KUNDU Fluid Mechanics 2 Episode 6 ppsx

In this section we shall discuss gravity waves at the free surface of a sca of liquid of uniform depth H, which rmiy be large or small compared to the wavelength h... The simplest case i

of a wave is Doppler.shifted by an amount U 9 K due to the mean flow Equation (7.20)

is easy to uiderstand by considering a situation in which the intrinsic frequency w is zero and the flow pattern has a periodicity in the x direction of wavelength 2n/k If this sinusoidal pattern is translated in the x direction at speed U , then the observed frequency at a fixed point is OJO = Uk

The effects of mean flow on frequency will not bc considered further in this

chaptcr Consequently, thc involved frequencies should be interpreted as thc intrinsic frcquency

In this section we shall discuss gravity waves at the free surface of a sca of liquid of uniform depth H, which rmiy be large or small compared to the wavelength h We shall assume that thc amplitude a of oscillation of the free surrace is small, in the sense

that both a / h and a / H are much smallcr than one The condition a / h << 1 implies that the slope of the sea surface is small, and the condition u / H << 1 implies that the instantaneous depth does not differ significantly from the undisturbed depth Thesc conditions allow us to linearize the problem The frequency of the waves is assumed large compared to the Coriolis frequency, so that the waves are unaffected by h e earth's rotation Hem, we shall neglect surface tension; in water its effect is limited

to wavelengths (7 cm, as discussed in Section 7 The fluid is assumed to have small

viscosity so that viscous effects are confined to boundary layers and do not affect the wave propagation significantly The motion is assumed to be generated from rest, say,

by wind action or by dropping a stone According to Kelvin's circulation theorem, rhe resulting motion is irivtariontil, ignoring viscous effects, Coriolis forces, and

stratification (density variation)

Formulation of the Problem

Consider a case where the wavcs propagate in the s direction only, and that the

motion is two dimensional in the xz-planc (Figure 7.4) Let the vertical coordinate z

be measured upward froin the undisturbed free surface The free surface displacement

is q ( x r ) Because the motion is ii-rotational, a velocity potential 4 can be defined

such that

Substitution into the continuity equation

gives the Laplace equation

Boundary conditions are to be satisfied at the [ne surface and at thc bottom The

condition at the bottom is zero n o d velocity, that is

at the free surface The forementioned condition can be written as

(7.25)

For small-amplitude waves both it and aq/a-r are small, so that the quadratic term

u ( a q / a x ) is one order smaller than other terms in Eq (7.25), which then simplifies to

(7.26)

We can simpllfy this condition still further by arguing that the righl-hand side C N ~ be evaluated at z = 0 rather than at lhc free surface To justify this, expand 8qb/az in a Taylor scries around z = 0:

Therefore, to the first order of acciuacy desired hen, a$/az in Eq (7.26) can be evaluated at z = 0 We then have

In addition to the kinematic condition at the surface, there is a dyncimic condition that the pressure just below the free surfacc is always equal to the ambient p i ~ m r e , with surface tension neglected Taking the ambient pressurc to be zero, the condition is

Equation (7.28) follows from thc boundary condition on t n, which is continuous

across an interface as established in Chapter 4, Section 19 As before, we shall simplify

this condition for sinall-amplitude waves Since the motion is irrotational, Bernoulli's

cquation (see Eq (4.81))

22 + i ( u 2 + 1 0 2 ) + + gz = F ( t ) ,

is applicable Here, the function F ( t ) can be absorbed in a # / a t by redefining 4

Neglecting the nonlinear term (u' + w') for small-amplitude waves, the linearized form of the unsteady Bernoulli equation is

As b e h e , for small-amplitude waves, the term &$/at can be evaluated at z = 0

rather than at z = r ] to give

- - g r ] at z = 0

a4

a t _ -

Solution of the Problem

Recapitulating, we have tq solve

IJI order to apply the boundary conditions, we need to assume a form for q ( x 1) The simplest case is that of a sinusoidal component with wavenumber k and frequency w,

lor which

One motivation for studying sinusoidal waves is that small-amplitude waves on a water surface become roughly sinusoidal some time after their generation (unless the water depth is very shallow) This is due to the phenomenon of wave dispersion discussed

in Section 10 A second, and stronger, motivation is that an arbitrary disturbance can be decomposed into various sinusoidal components by Fourier analysis, and the mpoiise of the system to an arbitrary small disturbance is the sum of the responses

to the various sinusoidal components

For a cosine dependence of q on (kx - ot), conditions (7.27) and (7.32) show that q5 must be a sine function of (kx - at) Consequently, we assume a separable solution of the Laplace equation in the form

The vclocity potential is then

q5 = (Ae" + Be-") sin(kx - wt) (7.33

The constants A aud B are now determined from the boundary conditions (7.24) and

by a Taylor serics expansion

Substitution of Eqs (7.33) and (7.35) into the surface velocity condition (7.27)

gives

(7.37)

k(A - E ) = (IO

The constants A and B can now be determincd from Eqs (7.36) and (7.37) as

The vclocity potential (7.35) then becomes

from which the velocity components are found as

sinhk(z + H) sinh k H

111 = UW sin(ks - or)

(7.38)

(7.39)

We have solved the Laplace equation using kinematic boundary conditions alone

This is typical of irrotational flows In the last chapter we saw that the equation of

motion, or its integral, thc Bernoulli equation, is brought into play only to find the prcssurz distribution, after h e problem has bcen solved from kincinatic considerations alonc In the present case, we shall find that application of the dynamic free surface

condition (7.32) gives a relation between k and w

Substitution of Eqs (7.33) and (7.38) into (7.32) gives thc dcsired relation

Thc phase speed c = w / k is related to the wave sizc by

I

~

This shows that the speed of propagation of a wave component depends on its

wavenumbcr Waves for which c is a function of k arc called dispersive because waves of different lengths, propagating at dZFerent spmds, “dispersc” or separate (Dispersion is a word borrowed from optics, whcrc it sigilifies separation of different colors due to the speed of light in a medium dcpending on thc wavelength.) A relation such as Eq (7.40), giving w as a function of k, is called a dispcwion relation because

it expresses the nature of the dispersive process Wave dispersion is a €undamental pmccss in many physical phenomena; its implications in gravity waves are discussed

in Scctions 9 and 10

5 Sornc? l~bt~~r~rurx of Sutfacc C m L i i Q - H%t~.?t?s

Scvcral featurcs 01 surface gravity wavcs are discussccl in tlus scction In particular,

we shall examine thc nature of pressure change, particlc motion, and the energy flow duc to a sinusoidal propagating wave Thc water depth H is arbitrary; simplitications that result from assuming the depth to be shallow or deep arc discussed in the next scction

Pressure Change Due to Wave Motion

It is sometimes possible to measure wave parameters by placing pressure sensors at the bottom or at some other suitable depth One would theEfore like to h o w how deep the pressure fluctuations pcnetrate into the water Pressure is given by the linearized Bernoulli equation

as theperturbation pressure, that is, the pressure change fromthe undisturbed pressure

a4

p l = -p-

at

On substituting Eq (7.38), we obtain

Paw2 cash k(z + H) cos(kx -

to the wavelength This is discussed further in Section 6

Particle Path and Streamline

To examine particle orbits, we obviously need to use Lagrangian coordinates (See, Chapter 3, Section2foradiscussionof theLagrangiandescriptionJLet ( x o + ~ , ZO+ f )

be the coordinates of a fluid particle whose rest position is ( X O , ZO), as shown in Fig- ure 7.5 We can use ( X O , ZO) as a “tag” for particle identification, and write &o, ZO, t )

and ((.TO, zo, r ) in the Lagrangian form Then the velocity components are given by

which rcpresents cllipses Both the semimajor axis n coshIk(z0 + H)]/sinh kH and

the semiminor axis a sinh[k(zo + fl)]/siiih RH decrcase with dcplh, the minor axis vanishing at LU = -H (Figurc 7.6b) Thc distance between foci remains constant with depth Equation (7.46) shows that thc phase of the motion (that is, thc argument

of thc sinusoidal term) is independent of zo Fluid particles in any vertical column arc therefore in phase That is, if onc of !hem is at the top of its orbit, then all particles at the same .VI) are at the top of their orbits

To find thc streamlinc pallern wc need to dctermiue thc streamfunction @? related

to the velocity components hy

Figure 7.6 Particle orbits of wavc motion in deep, intermediate and shallow seas

where Eq (7.39) has been introduced Integrating Eq (7.48) with respect to z,

we obtain

a o sinh k ( r + H)

cos(kx - ot) + F ( x , t ) ,

' = T sinhiiH

where F ( x , t ) is an arbitrary function of integration Similarly, integration of

Eq (7.49) with respect to J gives

' = - a" sinh k ( r -k

k sinhkH cos(kx - ot) + G(z, t ) , where G ( z , t) is another arbitrary function Equating the two expressions for @ wc see that F = G = h c t i o n of time only; this can be set to zcro if we regard $ as due

to wave motion only, so that 3 = 0 when a = 0 Therefore

aw sinhk(z + H)

e = - cos(kx - or)

k sinhkH Let us examine the streamline structure at a particular timc, say, t = 0, when

$ o sinhk(z + H)coskx

(7.50)

It is clear that $ = 0 at z = -H, so that the bottom wall is a part of the $ = 0 streamline However, $ is also zero at kx = f17/2, f3n/2, €or any z At these

T

u u

= O

Figure 7.7 Instantaneous strcanlinc pattern in ;s d a c c gravity wivc pmpagating LO thc right

values of k x , Eq (7.33) shows that q vanishes The resulting stremiline pattern is

shown in Figure 7.7 It is seen that the vebcio is in the direction qfpmpugation (and

horizontal ) ut all depths below the crests, rmd opposite to the direction qfpropagurioii

at all depths below truugh

Energy- Considerations

Surface gravity waves posscss kinetic encrgy due to motion of the fluid and potcntial

energy due to dcIbnnation of the free surface Kinetic energy per unit horizontal area

is found by integrating over the dcpth and avcraging over a wavelength:

Here the z-integral is taken up to : = 0, because the integral up to z = q gives a

highcr-order tcrm Substitution of thc velocity components from Eq (7.39) gives

+, Jd u2 sin2(kx - ut) d x lH sinh2k(z + H ) d z ] (7.51)

In tcrms of frcc su~facc displacemcnt q the x-integrals in Eq (7.5 I ) can be written as

a2 cos2(kx - w t ) d.r = a’ sin2(kx - w t ) dx

where 3 is the mean square displacement The z-integrals in Eq (7.51) are easy to evaluate by expressing the hyperbolic functions in terms of exponentials Using thc dispersion relation (7.40), Eq (7.51) finally becomes

-

which is the kinetic energy of the wave motion per unit horizontal area

Consider next the potenrid energy of the wave system, defined as the work done

to deform a horizontal fixe surface into the disturbed state It is therefore equal to the

djference of potential energies of the system in the disturbed and undisturbed states

As the potential energy of an element in the fluid (per unit length in y ) is pgz dx dz

(Figure 7.Q the potential energy of the wave system per unit horizontal area is

(7.53)

(An easier way to arrive at the expression for E, is to note that the potential energy increase due to wave motion equals the work done in raising column A in Figure 7.8

to the location of column By and integrating over halfthe wavelength This is because

an interchange of A and B over half a wavclength automatically forms a complete wavelength of the deformed surface The mass of column A is pq dx and the center of

gravity is raised by q when A is taken to B This agrees with the last form in Eq (7.53).) Equalion (7.53) can be written in terms af the mean square displacement as

(7.54) Comparison of Eq (7.52) and Eq (7.54) shows that the average kinetic and potential

energies are equal This is called theprinciple ofequipartition ofenergy and is valid in conservative dynamical systems undergoing small oscillations that are unaffected by

Figure 7.8 Cdculation of potential cnergy of a fluid column

planctary rotation However, it is not valid when Coriolis forces mz included, as will

be seen in Chapter 13 The total wave energy in the water columu per unit horizontal

m a is

(7.55) where the last form in terms of the amplitude u is valid if 9 is assumed sinusoidal, since the average of cos' x over a wave~ength is 1/2

Next, consider the rdtc of transmission of energy due to a single sinusoidal com- ponent of wavenumber k The energyJu.v across the vertical plane x = 0 is the pressnre work done by the fluid in thc region x < 0 on the fluid in the region x > 0 Per unit length of crcst, the time average energy flux is (writing p as the sum of a

pertiirbation p' and a background pressure -pgz)

The time average of cos'(kx - rut) is 1/2 The z-integral can be carried out by writing

it in tenns of exponcntials This tinally gives

(7.57)

The h s t factor is the wave energy given in Eq (7.55) Thereforc, the second factor must be thc speed of propagation of wavc energy of component k, callcd the group speed This is discusscd in Sections 9 and IO

6 , Ipprimirrintiims j&r llcep and Shallow Water

The analysis in the preccding section is applicable whatever the magnitude of )c

is in relation to the water depth H Inteizsling simplifications result for H / ) c << 1 (shallow water) and HIE, >> 1 (dcep water) The expression for phase speed is givcn

by Eq (7.41 j, namely,

(7.41) Approximations are now derived under two limiting conditions in which Eq (7.41) takcs simple forms

Deepwater Approximation

We know that tanhx * 1 for x + 00 (Figure 7.9) However, x need not be very large for this approximation to be valid, because tanhx = 0.94138 for x = 1.75 It follows that, with 3% accuracy, Eq (7.41) can be approximated by

(7.58)

for H > 0.28h (corresponding to k H > 1.75) Waves are therefore classified as

deepwater waves if the depth is more than 28% of (he wavelength Equation (7.58)

shows that longer waves in deep water propagate faster This feature has interesting consequences and is discussed fuaher in Sections 9 and 10

A dominant period of wind-generated surface gravity waves in the Ocean is 10 s,

for which the dispersion relation (7.40) shows that the dominant wavelength is 150 m The water depth on a typical continental shelf is e 100 m and in the open ocean it

is about -4 km It follows that the dominant wind waves in the Ocean, even over the

continental shelf, act as deep-water waves and do not feel thc effects of the ocean bottom until they arrive near the beach This is not true of gravity waves generated by tidal forces and earthquakes; these may have wavelengths of hundreds of kilometers

In the preceding section we said that particle orbits in small-amplitude gravity waves describe ellipses given by Eq (7.47) For H > 0.28A, the semimajor and

Y

Figun! 7.9 Bchavior of hyperbolic functions

2

semiminor axes or these ellipses each bccome ncarly equal to -aekZ This €allows from thc approximation (valid fork H > 1.75)

coshk(z + H) - sinh k(z + H ) ru - ek: .sinh kH sinh k H

(Thc various approximations for hyperbolic functions used in this section can easily be vcrified by writing them in tenus or exponeiitials.) Thcrerore for deep-water waves particle orbits described by Eq (7.46) simplify to

= a ek:l) sin(kx0 - of)

C = CI ek" cos(k.uo - of)

The orbits are themfore circlcs (Figure 7.6a), of which the radius at the surface equals

u , the amplitude of the wave The velocity components are

a t

at

aJ'

a t

I I = - = am& cos(kx - ut)

U I = - = umekz sin(ii:r - or),

whcre we havc omitted thc subscripts on (xu, io) (For sinal1 amplitudes the difference

in velocity at the present and mcanpositions of a pmicle is negligible The distinction between mean particle positions and Eulerian coordinates is therefore not necessary, unless finitc ainylitudc effects are considcred as we will see in Section 14.) The vclocirj vcctor therefore rotatcs clockwise (for a wave travcling in the positive x dircction) at kqueiicy o, while its magnitude remains constant at 1 1 ~ ~ ) e ~ ~ ~ l

For deep-water waves, the perturbation pressure given in Eq (7.44b) simplifies to

This shows that pressure clmgc due to the presence of wavc motion dccays exponen- tially with depth, reaching 4% of its surface magnitude at a depth of A/2 A scnsor placcd ai the bottom cannot thercrore detcct gravity waves whose wavelengths are smallcr than twice the water depth Such a sensor acts like a "low-pass filer,'' retaining

longer waves aiid Ejecting shorter ones

212 Cmuity I#ht.w

The approximation gives a better than 3% accuracy if H < 0.07A Surface waves are

therefore regarded as shllow-wurer wuves if the water depth is <7% of the wave-

length (The water depth has to be really shallow for waves to behave as shallow-water waves This is consistent with thc comments made in what follows (Eq (7.58)), that

the water depth does not have to be really deep for water to behave as deep-water

waves.) For these waves Eq (7.60) shows that the wave speed is independent of wavelength and increases with water depth

To determine the approximate form of particle orbits for shallow-water waves,

we substitute the followi& approximations into Eq (7.46):

These represcnt thin ellipses (Figure 7.6c), with a deplh-independ-lit semim jor axis

of a / k H and a semiminor axis of a(l + z / H ) , which linearly decrcases to zero at the bottom wall From Eq (7.39), the velocity field is found as

p' = pga cos(k:r - or) = pgq,

where Eq (7.33) has been used to express the pressure change in terms of q This

shows that the pressure change at any point is independent of depth, and equals the hydrostatic increase of pressure due to the surface elevation change q The pressure jield is therefore complerely Iiydrosmic in shullow-wafer waves Vertical accelera-

tions are negligible because of the small w-field For this reason, shallow water waves

are also called hydivstciric wcives It is apparent that a pressure scnsor mounted at thc

bottom can sense thesc wavcs

Wave Refraction in Shallow Water

We shall now qualitatively describe the commonly observed phenomenon of refmc-

smaller Consequently, the crest lines, which are pcrpendicular to the local direction

of c, tend to become parallel to the coast This is why we see that the waves coming toward the beach always seem to have their crests parullel to the caardine

An interesting example of wave refraction occurs when a deep-water wave with

straight crests approaches an island (Figure 7.11) Assume that the water depth

becomes shallower as the island is approached, and the constant depth contours are

circles concentric with the island Figure 7.1 1 shows that the waves always come in

towurd the island, even on the “shadow” side marked A!

The bcnding of wave paths in an inhomogeneous medium is called wuve refriic-

analogous phenomenon in optics is the bending of light due to density changes in its path

It was cxplained in Section 1.5 that the interface bctween two immiscible fluids is in a state of tension T h e tension acts as a restoring force, enabling the interface to support waves in a manner analogous to waves on a stretched membrane or string Waves due

to the presence of surface tension are called capillary waves Although gravity is not

nceded to support these waves, the existence of surface tension alone without gravity

is iincommon We shall therefore examine the modification of the preceding results for pure gravity waves due to the inclusion of surface tension

I I

Fiyre 7.11 Kcfraction of a surrace gravity wave approaching an island with sloping bmh Crest lincs perpeiiBcular to the rays are shown Note flint h e crest lines comc in toward thc island cvcii on thc shadow side A Reprinted with rhepemiission uf Mr.x Dorvthy Kinsmun Broiun: B Kinsman wild Waver

Prenticc-Hall Englewood Cliffs NJ, 1965

Figure 7.12 (a) Segment of a rree surface under he action of surface tension; (b) nct surkcc tcnsion kme on an clement

Let PQ = ds be an element of arc: on the free surfacc, whosc local radius of curvature is r (Figure 7.12a) Suppose pa is the pressure on the “atmospheric” sidc,

and p is the pressure just inside the interface The surface tension forces at P and Q,

per unit length pcrpendicular to the plane of the paper, are each cqual to cr and directed along the tangents at P and Q Equilibrium of forces on the arc PQ is considered in

Figure 7.12b The force at P is represented by scgment OA, and the force at Q is represented by segment OB The resultant of OA and OB in a direction perpendicular

to the arc PQ is reprcsented by 2 0 C 21 o d e Therefore, the balance of forces in a

direction perpendicular to the arc PQ requires

It follows that the pressure difFerencc is related to the curvature by

de cr

ds r

pa - p = 0- = -

The curvature l/r of q(x) is given by

when the approximate expression is for small slopes Therefore,

a29 ax2

pa - p = cr-

Choosing the atmospheric pressure Pa to be zero, we obtain the coiidition

Using the linearized Bernoulli equation

found by substitution of (7.33) and (7.38) into (7.64), to give

w = ,/k ( g + $) tanh kH (7.65)

Thc phase velocity is therefore

(7.66)

c- = /(E + $) tanh kH = ,/( + %) tanh 21r H

A plot of Eq (7.66) is shown in Figure 7.13 It is apparent that the eflect of surface

tension is to increase c above its value for pure gravity waves at all wavelengths This is because the free surface is now “tighter,” and hence capable of generating more restoring forces However, the effect of surface tension is only appreciable

4n

Figure 7.13 Sketch of phase velodty vs wavelength in B surfacc gravity wave

for very small wavelengths A measure of these wavelengths is obtained by noting that thm is a minimum phase speed at A = A,, and surface tension dominates for

A < A, (Figure 7.1.3) Setting d c / h = 0 in Eq (7.66), and assuming the deep-water approximation tanh(2aHlA) 2 I valid for H > 0.28A, we obtain

(7.67) For an &-water interface at 20 "Cy the surface tension is u = 0.074 N/in, giving

= 23.2 cm/s at A,,, = 1.73 cin (7.68) Only small waves (say, A < 7 cm for an air-wakr interface), called ripples, arc there- fore affected by surface tension Wavelengths t 4 rnm are dominated by surface ten- sion and are rather unaffectcd by gravity From Eq (7.66), the phase speed of these

pure cupillnr~ wai~es is

(7.69)

where we have again assumed tanh(2nHlh) 2: 1 The sinallcst of these, traveling

at a relatively large speed, can be found leading the waves generated by dropping a

stone into a pond

8 SEarrding Nams

So far, we have been studying propagating wwcs Nonpropagating waves can be gen- erated by superposing two waves of the same amplitudc and wavelength, but moving

in opposite directions The resulting surface displacement is

C O S ( ~ X - U t ) + u cos(kx + w t ) = 2n COS kx COS wt

Tt follows that q = 0 for k.r = f n / 2 , H n / 2 Points of zero siirface displacement

are called ~iudcs The free surface therefore does not propagate, but simply oscillates

up and down with frequency w, keeping the nodal points fixed Such waves are cdkd

srunding waves The corresponding streamfunction, using Eq (7.50), is both €or the cos(kx - ut) and cos(kx + wr) components, and for the sum This gives

A limited body of water such as a lake forms standing waves by reflection from

the walls A standing oscillation in a lake is called a seiche (pronounced “saysh”),

in which only ccrtain wavelengths and frequencies w (eigenvalues) are dowed by

the system Let L be tbe length of the lake, and assume that the waves are invariant

along y The possible wavelengths arc found by setting u = 0 at the two walls Because u = a+/az, Eq (7.70) gives

F i y e 7.14 Instantaneous stmainline paltcm in a standing surliice gravity wave If this is rnodc n = 0

ihcn two succcssive vertical stredincs are a dirlance L apart If this is rnodc n = I thcn lhe first and

third vcrt.icp;I srreamlines are distance L apart

-*

Figure 7.15 Normal modcs in a lab, showing dirtrihutions of u for h e first two modes This is consistent

with thc streamliiie pattern of F i p 7.14

The largest wavelength is 2L and the next smaller is L (Figure 7.15) The allowed

frequencies can be found from the dispersion relation (7.40), giving

(7.73)

which are the natural frequencies of the lake

9 Gmup I4?locidy and Energy Flux

An interesting set of phenomena takes place when the phase speed of a wave depends

on its wavelength The most common example is the deep water g m 7 i l - y wave, for which c is proportional to a A wave phenomenon in which c depends on k is called

dispemive because, as we shall see in the next section, tbe different wave components separate or ‘‘disperse“ from each other

In a dispersive system, the energy of a wave component does not propagate at the phase velocity c = w / k , but at the group velocity defined as cg = d o / d k To see

this, consider the superposition of two sinusoidal components of equal amplitude but slightly different wavenumber (and consequently slightly Werent frequency because

w = w ( k ) ) Then the combination has a waveform

r] = u cos(k1x - Ulf) + u cos(k2x - W t )

Applying the trigonometric identity for cos A + cos B, we obtain

9 Chup hliu!i@- tuut E t i ~ ~ p t - l k :

which has a large wavelength 4sr/dk, a large period 4sr/dw, and propagates at a spccd

(=wavelengWperiod) of

219

(7.75)

Multiplication of a rapidly varying sinusoid and a slowly varying sinusoid, as in

Eq (7.74, generates repeating wave groups (Figure 7.16) The individual wave com-

ponents propagate with the speed c = w / k , but the envelope of the wave groups

travels with the speed cg, which is therefon: called the group velocity Tf cg < c

then the wave crests seem to appear b i n nowhere at a nodal point, proceed forward

through the envclope, and disappear at the ncxt nodal point If, on the othcr hand

cg > c, then the individual wave crests secm to emergc from a forward nodal point

and vanish at a backward nodal point

Equation (7.75) shows that the group speed of wavcs of a certain wavenumber

k is given by the slope of the fangent to the dispersion curve w ( k ) Tn contrast, the

phase velocity is given by the slope of the radius vector (Figure 7.17)

A particularly illuminating example of the idea of group velocity is provided

by the concept of a ~ a v e packer formed by combining all wavenumbers in a cer-

tain narrow band Sk around a central value k In physical space, the wave appears

nearly sinusoidal with wavclength 2 x / k , but the amplitude dies m v q in a length of

Energy

Figure 7.18 A wave pnckct composed d ;L nmmw band or wavenumbcs Sk

order 1/Sk (Figure 7.18) Tf the spectral width Sk is narrow, then dccay of the wavc ,amplitude in physical space is slow The concept of such a wave packet is more real- istic than the one in Figure 7.16, which is rather unphysical because the wave groups

repeat themselves Suppose that, at some initial time, the wave group is represented by

q = a ( x ) coskx

Tt can be shown (see, for example, Phillips ( 1 977), p 25) that for small times the

subsequent evolution of the wave profile is approximately described by

q = a(x - c g t ) cos(kx - wc) (7.76) where cg = d o / d k This shows that the amplitude afa wuve packed rravels with the gr7)up speed It foUows that cg must equal the speed of propagation of energy of a certain wavelength The fact that cg is h c speed of energy propagation is also evident

in Figure 7.16 because the nodal points travel at cg and no energy can cross the nodal points

For surface gravity waves having the dispersion relation

the group velocity is found to be

2kH 1

" 2 - [I -k sinh2kH The two limiting cases are

cg = f c (deep water),

cg = c (shallow water)

(7.77)

(7.78)

The group velocity of deep-water gravity waves is h a t h e phase speed Shallow-watcr

waves, on the other hand, arc nondiupersive, for which c = cg For a linear nondis- persive system, any waveform preserves its shape in time because all the wavelengths that make up the wavcfonn travel at the same speed For a pure capillary wave, thc

p u p velocity is cg = 3c/2 (Exercise 3)