Deep Water Gravity Waves

Chapter 2 Deep water gravity waves 2.1 Surface motions on water of finite depth We now move on to consider motions in water that is not shallow, will allow the flow to vary with z within the water: z+dz u(z) z=O , we z х x+dx Figure 2.1: Configuration of deep water system. As before, the system is nonrotating and inviscid, and the water density р is assumed constant. Basically, we follow the same procedures as for the shallow ,vater case. However, rather tliaп consider balances for colнmns of water we mнst do so separately for elemental volнmes dx dz, such as tl1e Ьох CHAPTER 2. DEEP WATER GRAVITY WAVES in the figure. The horizontal and vertical momentum equations are 1 ди ди р-+ри­ дх дt дw ди pwдz др =--: дх, дw (2.1) др + pw= -- gp дz дz . Thus, the processes tending to accelerate the flow in the x-direction are: the x-gradient of pressure and advection of x-momentum; in the z-direction: the z-gradient of pressure, gravity, and advection of z-mornentum. We now consider conservation of rnass within the marked volume; its mass, per unit length in у, is m р dx dz, which for this incompressiЫe mediuш is constant with J•w u� u-� � dz .�w (x,z) dx Figure 2.2: Mass continuity. Ho\vever, there are fiнxes of mass across the voluшe, as shown in Fig. 2.2. Across the left face, into the Ьох, is а mass flux, per unit length in у, of р и(х, z) dz; there is an outward fiux of р и(х + dx, z) dz across the right face. Similarly, there is а flux р w(x, z) dx in through the lower face and 1Since the varaiЫes are now fur1ctions of (х, z, t), the substantive derivative is now d_д dt - дt + dх д dt дх + 2.1. SURFACE MOTIONS ON WATER OF FINITE DEPTH а fiux р w(x, z +dz) dx out through the upper face. Altogether, the net convergence of the mass fiuxes into the Ьох is с = рdz [и(х, z) - и(х +dx, z)] + рdz [w(x, z) - w(x, z dz)] дw) -+dxdz (ди дz дх = dxdz 'v · u where u is the vector velocity (и, w). Continuity of mass demands, since tЬere is no change of mass within the Ьох, that С Ье zero, whence our continuity equation (2.2) This tells us that incompressiЫe fiow is nondivergent. So we have 3 equations, the 2 momentum eqs. (2.1) and tl1e continuity eq. (2.2), in three unknowns и, w, and р. We also have boundary conditions. At the lower boundary z О, there can Ье no normal motion, wlience w О there2 . The motion will in general Ье nonzero at the free surface, since tliis can move. However, fiuid on the surface ( wliich is а rnaterial surface) must remain there----it cannot pass through, which tells us that а fiuid parcel on the surface must move with the local fluid speed along the surface, or dh dt (2.3) Finally, we also know that the pressure in the air immediately above the surface is atmospheric pressure р0 which, as before: we assume constant. Since pressure must Ье continuous across the surface (to Ье otherwise would imply an infinite pressure gradient and therefore infinite acceleration, wliich would Ье unphysical3 ). So our final boundary condition is (2.4) 2Note that we can impose no similar condition on и at tlie lower boundary, since we have assumed iпviscid flow and so cannot include viscous boundary effects. 3We are in fact neglecting surface tension here. If the surface is curved, as it will Ье in the presence of the wave, sшface tension will exert ап effective pressure оп the fluid beneath. Such effects are significant only for waves of wavelength а few сш or less (and sнch waves are known as сарШаrу waves). CHAPTER 2. DEEP WATER GRAVITY WAVES 2.2 Small-amplitude deep-water surface waves As before, we investigate the properties of small-amplitude disturbances to а stationary basic state, described Ьу =о' и w h D, (2.5) Ро + gp(D - z) . р Introducing small perturbations и', w', h', and р', the momentum eqs. (2.1) can then Ье written as ди' 1 др' ди' ди' -и/--w/дх дz ' дw' / / дw' -u--w-. дх дz -+-­ дt р дх 1 др' дw' +дt р дz Now, the terms in the RHS are nonlinear, in fact quadratic in the perturba­ tion quantities; we neglect these on the grounds that the perturbations are small, leaving ди' 1 др' -+-­ дt р дх 1 др' дw' -+-дt р дz о, (2.6) О. (2.7) The continuity eq. (2.2) becomes simply (2.8) We now proceed to derive а single equation in а single unknown from these three. In fact, it is very simple to do so: taking the x-derivative of (2.6), the z-derivative of (2.7), and adding gives _§_ дt ( ди' дх + дw' + � дz ) д2 р, ( 2 р дх + д2 р' = дz2 ) О. But, from (2.8), the term in the first parenthesis is zero, so д2 рl дх2 + д2 рl дz 2 = О . (2.9) 2.2. SMALL-AЛtJPLITUDE DEEP-WATER SURFACE WA1/ES As before, we focus attention on wave-like disturbances, for which р' (х, z, t) = Re [Р (z, t) (2.10) substitution into (2.9) then gives which has solutions of the form (2.11) (Note that tliis tells us that the vertical scale of the motion is k- 1, which is formally the same as tl1e horizontal length scale.) Now, the lower boundary condition tells us that w' О at z = О; therefore, from (2.7), др'/дz = О there. So, using (2.10), we have whence, in (2.11), Р1 Р2 р'(х, z, t) P(t)/2, say, and so (2.10) becomes Re [P(t) cosh kz eikx ] . (2.12) It is now straightforward, from (2.6) and (2.7), to oЬtain the form of the velocity perturbations и'(х, z, t) Re [-i;Q(t) cosh kz eikx ] (2.13) w'(x, z, t) Пе [-;Q(t) sinh kz i kx] (2.14) where dQ/dt = Р. Notice that, at this we have defined the spatial structure of the motions, but not the time dependence. То define tl1e latter, we have to include consideration of the surface dynamics. CHAPTER 2. DEEP WATER GRAVITY \-VAVES z=D z=D-d х ) Figure 2.3: Near-surface details. Consider the level z D - d, where d is just greater than lh' 1, so the level lies just beneath the wave troughs, as shown in Fig. 2.3. (Since our equations are valid only within the water, vve cannot choose z = D.) Now, since [eq. (2.4)] the surface pressure is р0, the total pressure (background + perturbation) is, using hydrostatic balance, р(х, D - d, t) Ро + gp(d + h') . But, since the basic state pressure is, from (2.5), р0 + gpd, the pertшbation is р'(х, D d, t) = gph'; taking the limit d-;. О (recall that h' is arЫtrarily small), we have (2.15) р' (х, D, t) gph', so, from (2.12), _!_Re [P(t) coshkD ikx ] 9Р (2.16) (Note that р'(х, D, t) is nonzero-it is the pressure perturbation at the surface z D + h' that is zero.) The material surface condition (2.3), neglecting nonlinear terms, is дh' = w'(x, D, t) . дt Using (2.15), др' дt(х, D, t) gpw'(x, D, t) . 2.2. SMALL-AMPLITUDE DEEP-WATER SURFACE WAVES Substituting from (2.12) and (2.14), we get dP dt(t) cosh kD = -gk Q(t) sшh kD or, since Р = dQ/dt, dz Q dt2 This has solutions Q + gk tanhkDQ =О. (2.17) + Q_e (2.18) = Q+e-iwt iwt provided w satisfies the dispersion relation w 2 = gk tanhkD. (2.19) Eq. (2.19) can Ье rewritten as w = ± �JkD tanhkD, (2.20) where с0 = -JgJ5 is the shallow water wave speed. (2.20) is plotted in Fig. 2.4. D 4 0,)Са о 4 6 8 kD 10 -2 -4 \ Figure 2.4: Dispersion relation for deep water gravity waves. Dashed lines show the shallow water relation. CHAPTER 2. DEEP WATER GRAVITY WAVES Note the departпre from shallow water behavior for kD > 1. In gen­ eral, the most important difference is that, for deep water waves, the phase speed с w / k is not independent of wavenumber--such waves are kno,vn as dispersive waves. 2.2.1 The long wave (shallow water) limit Note from (2.20) that, in the limit kD -+ О, w 2 gDk 2 , so we recover the shallow water dispersion relation (and, in fact, shallow water dynamics) in this limit when the wavelength is much greater than the depth. However, kD in fact has to Ье very small for this approximatioп to Ье valid: the wavelength 21r/ k must Ье greater than 14 times the depth before the shallow water result for с Ьесошеs within 3% of the correct value. From (2.13), since cosh kz -+ 1 as kD ---, О (note that z � D) the horizoпtal velocity Ьесошеs independent of z in this limit, as we assumed in our shallow water analysis. 2.2.2 The short wave (deep water) limit For kD---, оо, tanh kD---, 1, so (2.19) becomes (2.21) which is independent of D. In fact, the whole proЫem Ьесошеs iпsensitive to the background depth in this limit, as the waves do not tlie bottom. The vertical structure functioнs coshkz анd siнhkz appearing in (2.13) 1 (2.14) and (2.12) are coshkz sinh kz both of which can Ье approximated as ½ekz (except close to the bottom) as kD -+ оо. This means that the solutions decay downwards from the surface as ехр (-k (D z)), and becoming vanishingly small at depth, as shown in Fig. 2.5. Thus, though the waves can propagate great distances horizontally, they remain trapped near the surface and do not penetrate deep into the water. So, the water depth becomes irrelevant. This is because all the 2.3. BACKGROUND THEORY-DISPERSIVE WAVES / k(z-D) е Figure 2.5: Surface waves on deep water. dynamics of the waves-the effective elasticity that allows their existence-­ is tied up with the density discontinuity at the water surface. Thus, the surface acts as а wave guide, channeling the wave propagation. 2.3 2.3.1 Background theory---dispersive waves Dispersion The general dispersion relation for the frequency w of 1-D waves of wavenum­ ber k сап Ье written (2.22) w = w(k) and the phase speed is с= w (2.23) k Clearly, с is independeпt of k for all k only if w(k) =constaпtxk-this is the nondispersive case we discussed earlier and, as we saw, it implies that all disturbances, including localized ones, propagate without change of shape. This can Ье thought of in terms of' Fourier components. Any non-siпusoidal disturbance сап Ье described а sum of components of different waveпumber; if all these waves propagate at the same speed, so will the disturbaпce itself, and its shape will not change. For many kinds of wave motion, ho\vever (including surface waves on deep water), с varies with k, in ,vhich case the different wavenumber coшpo­ nents will have different speeds, а phenomenon known as wave dispersion. CHAPTER 2. DEEP WATER GRAVITY vVAVES Therefore the way they interfere with one another will change with tim�o the shape of the distшbance will change. 2.3.2 Group velocity There is one particularly important aspect of dispersioн, which concerнs the way that modulations propagate on а wave train. А monochromatic wave of the forrn х(х, t) Re [А ei [kox-w(ko)ti] with single wavenumber k0 has the simple spectrum of а 6-function AБ(k-k0) [Fig. 2.б(а)], and behaves in the way we have discussed, propagating with а speed с= w(ko)/ko. Ь} а) A(k) k k Figшe 2.6: Wavenumber spectra for (а) а morюchromatic wave, апd (Ь) an almost-monochromatic wave. Consider now an almost monochromatic wave, with а narrow spectrum [Fig. 2.б(Ь)]. This can Ье written x(x,t) = Re 100 A(k) ei [kx-u.1(k)t) dk' (2.24) where A(k) is rюnzero only for wavenumbers in the vicinity of k0. Writing k = k0 бk, we can rewrite this as 2.3. BACKGROUND THEORY-DISPERSIVE WAVES where Бw = w(k0 + Бk) - w(k0 ) � (дw/дk)Бk. At t х(х, t) = F(x) where 100 F(x) = Re A(k0 = О, this is simply ik x e o + Бk) ei Бk х dk is the modulating envelope of the wave train, which has carrier wavenumber k0, as illustrated in Fig. 2.7. тс/8k ТС/k ) " Figure 2.7: An almost-monochromatic wave packet, comprising many wave­ lengths. The phase of the carrier wave propagates at the phase speed, but the modulation envelope propagates at the group velocity. Now, for t > О, the wave packet behaves as х(х, t) { Re 100 A(ko F(x _ Cgt) + Бk) ei [Бk(x-cg t)]dk} ei [ko(x-ct)] , ei [ko(x-ct)] (2.25) where (2.26) = :� k=ko is the group velocity. Thus, while the carrier wave propagates at the phase speed, the modulation envelope propagates at the group velocity. This is an Cg 1 CHAPTER 2. DEEP WATER GRAVITY WAVES important concept, as it is the latter velocity that governs the propagation of information, as we sl1all see. Nondispersive waves have w = cok, with constant phase speed с0, and so their group velocity is the same as their phase velocity. But the group velocity of dispersive waves differs from the phase speed, so in а wave packet like that shown in Fig. 2. 7 the wave crests will шоvе at а different speed than the envelope. If с > с9 ( which, as we shall see, is the case for deep water waves), new wave crests appear at the rear of the wave packet, move forward through the packet, and disappear at its leading edge. We shall see some examples of this below. In general, it is easy to get а feel for both phase and group propagation graphically from the dispersion relation, as shown in Fig. 2.8. О) о Figure 2.8: The disperion relation w(k). The phase velocity at k - k0 is tan а, the group velocity tan fJ. For any wavenumber k, the phase velocity is just given Ьу the slope а of а line joining the point (w, k) to the origin, while the group velocity is given Ьу the slope tan fJ of the tangent to the curve at (w, k). 2.3.3 Group velocity in multi-dimensional waves For future reference, we note here that for 2- or 3-dimensional waves, the general dispersion relation is of the form w = w(k) (2.27) 2.4. SURFACE WAVE DISPERSION where k = (k, l, m) is tl1e (2-D or 3-D) vector wavenumber. The phase veloc­ ity4 is w w ) (2.28) с= k'l'm and the group velocity (и) ( дш дw дw ) дk' дl 'дrп (2.29) As we shall see later (and а comparison of (2.28) and (2.29) implies), not only may с9 and с Ье of different rnagnitudes, they may also Ье in different directions. 2.4 Surface wave dispersion 0) D 1 � 1 2 4 6 8 10 kD Figure 2.9: Phase speed c(slope of line A)and group velocity с9 (slope of line B)for surface ,vaves. Returning now to surface waves, and tl1e dispersion relation (2.19) slюwn iп Fig. 2.4, we can see from Fig. 2.9 that с9 =:; с for all wavenumbers (the slope of line В never being greater thaп that of line А. This is showп more explicitly in Fig. 2.10. The phase velocity is in fact not а vector, even though it has magnitude and direction. It does rюt transform like а vector under rotation-this stems from the fact that phase propagation has no meaning along the phase lines. 4 CHAPTER 2. DEEP tVATER GRAVITY WAVES оо 2 4 6 kD 8 10 Figure 2.10: Scaled phase and group velocities for surface waves. In fact, both group алd phase speeds are greatest алd equal in the long wave (shallow water) limit, when с, Cg - ,/riD kD-o. (2.30) Note that in this limit с becomes independent of k, i.e., the waves become nondispersive, as we saw in the shallow water case. However, as Figs. 2.9 алd 2.10 make clear, с and с9 differ significantly for kD � 1. In the short wave (deep water) limit, in fact, from (2.21), 00. (2.31) The difference between nondispersi ve long waves and dispersive short waves is illнstrated in the following. Consider an initial disturbance to the vvater surface of the form If Л is large, this has а length scale long compared with D (so it projects primarily onto waves with kD < 1) and, as shown in Fig. 2.11 for Л 4, the waves emanating from the distшbance are essentially noпdispersive. 2.4. SURFACE WAVE DISPERSION h' 1,00 0,75 о 0,50 0,25 10 -0,25 20 x/D 30 -0,50 -0,75 •1,00 Figшe 2.11: Dispersion of а large-scale initial distшbance 2 [ехр (x/4D) )on water of depth D. Behavior is symmetric about х О; only х > О is shown. Numbers on cшves are time in units of D / с0. (There is just а hint of dispersion; note the negative tail at t 20D/ с0. Note also that there is, for t > О, an identical distшbaпce, not shown, in х < О.) \ivЪen Л is smaller however, tl1e iпitial disturbance has а smaller length scale and hence а greater projection onto the dispersive short waves. This is illustrated in Fig. 2.12, for which the initial disturbance Л 0.5. CHAPTER 2. DEEP i-VATER GRAVITY WAVES 1.00 h' 0.75 о 0.50 0.25 20 20 -0.25 х/0 -0.50 .{).75 -1.00 Figure 2.12: Same as the previous figure , but for а small-scale initial distur­ bance [ехр(- (2х/ D)2 )]. The dispersion of the resulting ,vaves is evident. Note especially that (e.g., at t = 20D/c0) the leading edge of the wave train h.as the largest wavenumber, consistent with our observation that the long waves have the greatest group velocity and, in fact, should travel а distance х 20D Ьу this time). Wavelength becomes progressively shorter in the tail of the wave train. Both exaшples can easily Ье reproduced in а vv.1.,.,a,н.1x,.1. of water (one tliat is large enough to allow the dispersion to develop) or outdoors in а river or lake. If the water is sufficiently shallow, in response to а localized disturbance (e.g., from а small object dropped into the water) you will see а localized wave propagating away, with little or no apparent dispersion. If the water is deep, however, you will see а dispersing wave train, with the longest waves at the leading edge. If you look closely, you may Ье аЫе to see the wave crests appearing in the rear and propagating forward through the wave train (since с> с9 )5 . This effect can also Ье seen in а ship's wake (Fig. 2.13): 5Though in this case they never oveгtake the front of the wave train, since this consists of long ,vaves for which с= с9 . 2.4. SURFACE WAVE DISPERSION Figure 2.13: Waves in the wake of а sЬip. individual wave crests can Ье seen propagating to tЬе outside of the wake (cЬanging wavenumber in the process), and disappeariпg there. (Incidentally, in deep water the half-angle of the "wedge" made Ьу the wake is arcsin(½) 19.5° , independent of the speed of the ship. This сап Ье shown to follow frorn the fact that с9 / с = ½ for deep water; vve will not pursue that here but the proof can Ье found in many texts, such as "Waves in Fluids", Lighthill, Cambridge U Р, 1978; section 3.10.) The results that с9 S: с is made apparent in another common phenom­ enon, the wave train downstream of an obstacle in fiowing water (e.g., а river), as shown in Fig. 2.14. Figure 2.14: Disturbance produced Ьу fiow over an obstacle. There are two key statements to Ье made: (i) the wave train is stationary, апd so has с О·' CHAPTER 2. DEEP WATER GRAVITY WAVES (ii) it is located downstream-there is no disturbance ahead of the obstacle. Since our analysis has been for waves оп stationary water, let's shift our frame of reference to rnove with the flow. So now the obstacle and its wave train are пюving to the left with speed V. Since the wave crests are stationary with respect to the obstacle, they are also rnoving to the left with phase speed, in this frame of reference, с' V. Now, the energy is radiated from the moving obstacle radiates away at the group velocity, which is never greater than lc'I V--------so it must lag behind the rnoving obstacle. То put it another way, this moving frame of reference, jc�I � V (remember that с9 is negative in this case). So, in the frame in which the obstacle is stationary, the group velocity is с9 = с� + V � О. Thus, vvave energy can only radiate downstrearn and there are no upstream effects6. There is another aspect to this proЫern that is illustrative of tlie general characteristics of vvaves in fluids. The obstacle is subject to а force associ­ ated ,vith the flow across it. For а small, srnooth obstacle, this force is not primarily frictional (though there is а component of that) but is rnostly the result of form drag: the pressure оп the upstream side of the obstacle is greater than that оп the downstream side, and so the obstacle is subjected to а force to the right. (Оп F ig. 2.14, the free surface height is greater upstrearn of the peak of the obstacle than at а cornparaЫe position dowпstr-eam; ther-e­ fшe there is а positive pressure gradient on the obstacle.) Sirnultaneously, of course, the water mпst Ье subjected to an eqпal fшсе to the left (decelerating the fio,v). However, this is felt, not at the obstacle but downstream, within the wave train: this can happen Ьесапsе (like, e.g., electromagnetic waves) the waves, which as we have seen are сараЫе of transporting energy, can also effect transport of momentum. Thus, tl1e drag on the obstacle is relayed to remote parts of the water. This has several ramifications for atmospheric and oceanic dynamics. 6 The fact that this result is not trivial is underlined Ьу the observation that it does not al,vays apply. Capillary waves�those for which surface tension is crucial-can have cg > с анd so can and do travel upstream. If the obstacle is sшall а few сш in size) tl1ese waves а1:е important though, as they have sшall wavelength, they dissipate quickly and so may Ье hard to see. . ,,_ 2.5. PARTICLE Л10TIONS vVITHIN А WAVE 2.5 Particle motions within а wave Consider now the motion of (neutrally buoyant) marked particles in the wa­ ter, w h ich is of course t he same thing as the motion of the water itself. Let the instantaneous position of а particle Ье (х, z) (Х, Z) + (r/, (') ,vhere (Х, Z) is the undisturbed position (where the particle would Ье in the ab­ sence of t he wave), and r}' and (' are t he small perturbations in position associated with the wave motion. From the definition of velocity as rate of change of position, we have dx dt dz dt or (since (Х, Z) is fixed in time) и, w·' дr}' , дr/ , дr/ дt dz дх д(' д(' д(' -+и-+wдt дх dz -+и-+w- - и', f f w'. Linearizing, we have дr}' и' ., дt д(' w . дt Now, suppose we have а single propagating wave (single wavenumber, s ingle frequency). Then, from (2.13), (2.14), and (2.18), we have f 1 и1 (w Tlierefore Note: ) =R i �osh kz Q [-� р O ( sшh kz ) ei (kx-wt)] (2.32) Re [!':._Qo ( �osh kz ) ei (kx-wt)] i sш h kz wp (2.33) e 1. The displacements are oscillatory, so there is no net drift of the parti­ cles. Thus, even though tl1e wave pattern propagates, fluid parcels do not: they merely oscillate about their mean position. о CHAPTER 2. DEEP WATER GRAVITY WAVES с Figшe 2.15: Parcel orЬits in а water wave. Direction of motion is reversed for а wave traveling to the left. 2. Vertical and horizontal displacements are in quadrature ( 1r /2 out of phase). The marked particles perform elliptical orЬits, with, for real Q 0 > О (which we can insist on, Ьу defining the time origin accordir1gly) and k > О, ( r/ ) (' kQo coshkz cos(kx =_ [ wp х sinh kz sin(kx wt) wt) ] ' which implies that the parcels move clockwise aroнnd orЬit, for w > О (с > О) , and anticlockwise for w < О (с < О). See Fig. 2.15. Note that the ratio of vertical to horizontal axes of the ellipse is tanh kz, which increases from zero at the bottorn (where the boundary ensшes that (' -+ О) to tanh kD < l at the top. For waves in deep water (kD » 1), the orЬits become circнlar. 2.6 Wave generation Ьу wind It is common experience that waves on the ocean and lakes are usually weak on calm days, but strong on windy days, suggesting that much of the sшface wave activity is somehow produced Ьу the action of ,vind. It seems unlikely, though, that а wave of wavenumber k and fiequency w is directly forced Ьу winds as this would require that the wind itself (or the pressшe fluctua­ tioпs that accompany the wind) have а significant component at the same 2.6. WAVE GENERATION ВУ WIND frequency and wavenumber. There will always Ье some such component­ especially in the turbulent atmospheric boundary layer, the winds have а rich spectrum-but, at the short wavelengths and high frequencies cliaracteristic of surface water waves, these fiuctuations are generally weak. Moreover, laЬ­ oratory studies show that waves can Ье generated Ьу Ьlowing а steady air fiow across а water surface. (Тrу Ыowing across а glass of water.) The underlying process common to most wave generation is one of insta­ Ьility. That is, even though there may Ье no externally imposed "waviness" in the wind, there is а tendency to amplify any small perturbation on the water surface. Consider Fig. 2.16. Figure 2.16: Wave generation Ьу wind (schematic). А wave on the ocean sur­ face disturbs the air flow in such а -..,vay а to produce а pressure perturbation at the surface that reinforces the wave. In the absence of а surface wave, the air fiow is uniform and the pressure on the water surface is uniform; there is thus no tendency to force waves in the water. Ho\vever, in the presence of а small wave on the surface, the air fiow is disturbed and, like the "rock-in-the-river" proЫem7 , а perturbed pressure gradient is produced at the water surface which has the same wavenumber ( and frequency) as the surface perturbation. Provided the ,vater-air system сап get the phase relationships right (and it can) these pertшbations rein­ force will each other and grow, thus producing, eventually, а finite-amplitude surface wave from an initially infinitessimal perturbation. 7Except now the "госk" is the bump on the water surface and the "гivег" is the at­ mospheгe. CHAPTER 2. DEEP WATER GRAVITY WAVES 2.7 Wave breaking All our discussion thus far has been based on our linear (small-amplitude) approximation to the full proЫem. Using this approximation, we have been аЫе to explain many of the commonly observed properties of surface water waves. There is, however, at least one familiar aspect of these waves that cannot Ье explained Ьу linear theory: breaking, which is most commonly observed as waves ran up а beach. Figure 2.17: Steepening of а finite amplitude wave. This happeпs for two reasons. First, as the waves run up the beach, the energy in the \vave becomes focused iпto а shallower layer (once D :::; or so), thus concentrating the energy апd iпcreasing the wave amplitude. Second, when D:::; k- 1 or so, the phase speed becomes dependeпt on depth, being greater where the water is deeper. In а finite amplitude wave, the water is deeper at the wave crest than iп the wave trough. Hence the crests travel faster than the troughs; the crest therefore tend to catch up with and, eventually, overtake the troughs, as shown in Fig. 2.17. This produces tlie overturпing of the wave that is familiar in breakers. 2.8 Further reading See the suggestions given in Section 1.6.