY OF BRITISH COLUMBIA June, I 9 6 0 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r - s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f The U n i v e r s i t y o f B r i t i s h C o l u m b i a , V a n c o u v e r 8, C a n a d a . D a t e ABSTRACT A series of experiments has been undertaken i n which three major properties of a surface tension-gravity wave system have been c r i t i c a l l y examined. The results of these experiments have been compared with existing theories. The three properties are: viscous decay i n the absence of mean flow, which has been compared to the theory given in Lamb (1932) #348; propagation velocities i n the presence and absence,of mean flow, compared to Lamb (1932) #267, and the change of wave energy on crossing a stable Couette shear flow, compared to two theories - one obtained from the Navier-Stokes equations including terms up to second order in wave slope; the other, following previous authors, assuming any direct interaction of the waves with the shear flow to be negligible. According to the theory obtained from the Navier-Stokes equations the divergence of the rate of transport of surface wave energy i s equal to the rate of change of wave energy due to the interaction of the mean flow and the wave system plus the rate of change of energy due to viscous decay. An optical system was used to measure the maximum wave slope, the wavenumbers and the shear velocit ies . A grid of point source lights was set up to reflect off a previously chosen part of the shear flow into a properly oriented camera. A series of pulses of waves, generated by an electromechanical transducer, were then sent across this region of the flow. i i i For each pulse of waves a photograph was taken of the o s c i l -lat ing images of the lights with the exposure time longer than one wave period. The resulting streaks on the fi lm are proportional to the maximum wave slopes at the positions from which the undeviated l ight ref lects . A series of paral lel straight white strings were photographed with the aid of a flash unit when waves crossed the flow. This was then used to determine lines of constant phase from which the wavelength and hence the wav.enumber was measured at various positions across the flow. The viscous decay and part of the propa-gation measurements were obtained i n this way but with no mean flow. Results indicate that an anomalous region of wave properties exists for wavenumbers near 2.7 cm. 1 . For a set of data in which the wavenumbers were always less than 1.8 c m . ~ \ i t was found that the viscous decay rate and the propagation laws agree with theory to within the experimental error, and the interaction measurements f i t the theory with the non-linear term included rather than the traditional theories. iv TABLE OF CONTENTS page INTRODUCTION 1 EXPERIMENTAL METHODS 2 1. Flow Production 2 2. Flow Measurement 3 ( i ) Primary Flow 3 ( i i ) Secondary Flow 4 3. Wave Production 5 4. Wave Measurement 9 (i) Amplitude 9 ( i i ) Wave Number 10 ( i i i ) Projection System 11 (iv) Propagation and Dissipation . . . . . . . . . . . . . 12 5. Experimental Parameters 13 THEORY 14 1. Interaction 14 2. Viscous Dissipation 21 3. Wave Kinematics •• 22 ( i ) Zero'th Order Approximation 22 ( i i ) F i r s t Order Approximation 24 4. Secondary Flow 27 5. Analysis of Amplitude Photographs 28 RESULTS 29 1. Absence of Flow 29 ( i ) Propagation • 29 ( i i ) Viscous Dissipation • 29 2. Presence of Flow • 32 (i) Propagation 33 ( i i ) Interaction • ••• 35 V TABLE OF CONTENTS (continued) page DISCUSSION 37 SUMMARY 40 APPENDIX 41 BIBLIOGRAPHY 43 v i LIST OF ILLUSTRATIONS 1. Experimental Apparatus 2. Experimental Apparatus 3. Experimental Apparatus k. Wavemaker 5. Wave Production Circuit 6. Amplitude and Current Photographs 7. Phase Photographs 8. Energy Propagation Geometry 9. Wave Kinematics 10. Viscous Dissipation 11. Velocity Profile 12. Measured and Predicted Wavenumber Data 13. Maximum Wave Slopes i n Presence of Flow lk. Velocity Profile 15. Measured and Predicted Wavenumber Data l6a. Maximum Wave Slopes in Presence of Flow 16b. Normalized Comparison of Theoretical Curves with Measured Wave Slopes 17. Lines of Constant Phase 18. Lines of Constant Phase 19. Amplitude Analysis ACKNOWLEDGMENT I should l i k e to thank Dr. R.W. Stewart for his expert supervision and for the many-ideas that were contributed by him. I should also l i k e to thank Mr. J.C. Archer for his technical assistance. 1 INTRODUCTION The work discussed herein i s a c r i t i c a l experimental investigation of three major properties of surface waves i n the regime where both surface tension and gravity are important. These properties are: viscous dissipation i n the absence of any mean flow, propagation laws i n the presence and absence of mean flow, and the non-linear interaction of waves with a horizontal mean shear. The f i r s t two of these are given a theoretical treatment i n Lamb (1932), the last one has been investigated theoretically by J . Drent (1959)• The analysis given by Drent i s for the case of plane, long-crested waves and recti l inear shear flow i n the absence of surface tension and viscosity. The results indicate that there are two major causes of amplitude change as the waves traverse the flow: one i s due to the refraction of the waves, causing a spreading or concentration of energy; and the other i s due to the radiation pressure, associated with the waves, interacting directly with the shear flow. For waves running directly into a converging flow these two effects are about equal; for waves crossing a lateral shear at an angle the former appears to be 2 to 3 times larger than the la t ter . Previous theories have consistently neglected the direct interaction effects. For example, see 2 Johnson (1947). This thesis i s a report of a laboratory experiment designed to measure the three properties l i s t e d above and a comparison of the results with the respective theoretical predictions. The wave energy measurements i n the presence of mean flow are compared with two theoretical models - one including the interaction effects, following Drent, and the other omitting the interaction effects following the previous theories. EXPERIMENTAL METHODS 1. FLOW PRODUCTION A stable Couette-type flow i s created between the outer wall of a circular tank s l ight ly greater than 2.5 meters i n diameter and an inner wall imposed by a raised centre portion 1.52 meters i n diameter, (see figures 1, 2 for two different views of the actual apparatus and figure 3 for a scale diagram). A hollow annular r ing, 2.44 meters i n dia-meter, floats freely, concentric with and about 2.5 cm. radial ly inside the outer wall of the circular tank (figures 1, 2, point A). A series of six drive jets are situated i n the outer wall of the tank and are connected by rubber hoses to the water mains (figures 1, 2, point B). These jets are oriented so that water passing through them impinges on the wall of the annular ring at an angle of 145°, thereby making i t rotate. In order to keep the annular ring rotating 3 about the centre of the tank, three foam-padded castors, spring-mounted, are situated on the wall of the tank and bear gently on the outside of the rotating ring i f i t i s not i n the centre of the system (figures 1, 2, point C) . It i s found that i f these castors are not used the ring gradually shifts i t s axis of rotation u n t i l i t hits the wall of the tank, creating a large disturbance on the surface of the water. It was necessary to construct a wooden form to f i t snugly around the annular ring i n order to hold i t to a circular shape (figures 1, 2, point D). With the wooden form, the annular ring i s circular to within 0.6 cm. i n i t s 2..kk meter diameter. 2. FLOW MEASUREMENT (i ) Primary flow Measurement of the horizontal shear flow profi les i s performed photographically during the experiment i n the following way: ten small circular pieces of white paper, 0.6 cm. i n diameter, are simultaneously dropped onto the surface of the water i n a l ine approximately at right angles to the flow i n the area under consideration. A photograph i s taken of these with the camera shutter open for .98 seconds (1 second on camera settings). The lengths of the resulting streaks on the fi lm are proportional to the velocities at the points where the pieces of paper dropped. This i s done ten times i n order to achieve a smooth profi le curve. An example i s shown i n figure 6. k Depth profi les were obtained with a midget propellor type current meter at three positions radially across the shear flow. However, since the waves that are used are only about 2.5 cm. i n wave length, i t appears that the current meter could not really get near enough to the surface to examine the important region. Visual observations were ob-tained comparing the flov; at a few centimeters depth and the flow right on the surface. A small plastic float about ^ cm. i n diameter was attached to a relat ively high drag body of about the same dimensions by a piece of thread a few centi-meters long. The rate of movement of this unit when placed i n the water was found to be not s ignificantly different from the rate of movement of a plastic float by i t s e l f at the same radius. Because there i s nothing to produce a surface stress except for the negligible effect of the a i r , there i s l i t t l e reason to believe that a depth-wise velocity gradient should exist . Any such gradient would be associated with secondary flow. ( i i ) Secondary flow The secondary flow generated i n the tank manifests i t s e l f mainly in changing the curvature of the radial profi le (see Theory). For a l l intents and purposes the value of the secondary flow i s negligible, i . e . the f l u i d particle trajectories are circular on an average. The strength of the secondary flow was estimated by observation of the paths of the floats described above. On an average they tended towards the outer wall of the system but the rate of progression i n 5 this sense was of the order of 1/200 or less of the average current. 3. WAVE PRODUCTION Because of the limited surface area of the tank i t i s necessary to use either a wave absorbing system on the walls or to use a pulsed wave t ra in . The former was attempted but soon abandoned i n favour of the la t ter . The energy absorbing device tried was a 'beach1 formed by a s tr ip of sheet metal with ends joined to form a truncated cone and attached to the inside of the annular ring at the water l e v e l . Unfortunately, the presence of the rotating beach i n the water produced a vert ical shear of the horizontal velocity because the sub-merged part of the beach acted as a boundary not i n accord with the flow pattern set up by the outer and inner walls of the tank. This vert ical shear zone produced more turbulence than could be tolerated, i t s effects showing up i n the form of local surface depressions and horizontal velocity f l u c -tuations and possibly a change i n the dissipation rate of the wave energy. The most serious drawback with the pulse system i s that only the f i r s t part of the pulse i s available from which to obtain data. The part of the shear flow that i s used extends from the edge of the inner wall to approximately 20 cm. from the outer wall , so that the reflected wave arrives at the edge of the region under examination i n % to 1 second. 6 Therefore a steady state wave pattern must be created i n less than }6 second after the pulse has been applied to the wavemaker. The device used to produce the waves i s as follows: (see figure k) a circular wooden disc with i t s edge bevelled from 15.2 cm. i n diameter to 11.5 cm. i n diameter i s firmly attached to the cone of a Jensen #B69V e l l i p t i c a l loudspeaker. This assembly i s situated on the top of the raised centre piece of the tank i n such a way that excursions of the cone are perpendicular to the surface of the water. The disc extends about 1 cm. under the surface. The loudspeaker i s e lec t r ica l ly connected to a low frequency osci l lator via an automatic switch and a matching transformer (see figure 5). The voltage being applied to the wavemaker i s continuously monitored by a single beam oscilloscope. The switching rate used i s about one per minute with an "on11 duration of about ten seconds. This gives ample time to take the necessary data and enables the waves to die out between pulses. Two of the prime d i f f i c u l t i e s of wave production by this method arise from lobal pattern and transient modulation characteristics. Asymmetry of the former arises because of the d i f f i c u l t y of properly orienting the wooden disc with respect to the water. It appears that the bottom of the disc should be accurately paral le l to the surface of the water and that the motion of the disc should be accurately perpendicular to i t . Transient modulation occurs because a 7 resonant system i s being driven by pulses of off-resonance frequency. Even though the wave amplitude on the scope i s constant to within 1 per cent, there s t i l l i s evidence of a transient modulation phenomenon of 2 to 5 per cent, existing i n the waves themselves. In order to reduce these modulation effects the wave amplitude measurements were taken at different parts of each pulse. An electronic counter was set up across the osci l lator to determine the time interval between the application of the pulse to the loudspeaker and the instant when the amplitude or phase data was taken. A neon bulb was set to flash every fourth cycle of the frequency used. By using one half of the flash interval as a basic time interval change, the time at which the data i s taken with respect to the front of the wave pulse i s varied by three or four steps over the available time, depending on the frequency used. One third or one quarter of the total data was obtained at each of these time intervals . The reflected waves begin interfering after a time interval of approximately one second after the i n i t i a l part of the pulse has passed the area i n question. Two methods were employed to measure or control the osci l lator frequency. One of the methods applies when the frequency desired i s some integral or half- integral fraction of the 60 cps. , such as 8 cps. , 8.57 cps. , 10 cps. , etc . , taking advantage of the fact that i n this geographical area the mains frequency i s held very precisely to 60 cps. The 8 sync, on the scope i s set to ' l i n e ' and 60 cps. mains i s displayed on i t s face. The sweep frequency i s set so that the trace i s triggered at the same rate as the desired frequency on the oscil lator or a simple multiple of i t . The time interval between sweeps i s given exactly by the time taken for the mains to display the integral or half- integral number of 60 cps. waves, including the delay time required i n order that the triggering always be instigated at the same phase of the 60 cps. The mains signal i s now removed and the osci l lator output i s displayed. The frequency i s varied unt i l a stationary one or two cycle pattern i s observed, depending on whether the fraction i s integral or half - integral . The other method i s used when the frequency of the osci l lator i s already set by some cri terion other than convenience of measurement. In this case the sync, on the scope i s set to ' internal ' and the osci l lator output displayed. The sweep frequency i s then varied u n t i l a conveniently countable number of cycles occurs between triggerings. With a stop watch the sweep rate i s then measured as accurately as desired. The oscil lator period i s given by the sweep period divided by the number of cycles per sweep. An attempt to use plane waves was undertaken but i t fa i led because of the end effects of the osci l la t ing rod. Two loudspeakers, spaced apart by a straight glass tube r i g i d l y attached to their cones were used; however, even with a tube 1.52 meters long, the diffract ion effects were too large. 9 4 . WAVE MEASUREMENT An optical system was chosen to measure the wave amplitudes, wave-number and wave direction. A board contain-ing sixty-four l ight bulbs, # 4 3 , i s fastened to the ce i l ing so that the l ight from these effective point sources reflects off the area of the shear flow to be studied into a properly oriented camera. The l ights are in a square array, each 7 .6 cm. apart. The board i s approximately 3 meters vert ical ly above the surface of the water and approximately 3 meters horizontally from the reflection points on the surface of the water (figure 3 ) • The camera that i s used i s a 35 mm. Agfa colorflex. It i s positioned approximately 1 meter vert ical ly and horizontally from the reflection points on the water surface and angled so that the image of the ' l ight board' appears in the centre of i t s f i e l d of view, (i) Amplitude As the waves cross the image of the board, the image of a l ight undergoes oscillatory excursions, the max-imum of which i s a direct measure of the maximum wave slope at the position on the surface of the water from which the undeviated l ight ref lec ts . The camera shutter i s opened for somewhat longer than one wave-period. A grid of streaks i s thus obtained on the f i l m , whose lengths are a measure of the maximum excursions of the images of the l ights . A small alignment mirror mounted just above the surface of the water and to one side of the image of the l ight board i s oriented to reflect one column of l ights into the camera, thus pro-ducing a constant reference line from frame to frame. (Examples of these photographs are given i n figure 6.) ( i i ) Wave number The l ight board also has twenty-three straight, white nylon strings mounted on i t , each i n a vert ical plane (figure 3)' These strings are photographed, u t i l i z i n g an electronic flash unit . (Examples of these photographs are shown i n figure 6.) This enables l ines of constant phase to be determined, from which the wave number i s measured as a function of position on the shear flow. The flash unit i s positioned about k$ cm. from and sl ightly below the board for half the phase pictures and 30 cm. in front of the centre of the board for the other half of the phase pictures. This ensures measurability of the phases over the whole area of the board. It i s necessary to use the smallest aperture stop that i s available on the camera because the surface reflecting the l ight being photographed i s local ly curved. A bundle of rays of l ight emanating from a point on the board, reflecting off the wavy surface and into the camera lens, becomes de-focussed or extra heavily concentrated depending on the curvature at the point of ref lect ion. For a l l aperture stops larger than f/22 this effect makes deter-mination of constant phase lines impossible. As the waves cross the area i n question, the images of the strings are distorted into sinusoids. The procedure for determining the lines of constant phase from 11 a photograph of this nature i s as follows: a photograph i s f i r s t taken of the string with no surface waves i n the area, then photographs are taken when waves are present. When the f i r s t photograph i s projected, the images of the straight strings are traced onto a piece of graph paper on the projec-tion board and the images of the l ights i n the alignment mirror are traced out. Then the photographs with waves are projected onto this same piece of graph paper. When the images of the l ights i n the alignment mirror correspond with the traced images, the traces of the straight strings form a baseline for each of the sinusoidal string images. The places where the sinusoidal image crosses i t s baseline are marked. Joining up these marks along wave fronts then produces lines of constant phase. ( i i i ) Projection system In order to fac i l i ta te measurement of the data on the films, a t i l t e d board projection system i s used with the focal length of the projector the same as the focal length of the camera. If the board onto which the fi lm i s projected i s t i l t e d with respect to the projector, to the same angle that the surface of the water makes with the camera, the measure-ments obtainedl correspond directly to the plane of the surface of the water to within a uniform scale factor. Details of the setting up procedure are given under section 5 , Experimental parameter. 12 (iv) Propagation and dissipation The propagation laws governing the transmission of wave properties i n both the presence and "absence of flow have been experimentally examined. In the absence of flow i t i s necessary only to know the surface tension and either the wave number or the frequency to determine the velocities of propagation. In the presence of a known steady state flow i t i s necessary to invoke geometrical arguments i n conjunction with a basic assumption regarding the interaction of the two motions (see Theory). The surface tension was determined by measuring the height of rise of the water i n a capillary tube. The bore of the tube was measured by a travelling microscope. The rate of dissipation of wave energy by viscosity i s changed severely by the presence of a surface film of o i l or other immiscible f l u i d material. Because of the i n a b i l i t y to include theoretically the effect of this f i lm, a siphon was set up to continuously remove the surface f l u i d . For purposes of determining the kinematic viscosity of the water, the surface temperature was measured before and after each experiment. The values used were taken from the 'Handbook of Physics and Chemistry'. In order to measure the rate of dissipation of energy i n the absence of any surface films or flow, the camera was moved back and up from the tank i n order to spread the reflection points of the l ights apart as much as possible and to move the location of the reflection points to a region where the time between the arrivals of the direct wave pulse and the reflected wave pulse was a maximum. To further re-duce the effect of the reflected information, a short beach was placed on the annular ring at the position from which the f i r s t reflected energy arrived. A series of photographs of wave amplitudes were taken at 8 cps. , 8.57 cps. , and 10 cps. , these frequencies possessing wave-numbers representative of those measured i n the presence of flow. The photographs were taken about five seconds after the i n i t i a l part of the wave pulse had begun to affect the images of the l ights , thus minimizing transient modulation effects. 5. EXPERIMENTAL PARAMETERS In order to compute the wave characteristics from the measured values on the f i lm, i t i s necessary to know the vert ical and horizontal distance from each l ight to i t s re-f lect ion point, of each reflection point from the entrance pupil point of the camera, the angle to which the projection board must be t i l t e d , and the uniform projection scale factor. The distances were a l l measured by triangulation with a survey-ing transit . A piece of graph paper was firmly glued onto a pane of glass and the whole assembly held about 1A cm. under the water surface i n the area of the reflection points and 14 such that the plane of the graph paper was paral lel to the surface. With the help of the transit , the graph paper was situated so that one of i t s ruled l ines f e l l i n the vert ical plane determined by the camera entrance pupil point and the centre of the tank. A photograph was taken of the images of the l ights superimposed on the graph paper. The transit was now used to measure the distances from each l i g h t , from the centre of the tank, and from the camera entrance pupil point, to a point on the graph paper previously marked. This photograph, together with these measurements, now serves as a calibration unit from which the projection system i s set up properly, i t s scale factor determined and a l l the other necessary parameters obtained. The one second camera shutter speed was accurately measured by photographing the trace on the oscilloscope after the sweep had been calibrated. The sweep rate was determined from the osci l lator after the oscil lator was adjusted i n conjunction with the 60 cps. , as explained above. The flash connections on the camera were used to short a small steady voltage onto the vert ical input of the scope while the camera shutter was open. THEORY 1 . INTERACTION DYNAMICS Consider the Navier-Stokes equations i n cyl indrical polar coBrdinates for an incompressible f l u i d : 15 = total velocity i n radial sense = total velocity i n angular sense w^ , = total velocity i n vert ical sense p ,^ = total pressure r = radius from centre of system & = angle measured about centre. (lc) ^ ^ t ¥ ^ t ^ ' ^ ' i ^ ^ ( v 2 i r - % - Z-M) (la) ^ ^ ^ ^ = density V = kinematic viscosity. For the experimental set-up neglecting secondary flow, u ,^ = u, particle velocity due to wave action, Vfj, = v + V, particle velocity due to wave action plus current w^ , = w, particle velocity due to wave action p = p + P, fluctuating pressure due to wave action and mean pressure due to mean flow and hydrostatic head. The centre of the coordinate system i s placed at the centre of the tank. The viscous terms are assumed to have no inter-16 a c t i o n with the n o n - l i n e a r terms, i . e . the v i s c o s i t y a c t s to damp out the waves i n the same manner as i f no mean flow were present, and i t i s the agency which makes the shear flow of t h i s type p o s s i b l e . S u b s t i t u t i n g v e l o c i t y components and m u l t i p l y i n g each e q u a t i o n by i t s r e s p e c t i v e v e l o c i t y component, + v i s c o u s / <?>- (2a> + VISCOUS (2b) / fe A d d i t i o n r e s u l t s i n : VISCOUS (2c) v i s c o u s (3 ) 2 2 2 2 where q = u + v + w . sur Now i n t e g r a t e from bottom of tank, i . e . to face of the water, +£, where ^ i s the v a r i a b l e d e s c r i b i n g 17 the surface elevation under influence of the waves. In considering the orders of magnitude of the various terms i t i s seen that for very small waves, i . e . p, u, v, w,_£, small compared to V and P, the terms become divided into f i r s t order, second order and third order terms. It w i l l be seen presently that a l l f i r s t order terms w i l l cancel so considerations w i l l be taken to second order. A l l third order terms shall be considered negligible i n the present treatment. Therefore the upper l i m i t of integration when applied to second order terms w i l l be set to zero, the wave-free surface l e v e l , since the re-mainder i s a third order term. A l l f i r s t order terms when integrated remain sinusoidal i n nature. Under these con-siderations the equations are now averaged with respect to time over a large number of waves and a steady state assumed. 18 To second order: -cJb + viscous (5 ) employing the continuity condition lot , / 7ir , 9hj _ Vr + r T fa - ~r t   lir j_9*J _ _ 66 ( 6 ) and where the bar indicates the average with respect to time. However, by the equation of mean motion of the v component i t i s seen that Therefore, - vj^bc/* = ^jy-^j (my d^. v i S c o u S ( 8 ) 19 It i s assumed that the local wave properties are given by an irrotat ional velocity potential, or, the local change of the character of the wave motion i s f e l t at higher than second order so that an irrotat ional wave model may s t i l l be employed.to this approximation. If the velocity potential with respect to the moving f l u i d i s given by <p, Bernoulli 's theorem i n the absence of viscosity produces: P T lQ . (9a) ~ = — - gz - #q + const. or, | - g(z - i) (9b) Thus, the last term on the l e f t hand side of equation (8) Ml, becomes — ps^(potential energy per unit surface area of wave motion), which, combined with the second to last term, becomes "f-s^fi (total energy per unit area of wave motion). The f i r s t term becomes the negative of the divergence of the total power represented by the wave motion plus third order terms, representing the transport of the wave potential energy by the fluctuating motions. Thus, the terms on the l e f t side combine to form the negative of the divergence of the total power of the waves, as determined by an observer stationary with respect to the coBrdinate system. The right side of the equation becomes - | s i n 2 ( c < + e ) ( | l . 2) where vX i s the angle between the arbitrary zero angle direction and the normal to the wave front at the point in 20 question. The angle <* i s considered positive i f measured i n a clockwise sense; the angle & i s considered positive i f measured i n a counter-clockwise sense (see figure 8 ) . Since the total energy of a wave group i s propagated by wave action at the group velocity, , the f i n a l equation can be written most simply i n vector form as follows: div E(c* + " t ) = TrEsin2(ol + 0 )(—--) - viscous r (10) — H- dr r where " ? i s i n the direction of advance of the wave and normal to i t s front, and V i s tangential to the mean streamline and i n the direction of positive 0 . The path obtained by integrating from the inner edge of the flow outwards with the direction of ~c + V , a function of r and & , tangent at every point to the path shall be called a "group ray". A path line defined this way possesses the property that no wave energy crosses i t . That i s , given any two group rays, the rate that energy flows across any line joining these rays i s the same as the rate that energy flows actoss any other l ine joining them i n the absence of dissipating mechanisms and non-linear interactions. There-fore, by considering the configuration of the group lines on the water surface the effect of the refraction and spreading of energy can be taken into account. Equation (10) can be reduced to simpler terms by integrating i t over the area contained between two d i f f e r -ential ly separated group rays and any two arbitrary points on the rays: 2 1 = ^1 I Eein2(ot + - ^)dids - viscous ( 1 1 ) or 4 ( E | £ + - V U ) . j - ^ = ^ E s i n 2 ( o t + © ) ( £ ! _ I) - viscous ( 1 2 ) where Q. i s the perpendicular distance separating the two group rays and i s an element of length along the group rays. The theoretical curves used for comparison purpose with the experimentally determined data have been obtained by a numerical integration of the equation ( 3 ) including viscosity, along a previously determined group ray: the interaction curve including the term i E s i n 2 ( o C + £ ) ( — • - — ) , •* dr r the non-interaction curve omitting i t . 2 . VISCOUS DISSIPATION The viscous decay term that has been used i n the determination of the above-mentioned theoretical curves i s essentially the same as that given i n Lamb ( 1 9 3 2 , # 3 ^ 8 , 3^9). If the assumption i s made that i rrotat ional waves maintain their i r rotat ionali ty even though decaying by means of a rotational agency, the rate of energy decay i s given by: where k i s the instantaneous wave number. By considering only a group ray again, i t i s possible to change the time derivative to a spatial derivative by the following trans-22 formation: Therefore, | | = - E ( 1 5 b ) (For a par t ia l just i f icat ion of this equation experimentally, see Results - Viscous decay.) 3. WAVE KINEMATICS In figure 9» part 1, A and B are two consecutive crests. For the condition of a steady state, the point E traverses to point F i n the same time that the point C traverses to point D. 1 * Therefore, j = c - Vsin( + * ) (14a) o where fQ i s the frequency of the oscil lations of the wave-maker, 7^  i s the wavelength of the wave at G, defined as the minimum distance between consecutive crests, and C i s the phase velocity of the wave relative to the water. Thus, using k = ^  ,CO = 2TTfQ kc -u> = kVsin(ot+£ ) (l4b) o (i) Zero' th Order Approximation (See figure 9 , part 2.) Clearly, i f a l l wave crests make the same angle with respect to the shear flow when entering the region of flow, then whatever happens to one wave crest w i l l precisely happen to i t s neighbours 23 except for a constant angle of rotation about the tank centre. The trajectory of a wave crest shal l be called a PHASE RAY and i s defined i n the same way as a group ray except that the phase velocity i s used i n place of the group velocity. The tank centre, to this approximation, i s a point of isotropy with respect to these phase rays. It can be seen that the distance along arc A between the two crests i s given by l^ji, thus 9VQ = rQp sin(c< + 0 ) q (15) For steady state, when these two crests reach arc B the distance separating them along B i s 1^3, thus 9\ = rjl sin(ot + B ) therefore, using k = krsin(ol+0) = const. (16) This approximation becomes exact for the phase ray which joins the wavemaker at a right angle with respect to the l ine between the wavemaker and the tank centre. Thus, the constant in equation (16) i s equal to k^L, where k Q i s the wave-number in the absence of flow and L i s the distance from tank centre to wavemaker. From equations (l4b) and (16) i t i s possible to eliminate (od + # ), leaving k as a function of r only. k VL k c ~H = "f- ( 1 7 ) 2k using C = (18) (Lamb, 1932, #267) where T i s the surface tension of the water and g i s the acceleration of gravity; and, using V as measured, values of k have been determined and are com-pared with measured values. In order to determine the rate of spreading of the energy, i . e . jl defined i n equation (12) as a function of r , employing zero' th order theory, i t i s only necessary to use the fact that a l l the group rays are geometrically similar with respect to the tank centre. If the distance along an arc about the tank centre between two group rays at the edge of the flow i s cf0 = f ^ ^ t h e n the separation be-tween these same rays, measured along an arc about the tank centre, at any radius r i s (j = f% but, Jl = J COS ((p^ +• 0) where (JJj i s the angle that the tangent to the group ray makes with the arbi t rar i ly chosen zero angle l i n e . Therefore, J ^ f f i + f j ( i i ) F i r s t Order Theory In order to reduce the errors introduced by the use of zero'th order theory, a f i r s t order theory has been devised. It i s convenient to consider the total rate of energy transport between a real group ray and an a r t i f i c i a l one constructed by rotating the real one about the tank centre by a dif ferent ia l angle 66. If this second ray i s i n 25 fact a real group ray, zero'th order spreading applies. If the second ray i s not a real xgroup ray, the f i r s t order effect can be obtained by determining how much energy actually flows across a l ine drawn perpendicularly between the two rays at an arbitrary radius compared with how much would flow across the same line on the basis of zero' th order theory, i n the absence of viscosity and interaction. The total rate of propagation of energy can be resolved into two perpendicular components: one along the radius at the point i n question and the other at right angles to i t . The former component i s given by Eccos (oC+# ) with dimensions of energy per unit time per unit arc length, the latter i s given by E^csin ( e( + 0 ) - v) with dimensions of energy per unit time per unit radial length. The following discussion of the group rays proceeds assuming ray I at the l e f t of ray II with the current and increasing i n the direction from I to II . Given that ( o i + £ ) i s the angle that the normal to the wave front makes with the radius at a given radius on ray I, then ( d l + £ ) 2 = (eC+0 ) x » ^ ( B ^ / } (20) i s the angle that the wave crest normal makes with the radius at the same radius on ray II to f i r s t order. The effect of the dependence of ((* + •£) on # i s made evident by the rate at which energy i s actually crossing ray II . This rate i s given by Ecfsin (<* + •£ ) £ - sin (<* +8 ) j , or 26 Excess rate of flow of energy between rays I „ ^O^^BJ •» r j A \ and II per unit time — per unit radial length (21) Therefore, since Ar =4scos(^.+ £ ), where s i s distance measured along a group ray, Excess energy flowing f per unit time between = JEccos(c< +& )cos(<p6 + Q ) 40ds r6 7)6 the two rays J (22) The rate at which energy flows across a l ine drawn perpen-dicularly between these two rays at r Q i s given by . E |"c + ~vl r cos( (P, + 0 ) £ 0 . o| — | o o »« o Thus, i n the absence of any dissipation or production of energy, the rate of flow of energy across a perpendicular between these rays at any other point w i l l satisfy the equation E|l + v| rcos(<pc+*> = Eo|l+t|orocos(qP 6 r +a) 0 + jEc c o s (oc+a ) c os (<p& +e)^fg&ks s° c + 1 fkcosioc^o c o ^qk^Us r o J E o ^ o C o s ( o l + ^ o Y 6 ™ (23) Between any two real group rays i t can be seen from equation (12) that E J " £ + " V | Jt i s a constant i n the absence of diss ip-ation or interaction. Equation ( 2 3 ) , then, may be taken as defining S. as a function of r i f , except for ^ — , zero' th order values are used to evaluate the integral . In this case, 27 p - o r c o s ( ^ + & ) f i S ^ I I Eccos(ot + 0) ?{oL+ » ) c o s ( Ol+ ff)ds{ + ' „ / « o 2 o o . . ( * + « ) 0 — 7 5 - r a J C 2 , t ) In order to determine the necessary values of — — i t i s assumed to be constant along any given phase l i n e , thus i t has the value at any point given by i t s evaluation at the location where the phase ray that goes through that point enters the region of flow. In the zero'th order case this assumption i s ")(aJ ) exactly true since — i s everywhere zero. It i s assumed, therefore, that variations of < ^ C ^ * & ^  along a phase ray w i l l exist only to f i r s t order compared with i t s value. Evaluation i s obtained from the following equations: In the absence of flow (figure 7)» rsin(o<+ 0) = Lsin(o(+Y") (25) From which i s obtained ?(* + 6 ) tan(o< + 4 ) , g . 1$ " tan(ox +0 ) - tan(<* + ^ ) Equations (24), (25) and (26) permit calculatiomJof the f i r s t order spreading. As w i l l be seen (page 37)> the difference between the zero' th and the f i r s t order spreading i s so small that higher orders need not be considered. k. SECONDARY FLOW The mean flow generated by the tank system includes a secondary flow because of the proximity of the bottom. If the vertical component of the secondary flow i s assumed not important compared to the radial component in regions near 28 the surface and not too near either wall , the equation „dv Uv , / d v 1 dv v J Ud7 + — =^[^2 + r d? " ;2/ ( 2 7 ) represents the flow very closely, with U as the radial com-ponent, V as the angular component and i) as the kinematic viscosity. Total derivatives are used because under these assumptions V i s a function of the radius only. For a region not too near the walls and at the surface the functional form of U that has been chosen i n order to solve equation (27) i s a source-like one, i . e . U = — where A i s an as yet unevaluated constant. Equation (27) now has a solution given by n+1 V = B(r n - ^ r ~ ) ( 2 8 ) A with r as the radius of the inner wall and n = 1 + — . o V It i s interesting to note that the curvature i n equation (28) changes from negative to positive for r > r Q with a change of n from 1 to 2: that i s , a secondary flow changing from zero to —. The value of — for r - 1.0 meters i s about r -r _ ^ 10 cm./sec. This i s negligible i n terms of interaction dv v phenomena yet i t changes the position of maximum — - — from near the inner wall to near the outer wall , and by equation (3) the position where this i s maximum i s also the position of maximum interaction. 5 . ANALYSIS OF AMPLITUDE PHOTOGRAPHS The theory used to obtain the equations which relate 29 streak lengths on the film to actual wave slopes i s given i n the Appendix. It i s possible to determine exact relationships but for the purpose of analysis, modified linear equations were used. The errors introduced by linearization are less than 1 per cent, for waves of the size used (*•" .008 cm. in amplitude). Since absolute wave amplitudes are never required, the effective error i n the f i n a l comparison curves w i l l be less than 1 per cent, by a factor of 2 or more. RESULTS 1. ABSENCE OF FLOW (i) Propagation On the assumptions of i r rotat ional i ty , incompress-i b i l i t y and infinitesimally small amplitudes, inviscid wave theory produces the result given i n equation ( l 8 ) , and since 0J= kc, 2 . k 5 T 0) = gk + -j-For a wave frequency of 8.00 cps. and a surface tension of 6 7 - 4 dynes/cm., this equation yields k = 2.015 cm. - .02. In two experimental determinations at 8.00 cps. k was found to be 2.028 cm. - .02 and 2.053 cm. - .02. ( i i ) Viscous dissipation Results of the viscous dissipation experiments are 30 presented i n graphical form in figure 10. The theoretical curves plotted are taken directly from equation (4) written JT » i n terms of amplitudes and with V =.0. A term i n / / — i s 'o included to account for spreading of energy: •i/F -2Vk2s wave slope _ / _o c , . wave slope " = \ ?* The parameters Wave Slope and S are taken as the average o o measured wave slope at that particular frequency and the dis-tance from the wavemaker at which that average occurs. Thus, discrepancies from the theory show up when the slopes of the sol id l ines do not agree with the slopes of the measured points. The 8 cps. data i s seen to agree quite well with the theory except for the wavy characteristic of the measured points. This waviness i s thought to be due to the presence of small amounts of reflected energy. At 8 . 5 7 cps. the theoretical line disagrees " appreciably with the measured points. The reason for or significance of this result i s as yet unknown. The 10.0 cps. data was taken with the aim of measuring the viscous dissipation of a wave whose phase velocity i s exactly the same as that of a wave with twice i t s frequency. Unfortunately, i t appears that this condition has not been achieved. From the photographs used to measure the wave slopes i t can be seen that an extra wave train i s 31 present, but in the distance covered by the measurements the phase of this extra wave with respect to the 10.0 cps. wave has changed by 90°. It i s thought that the apparently rapid decay over the l e f t side of the graph in figure 10 i s a com-bination of two factors. One i s the fact that any wave of a frequency of 20.0 cps. which has appreciable amplitude that far from the wavemaker must have a continuous source of energy as i t propagates. One source of energy i s a non-linear wave-wave interaction with the fundamental. Between any crest of the fundamental and the preceding trough i s a region which i s undergoing a continual rate of compression by the f l u i d particle veloci t ies . If a part of an extra wave happens to coincide with this region for any length of time i t w i l l be increased in energy by the same sort of interaction as that between waves and a mean shear. (Longuet-Higgins and Stewart, i960, "Journal of Fluid Mechanics", i n the press.) The larger velocity gradients present i n the 20.0 cps. wave result i n a much faster decay of energy to viscosity; so, presumably a balance i s achieved between non-linear input and viscous decay. The fundamental thus experiences an increased rate of loss of energy. A phase photograph of a wave at 10.5 cps. i s given i n figure 7, clearly showing the complexity of the wave form. The other contributing factor to the anomalous measured decay rate of the 10.0 cps. waves i s due to the change of phase that occurs between the two types of waves. The 32 shape of the parts of the wave from which the l ight reflections take place can be inferred from the character of the streak density obtained from an amplitude photograph. From an exam-ination of this type i t appears that the parts of the wave which give the maximum excursions of the reflected l ight change from having a slope more nearly l ike that of 20.0 cps. to that characteristic of a 10.0 cps. saw-tooth wave. The peculiarit ies of waves i n the neighbourhood of 10 cps. have been noted previously (Wilton, 1915; ;Pierson, i960). 2. PRESENCE OF FLOW Two complete sets of data were taken i n the presence of flow. Set I had an oscil lator frequency of 8.23 cps. and a velocity profi le as shown i n figure 11. Set II had an oscil lator frequency of 6.00 cps. and the velocity profi le given i n figure 14. For both sets of data, one hundred amplitude photographs were taken and the 'best' twenty i n each were measured. The cri terion used to judge which frames were to be measured was based on overall smoothness of the streak lengths within the frame being judged. (See figure 6 for an example of a photograph rejected on this basis compared with two that were not.) The judgment was a purely visual one, i . e . i t was performed before any of the streak lengths were actually measured. These anomalous streak variations are caused by the accumulative effect of fluctuations i n the horizontal velocity that occur during the time required for the energy to propagate from the edge of the shear flow to 33 the position of examination. The major effect of a local velocity fluctuation i s a change of the rate of spreading from that point onwards along the group rays affected. For Set I, a total of forty phase pictures were taken, of which five were chosen and measured. The judgment in this case was visual and was based on the smoothness of the sinusoidal string images. In this case, however, the cause of the discrepancies was due only par t ia l ly to velocity fluctuations and mostly to the presence of any waves of a period different from the main 8.23 cps. component. Waves of different period exist as i s shown by the presence of trans-ient modulation. If any waves of different period do exist they w i l l be refracted differently than the 8.23 cps. wave according to equations (14b) and (16). For Set II , an attempt was made to take forty photographs but because of a faulty camera aperture mechanism only ten usable ones resulted. On the whole, these ten were much freer from discrepancies than those of Set I, so no d i f f i c u l t y was encountered i n choosing five frames. Figures 17 and 18 contain typical l ines of constant phase from sets I and II respectively. Only data from two out of the five frames i s shown in each. (i) Propagation It can be seen that the values predicted for wave-number on the basis of equation (17) depend on the value of 34 the phase velocity at that wavenumber. Therefore, a com-parison between predicted and measured wavenumbers w i l l determine how accurately equation (18) i s obeyed, assuming zero' th order kinematics to hold. The theoretical curves shown i n figures 12 and 15 are plotted from equation (17) and are determined from Set I information and Set II information respectively. The measured points are obtained from the respective ' l i n e s of constant phase' diagrams i n the following manner: a series of seven straight l ines are drawn on the constant phase dia-grams paral le l to the zero baseline for angles. They are spaced equally apart and cover a l l the information. Wave-lengths are then measured both forward and backward from a l l the positions where lines of constant phase cross the seven straight l ines , and the respective distances of these pos-i t ions from the tank centre are determined. The data sub-sequently used i s the average of the forward and backward measurements at any given point. The measured points shown i n figures 12 and 15 comprise a non-selective choice of one-f i f t h of the information obtained along one of the seven l ines regarded as typical i n each case. This method of presenting the data was chosen i n order to show the amount of scatter i n the measurements without overcrowding the diagram. Smoothing of the measured points i n figure 12 was done v i s -ual ly . As can be seen from figures 12 and 15, Set II data 35 agrees very well with equation (7) whereas Set I data differs significantly from i t . The maximum discrepancy i n figure 12 (approximately 5 per cent.) occurs at k — 2.7 cm. and, since the wavenumber that propagates at the same phase velocity as the wave at twice the wavenumber has a value of 2.71 cm. \ calculated from equation (18), i t i s thought that the two phenomena are intimately linked, though exactly how i s not yet understood. In order to estimate the maximum error introduced by the use of equation (16), equation (l4b) was used to determine that 9 k / n c. -1 Therefore, since A£ i s less than 0.1 radians measured from the phase line for which the zero'th order theory i s exact to the positions at which k was measured, the error in k pre-dicted from equation (17) w i l l be less than .005 cm. \ i . e . negligible. The only poss ibi l i ty , i t seems, i s that the value of the phase velocity i s i n error. If this i s so, the pro-pagation speed in the vic ini ty of k = 2.7 cm. differ from those obtained on simple theory by approximately 5 per cent. ( i i ) Interaction In order to determine the results of the inter-action measurements several calculations must be made. One of the f i r s t of these i s the determination of a group ray along which equation (12) i s to be solved numerically with and 36 without the interaction term. The group rays that are used were obtained simply from a graphical integration of the equation - ^ f f = tan < £ + & ) (30) or, (see figure 8) " ^ f f = t a n ( o i + * } " c" sec(oUtf ) (31) The angle (©<+•#) was i n a l l cases determined from equation (l6) using the measured values of k. The group rays so obtained are shown in figures 17 and 18 for sets I and II respectively. (The circles shown i n these figures are the positions of reflection of the l ights . ) Eight wave slopes are then determined for each frame of amplitude photographs chosen to be measured - one for each row of l ights . If the group ray happens to pass between two l ights , the value of the slope at the ray i s obtained by interpolation between the l ights . Each frame i s then normalized by setting the average wave slope along the group ray for each frame to unity. The normalized measured values for each point on the group ray are then averaged over a l l twenty frames. The standard errors of each point are then calculated from the normalized data. The f i n a l averages for Set I are shown as the measured points i n figure 13 with the vertical bars indicating one standard error on each side. The theoretical curves plotted in figure 13 were calculated on the basis of zero' th order kinematics -inclusion of f i r s t order effects produces a negligible change. They incorporate the smoothed wavenumber data of figure 12 i n the conversion of energy to slope. They also are normalized to unity at the same position along the group ray at which the averaged wave slope occurs. The theoretical curves used for Set II incorporated f i r s t order spreading theory. The maximum deviation from zero' th order theory encountered was 1.7 per cent, i n wave slope. Figure l6a represents the data obtained i n Set II, plotted in the same manner as figure 12. The upper points i n figure l6b are plotted as the difference between the measured points and the normalized theory omitting the interaction term. The lower points i n figure l6b are plotted as the difference including the interaction term. In both cases, the vert ical bars indicate one standard error on each side of the points, and the dashed straight l ine was obtained by least squares regression through the points. The interaction effect results i n an added energy change of 12 per cent, over the distance examined. The change i n energy due to spreading i s about 30 per cent. DISCUSSION The optical methods used in this experiment were entirely satisfactory - i t i s estimated that an accuracy of 0.5 per cent, could be achieved i n a l l the optical measure-ments i f extreme care i s exercised. The method used to create the shear flow, though 38 extremely simple and adequate, possessed one major drawback -unsteadiness. Because of up-welling at the inner wall , a region of turbulence existed at the edge of the flow. This region produced most of the random variations of wave pro-perties that were subsequently dealt with by s t a t i s t i c a l analysis. Considerable time was spent trying to produce waves with very l i t t l e transient modulation. Unfortunately the problem was not entirely solved. The attempt to random-ize the phase of the modulation from pulse to pulse appears to have succeeded for Set I data but not completely for Set II . It appears that a further experimental investigation i s warranted into viscous dissipation and propagation pro-perties of waves near k = 2.7 c m . - 1 , both i n the presence and absence of mean flow. The interaction measurements as presented i n figure 13 are definitely anomalous. The difference in decay rate between the measured points and the theoretical curves i s i n the same sense and of approximately the same magnitude as the discrepancies noted i n the viscous decay between 8.57 cps. and 10.0 cps. It i s perhaps significant that a sudden decay of energy occurs very near the position along the group ray, 12 cm., at which k = 2.7 cm. 1 occurs. No anomalies were found i n the information taken in Set II and figures l6a and l6b clearly indicate that interaction theory i s well favoured over non-interaction theory. ko SUMMARY Except for the wavenumber region near k = 2.7 cm. \ i t has been found that: (a) viscous dissipation i s not significantly different from the theoretical predictions given i n Lamb (1932); (b) propagation laws for infini tesimal , invisc id waves are obeyed; (c) interaction theory taken to second order i n wave slope i s not significantly different from the measurements, whereas theoretical predictions based on the omittance of the non-linear interaction term are significantly different . The viscous decay and propagation properties appear to be anomalous for wavenumbers near k = 2.7 cm. APPENDIX Amplitude Analysis Consider figure 1 9 • Let the point J be situated on the part of a wave whose slope i s tan £ . Subsequent analysis w i l l assume that the elevation of point J from the plane GDH i s negligible. (For waves of approximately . 0 0 8 cm. i n amplitude, the error of this assumption i s of order 1 0 per cent.) Thus, by the law of specular reflect ion, angle AJB = angle BJC. Let € and JU be the angles described by the projection of 8 on the planes AGH and AGD respectively. By considering triangles AJC and AEC i t can be determined that sinp sinx J(hl + U - X)l + (d * x)l + ^) (Al) 1/ \ b 2 + (a - x ) 2 + yZJ\c + (d + x r J Similarly, by considering triangles AHL and AJC smT sine* l / \ b + y y l c + d + y J (A2) Using maximum values for x and y and measured values for a, b, c, d: sinj> = . 9 9 9 3 5 sinK sinx = . 9 9 9 9 4 sino^ therefore, j» =K , T = <T » with an error of less than . 1 per cent. 42 Let angle AEG = ^ , angle CED = y . Thus, y + K - y u = ' A + ^ + / J . (A3) or 2yu= V - A (A4) Similarly, let angle AHG = Y* , angle LHG = ji then, TT-T-e+a-=p + €+x (A5) or 2€ = TT -Y ~ CA6) Now, tan y = b and tan A =-a - x c d + x therefore, tan 2 ju = * , — ° - ^ v **' ^ ' / (a - x)(d + x) + be or tan 2/u = . ( b + . c ) x . s 2 (A7) / ad + be + (a - d)x - x c b Also, tany* = — and tan /$ = — Therefore tan 2€ = ( b * ° ] y ( A 8 ) be - y Tan S i s obtained from equations (A8) and (A7) as follows: 2 2 2 tan S = tan yu. + tan 6 ( A 9 ) The actual measurements indicate that equations (A7) and (A8) can be linearized with a maximum error of approximately .7 per cent. Therefore: (b + c) t a n ^ = 2(ad + be) x (A10) *«»« - . ( A 1 1 > If the measured values of x and y are taken as the components of the total length of the streak an additional factor of 2 must be incorporated in the denominators i n equations (A10) and ( A l l ) . 43 BIBLIOGRAPHY Drent, J . , 1959, "A Study of Waves i n the Open Ocean and of Waves on Shear Currents", Doctoral Thesis, University of Br i t ish Columbia. Handbook of Physics and Chemistry, 4lst edition. Johnson, J.W., 1947, Trans. American Geophysical Union, V o l . 28, No. 6. Lamb, H . , 1932, Hydrodynamics, 6th ed. , Cambridge University Press. Longuet-Higgins, M. , and Stewart, R.W., I960, J . F luid Mech., i n the press. Pierson, W.J., and F i f e , P . , I960, "Some Properties of Long Crested Periodic Waves with Lengths near 2.44 Centimeters", Technical Report, Dept. of Meteorology and Oceanography, Research D i v . , College of Engineering, New York University. Wilton, J .R . , 1915, Philosophical Magazine, V o l . 29, Ser. 6. EXPERIMENTAL a n n u l a r r i n g L — o u t e r t a n k f i9 3 L I G H T B O A R D W A V E P R O D U C T I O N C I R C U I T L.F [r\j\ Osc i l l a to r * M a t c h i n g T r a n s f o r m e r o o o o o o o o o o o o o o o o A u t o m a t i c S w i t c h 115 V . , 6 0 c p s . D.RS.T 115 V. , A . C . R e l a y E l e c t r o n i c C o u n t e r W a v e m a k e r O s c i l l o s c o p e R » I 0 o h m P o t e n t i o m e t e r f i g . 5 With Flow ( 6 c p s . ) W i t h Flow (6 c p s . ) E N E R G Y P R O P A G A T I O N G E O M E T R Y T a n k C e n t r e W a v e m a k e r E d g e o f F l o w K I N E M A T I C F R E Q U E N C Y C O N D I T I O N W a v e m a k e r E d g e o f F l o w (a + 0 ) ( P a t h I I / / P a t h I B ( a + 9) II. Z E R O ' T H O R D E R K I N E M A T I C A P P R O X I M A T I O N fig. 9 D I S T A N C E F R O M C E N T R E O F T A N K ( c m . ) f i g . I I M E A S U R E D ANO P R E D I C T E D W A V E N U M B E R D A T A 3.0 OS 10 00 3 _ 2 '6 u UJ — > < 2.0 • — M e a s u r e d P o i n t s S m o o t h e d E x p ' t l . D a t a — Theoretical Curve JL 100 200 300 H O R I Z O N T A L D I S T A N C E A C R O S S FLOW ( I u n i t = . 3 6 mm.) 400 f i g , "2 MAXIMUM WAVE S L O P E S IN P R E S E N C E OF FLOW MAXIMUM WAVE S L O P E S IN P R E S E N C E OF FLOW DIFFERENCE OF MEASURED WAVE SLOPE FROM INDICATED THEORY ( I Unit = 1 % of average wave slope along group ray) + + I I o + ro ro ro H 1 I 1 1 r-I h I ro - h 01 o CO H > z o m > r -o z o o JO o c Tl JO > a 3 ro o ro 0> cr O o a —I r-Q O (A C - * lO =>• c/> 2 c » o o at a> c « lO TJ -t o <e — (A » 2 m a M c co o 3 a. a o z c o JO JO < m > CO r-N m o —i ' X o O TJ m > > CO c CO O m z o CO o > n < m H X CO m r- o O JO TJ m m H CO O > 1 I I I I I I r— I I + H z o I _ a o L I N E S O F C O N S T A N T P H A S E i 1 I Cm. on W a t e r S u r f a c e f i g . 17 L I N E S O F C O N S T A N T P H A S E I — | I Cm. on Water S u r f a c e f i g . - 18 LINES OF C O N S T A N T P H A S E 

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