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Chap4 Waves Basics

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Part II PROCESSES 69
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Part II

PROCESSES

69

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Chapter 4

Waves

SUMMARY: The attention now turns toward specific types of motions that existin natural fluid flows, beginning with waves. All waves need a restoring force, andto every restoring force corresponds a different type of wave: surface water wavesunder the action of gravity, internal waves under the action of buoyancy, topographicwaves under the action of a vorticity gradient, etc. The study of wave dynamicsalso prepares for the study of instabilities, which in turn is a prelude to the studyof turbulence.

4.1 Surface Gravity Waves

4.1.1 Mechanism

Gravity waves on the surface of water are one of the most visible manifestationsof fluid motions and one with which we all have a certain experience (Figure 4.1).The process at work is relatively easy to comprehend: A fluctuation causes waterto rise above the equilibrium surface level, gravity pulls it back down because wateris heavier than air, inertia acquired during the falling movement causes the waterto penetrate below its level of equilibrium, and a bouncing motion results. Theoscillation is similar to that of a spring that has been stretched and released. The‘spring’ action in a surface water wave is gravity, hence the name of surface gravitywave.

What is somewhat less intuitive is why gravity waves propagate horizontally. To

understand this, one needs to consider the horizontal forces at play. When a parcelof water rises somewhere above the surface, the added weight of this water createsa pressure that is locally higher than normal, and this pressure anomaly accelerates(pushes, so to speak) the fluid away from that place and piles it up a little further,generating another surface rise some distance away. The net effect is a translationof the disturbance, hence a traveling wave.

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72   CHAPTER 4. WAVES 

Figure 4.1: Surface gravity waves on the sea approaching the coast. [Piha, NewZealand, photo  Practical Ocean Energy Management Systems, Inc.]

Water motion under a surface wave is very nearly oscillatory, with almost no netdisplacement. Thus, surface waves, like most other fluid waves, are a mechanismby which the fluid moves energy from one area to another without involving anysignificant movement of the fluid itself. Energy and information are carried with thefluid acting as the support medium rather than as the messenger. That propertyhas a fundamental implication: Surface waves by their very nature are unable totransport any mass, including dissolved pollutants and suspended matter. This factis clearly manifested in the behavior of a floating object (such as an autumn leaveon a pond) in the presence of surface waves: The waves pass by, but the object onlybobs up and down.

The energy carried by surface waves, however, must eventually be dissipatedsomewhere and will affect the water contents there. For example, wave energycan be converted into turbulent mixing under wave breaking, and the resultingmixing can stir the local water contents, such as pollutants, biological matter andheat. Wave energy can also be dissipated by bottom friction under wave-induced

oscillatory flow, and this friction can in turn create a shear stress sufficiently strongto entrain sediments into suspension. In sum, waves do not contribute directly totransport and redistribution of fluid-borne elements along their travel but can beeffective means by which a remote source of energy can affect the concentration of dissolved and suspended matter at a distant location. This remark holds true formost types of waves.

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4.1. SURFACE GRAVITY WAVES    73

4.1.2 Linearization

Because of the oscillatory motions that they generate, surface gravity waves can bereasonably well described by a linear analysis. This is mathematically justified byrestricting the attention to small wave amplitudes and weak accompanying motions.In the momentum equations (3.10)–(3.12), all terms linear in the velocity and pres-sure are then assumed to be small, while the nonlinear terms are assumed to be evensmaller and therefore negligible. Among other terms, we neglect the first advectiveterm  u∂u/∂x  next to the relative acceleration term  ∂u/∂t, under the assumptionthat the velocity, length and time scales,  U ,  L  and  T , meet the following criterion:

U 2

L  <<

  U 

or

U <<   LT 

  .   (4.1)

The ratio   L/T , representing a distance of influence over a characteristic time of the phenomenon, provides a scale for the wave speed. Hence a corollary of theapproximation made to justify the linearization of the equations is that we restrictour attention to wave motions in which the velocity of the fluid is much smallerthan the speed of propagation of wave crests.

Example 4.1

Consider a 20-m long wave traveling on the surface of a 5-m deep water body.The shallow-water wave theory below tells that the period of this wave is the wave-length (20 m) divided by the wave speed (

√ gH   = 7.00 m/s), which is equal to

2.86 seconds. If the wave amplitude (height from crest to trough) is 20 cm, themaximum water velocity is  U  = 14 cm/s. This is indeed significantly smaller thanthe ratio L/T  = (20 m)/(2.86 s) = 7.00 m/s, the wave speed. The wave propagatesmuch faster than the water moves.

4.1.3 Theory

Surface waves owe their existence to the large density difference between waterand overlying air. We can therefore study gravity waves on the surface of a body of water without considering that either the water below or the air above are stratified.Mathematically, we choose to take the density of the water equal to  ρ0   everywhereand that of the air as zero.

As we shall note   a posteriori , vertical accelerations can be important in grav-ity waves at high frequencies. Indeed, when the frequency is high, fluid particlesexecute rapid vertical oscillations, and the vertical acceleration (∂w/∂t) to whichthey are subjected may be significant compared to the gravitational acceleration(g). Thus, we ought to use the non-hydrostatic form of the governing equations.After linearization, these equations are:

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74   CHAPTER 4. WAVES 

Figure 4.2: Sketch of a surface wave and the attending notation.

∂u

∂t  =   −   1

ρ0

∂p

∂x  (4.2)

∂v

∂t  =   −   1

ρ0

∂p

∂y  (4.3)

∂w

∂t  =   −   1

ρ0

∂p

∂z  −   g   (4.4)

∂u

∂x  +

  ∂v

∂y  +

  ∂w

∂z  = 0.   (4.5)

This set of equations forms a 4–by–4 system for the flow variables   u(x,y,z,t),v(x,y,z,t),  w(x,y,z,t) and  p(x,y,z,t). An accessory variable is the surface eleva-tion a(x,y,t) (Figure 4.2).

At the top of the fluid layer, along the deformable surface (z =  H +a), we imposethat fluid parcels move with the surface (w =  da/dt =  ∂a/∂t + u∂a/∂x + v∂a/∂y)and that the pressure is atmospheric [ p(z  =  H + a) =  pa]. Along the bottom, whichwe take as horizontal (z  = 0), the boundary condition is that there is no verticalvelocity (w = 0). After linearization (|a| ≪ H ), the boundary conditions become:

z  =  H    :   w =  ∂a∂t

  and   p =  pa + ρ0ga   (4.6)

z  = 0 :   w = 0 (4.7)

where H  is the height of the undisturbed surface, and  a(x,y,t) the surface displace-ment caused by the wave.

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4.1. SURFACE GRAVITY WAVES    75

The linear equations permit us to seek a periodic solution of sinusoidal shape.Taking λ  as the wavelength (the distance from one crest to the next crest at a given

time) and   T   as the period (the interval of time during which a crest travels onewavelength), we introduce for convenience the   wavenumber 

k   =  2π

λ  (4.8)

and the  angular frequency 

ω   =  2π

T   .   (4.9)

Then, we seek a wave solution of the form:

u(x,z,t) =   U (z) cos(kx − ωt) (4.10)v(x,z,t) = 0 (4.11)

w(x,z,t) =   W (z) sin(kx − ωt) (4.12)

 p(x,z,t) =   pa   +   ρ0g(H − z) +   P (z) cos(kx − ωt) (4.13)

a(x, t) =   A   cos(kx − ωt).   (4.14)

With this notation, the vertical displacement of the surface varies between −A   ina trough to +A  at a crest. The choice of  w   in quadrature with the other variablesis to ensure a match of trigonometric functions after substitution in the equations.To simplify the analysis, we are not considering variations in the second horizontaldirection (y-direction) and any transverse velocity (v = 0).

Substitution in the governing equations yields the reduced equations:

ωU (z) =  k

ρ0P (z)

ωW (z) =  1

ρ0

dP (z)

dz

−  kU (z) +  dW (z)

dz  = 0,

and subsequent elimination of  U (z) and W (z) yields a single equation for the pres-sure amplitude  P (z):

d2P 

dz2

  =   k2P,

of which the solution is:

P (z) =   P 1   e+kz +   P 2   e−kz ,

with  P 1  and P 2  two constants of integration to be determined.

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76   CHAPTER 4. WAVES 

The boundary condition at the bottom demands  W (0) = 0, that is,  dP/dz  = 0at   z   = 0, which yields   P 1   =   P 2. Then, the boundary conditions at the surface

impose:

W (H ) =   ωA   =⇒   kP 1   e+kH  −   kP 1  e−kH  =   ρ0ω2A

P (H ) =   ρ0gA   =⇒   P 1  e+kH  +   P 1  e−kH  =   ρ0gA.

In the presence of a non-zero amplitude (A = 0), the ratio of these last two equa-tions,

k   tanh(kH ) =  ω2

g  ,   (4.15)

imposes a relationship between the frequency  ω  and the wavenumber k :

ω   =   ±  gk   tanh(kH )  .   (4.16)

This equation bears the name of  dispersion relation .From this follows that all wave functions can be determined in terms of the

surface amplitude  A, which itself remains arbitrary:

u   =   ωA  cosh(kz)

sinh(kH )  cos(kx − ωt) (4.17)

w   =   ωA  sinh(kz)

sinh(kH )  sin(kx − ωt) (4.18)

 p   =   pa   +   ρ0g(H − z) +   ρ0gA  cosh(kz)

cosh(kH )  cos(kx − ωt) (4.19)

a   =   A   cos(kx − ωt).   (4.20)

Let us now consider properties of these waves. First, every variable is a functionof  kx− ωt  =  k(x− ct), where c  is defined as

c   =  ω

k  =   ±

 g

k  tanh(kH ) .   (4.21)

A fixed surface elevation (a crest, for example) thus progesses in the  x−direction atspeed  c, called the  phase speed  of the wave. The duplicity in sign means that thereare actually two waves, one (with the + sign) traveling in the +x direction and theother (with the − sign) in the −x direction. Since the speed |c| is the same in bothdirections, gravity waves progress at the same speed regardless of their direction of 

propagation and are said to be   isotropic .The speed of the wave, however, depends on its wavelength. Indeed, Equation(4.21) written in terms of the wavelength,

c   =   ± 

2π  tanh

2πH 

λ

 ,   (4.22)

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4.1. SURFACE GRAVITY WAVES    77

Figure 4.3: Variation of thewave speed of a surface grav-ity wave as a function of wave-length.

reveals that   c  increases with  λ   (Figure 4.3). Longer waves propagate faster thanshorter waves, and an initial group of waves with a mix of different wavelengthswill gradually disaggregate as the longer waves overtake the shorter ones. Thisphenomenon is called wave dispersion  and explains the name given to relation (4.16).

Across the fluid layer, particles execute ellipses with horizontal and vertical di-mensions of 2A cosh(kz)/ sinh(kH ) and 2A sinh(kz)/ sinh(kH ), respectively. Thereis attenuation with depth.

It is worth also comparing the vertical acceleration (∂w/∂t) to the gravitationalacceleration (g). The maximum vertical acceleration occurs at the surface (z  =  H )and according to (4.18) and (4.15), its maximum value over the wave period is ω2A =g(kA) tanh(kH ), which can exceed   g  at high wavenumbers (short wavelengths).Only when the vertical acceleration is much less than the gravitational acceleration

[(kA) tanh(kH ) ≪ 1] is the wave motion in near hydrostatic balance.

4.1.4 Deep-water waves

Surface waves possess two asymptotic limits depending on whether the wavelengthis much shorter or much longer than the water depth. When λ ≪  H  (in practiceλ < H  is enough), the wave is said to be a deep-water wave. Mathematically, thewavenumber k , is very large (to the point where  k H  ≫  1 and tanh(kH ) ≃ 1), thedispersion relation (4.16) may be approximated to

ω   =   ± 

gk .   (4.23)

The corresponding wave speed is

c   =  ω

k  =   ±

 g

k  =   ±

 gλ

2π  .   (4.24)

It is clear from this last expression that longer waves propagate faster thanshorter ones. This is why after a stone has been thrown in a pond (Figure 4.4) thelonger waves radiate outward faster than the shorter ones.

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78   CHAPTER 4. WAVES 

Figure 4.4: Ripples following the throwing of a stone in a quiescent body of water.Note how the circles of the shorter waves lie inside those of longer waves becauseshorter waves travel at lower speeds. [Photo by the author]

Motion in the water below a deep-water wave is vertically attenuated, that is,the horizontal and vertical velocities decrease with depth, according to

u   =   ωA  e−kh cos(kx − ωt) (4.25)

w   =   ωA  e−kh sin(kx−

ωt),   (4.26)

with h  =  H − z  being the depth measured dowmward from the mean water surfaceand A  the wave amplitude.

Example 4.2: Swell

Swell on the sea is an example of deep-water waves. These waves are generatedby the wind during a storm and propagate almost without dissipation over longdistances, far away from their place of origin. They eventually crash as surf uponencountering a beach. For example, the cause of surf on Hawaiian beaches is oftenwind waves generated during a storm near Alaska, which then travel as swell overlong distances across the North Pacific Ocean.

When a 25-m wavelength swell originates South of Sitkinak Island (56◦30′N,154◦07′E) of the Aleutian Island Chain and travels to the north shore of OahuIsland (21◦35′N, 158◦07′E) in Hawai’i, it covers a distance of 3885 km at a speed of 6.25 m/s. This takes 7 days and 5 hours. The Alaskan storm has long disappearedbefore its effect is felt along the Hawaiian shore.

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4.1. SURFACE GRAVITY WAVES    79

4.1.5 Shallow-water waves

When   λ >  20H , the wavelength is much longer than the depth, and the wave issaid to be a shallow-water wave. Mathematically,  k  is so small that  kH  ≪  1 andtanh(kH ) ≃ kH . The asymptotic expression for the frequency is

ω   =   ± 

gk2H ,   (4.27)

and the corresponding wave speed is

c   =  ω

k  =   ±

 gH ,   (4.28)

which is independent of the wavelength. Thus, all long waves travel at the samespeed and do not disperse.

It can be shown (see Problem 4.3 at the end of this chapter) that the horizontal

velocity u   under a shallow-water wave is nearly uniform in the vertical. In otherwords, there is almost no attenuation with depth, in contrast to deep-water waves.Shallow-water waves are thus capable of bottom erosion and sedimentation. Aparticularly troublesome situation in the coastal ocean is the covering of old seamines by sediments and their occasional uncovering by large, storm-induced waves(Fowler et al., 1993). Needless to say, this poses a significant hazard to navigationin former coastal war zones.

4.1.6 Seiches, tides and tsunamis

Seiches, tides and tsunamis are examples of shallow-water waves. A seiche is astanding wave , formed by the superposition of two waves of equal wavelength andpropagating in opposite directions. Such situation occurs in confined water bodies,

such as a lake, by reflection on lateral boundaries.Consider for example, a rectangular basin of length   L   and with flat bottom

at depth  H . The long-wave approximation of the horizontal velocity (4.17) is  u =(ωA/kH ) cos(kx−ωt), and the superposition of two such waves, one with amplitudeA1  propagating in one direction (say with + sign chosen for  ω) and the other beingits reflection, with amplitude  A2  and propagating in the opposite direction (− signfor ω) yields

u   =  ωA1

kH   cos(kx − ωt) −   ωA2

kH   cos(kx + ωt),   (4.29)

where ω  = 

gk2H  is the (positive) angular frequency and  k  the wavenumber. Withthe x-axis running from one end of the lake at x = 0 to the opposite end at x =  L, the

boundary conditions on the horizontal velocity are  u(x = 0) = u(x =  L) = 0. Thefirst condition implies  A1  = A2, allowing us to drop the subscript. The horizontalvelocity field can be then rewritten as

u   = 2A

  g

H   sin(kx)sin(ωt).   (4.30)

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80   CHAPTER 4. WAVES 

The second boundary condition,  u(x  =  L) = 0, yields sin(kL) = 0, which impliesthat the wavenumber  k  can only take one of the following discrete values:

k   =  π

L  ,

  2π

L  ,

  3π

L  ,   etc.

Since k  = 2π/λ, the wavelength is likewise restricted to a set of discrete values:

λ   = 2L,  2L

2  ,

  2L

3  ,   etc. (4.31)

Thus, the longest wavelength allowed is twice the basin length and the secondlongest is exactly the basin length.

Finally, because the frequency is related to the wavenumber by  ω   = 

gk2H ,the wave period T  = 2π/ω  takes on one of the following values:

T    =  2L

√ gH   ,

  L

√ gH   ,

  2L

3√ gH   ,   etc. (4.32)

The first value is the largest, thus corresponding to the gravest mode. The secondperiod is half as long and corresponds to the second mode, etc.

In environmental applications, a quantity of interest is the maximum bottomvelocity, because if it exceeds a certain threshold value, resuspension of sedimentstakes place. The maximum bottom velocity (in absolute value) is that obtainedfrom (4.30) when sin(kx) and sin(ωt) are each equal to ±1:

umax   = 2A

  g

H   ,   (4.33)

where 2A   is the crest-to-trough amplitude of the wave on the surface and  H   thewater depth.

Example 4.3: Seiches in Lake Michigan

Figure 4.5 shows the first two seiche modes of Lake Michigan, calculated by asimple numerical model with realistic bottom topography (Mortimer, 1979). Thefirst mode, with a period of 8.9 hours, corresponds to a seesaw motion, with a hingeline at mid-length and out-of-phase maxima at each end. The second mode, witha period of 3.7 hours, exhibits a double-hinge motion, with water rising (falling) inthe middle of the basin when the water level falls (rises) at the extremities.

Depending on the strength of the wind storm that generates the sloshing motionin the lake, a seiche of Lake Michigan can reach several meters. An occurence of a2.44-m seiche killed eight people in Chicago on 26 June 1954 (Hughes, 1965).

A tide is a long (i.e. shallow-water) oceanic wave driven by the periodic gravi-tational attraction of the moon or sun. Tides hold little interest in environmentalfluid mechanics, except in the study of estuaries.

A tsunami is a half wave triggered by an underwater earthquake. With wave-lengths ranging from tens to hundreds of kilometers, tsunamis behave as shallow-water waves. What makes tsunamis disastrous is the gradual amplification of their

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4.1. SURFACE GRAVITY WAVES    81

Figure 4.5: Seiche modes of Lake Michigan. Numerical values indicate phase lagin degrees, relatively to the phase of the oscillation at the northeastern tip of thelake (Mackinaw City). Thus, points with phase lags differing by 180◦ are such thatwhen the water level is highest at one point, it is lowest at the other, and viceversa. Acronyms in circles indicate cities along the shore: CH = Chicago, MI =Milwaukee, ST = Sturgeon Bay, MC = Mackinaw City, LU = Ludington, HO =Holland. [Adapted from Mortimer, 1979]

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82   CHAPTER 4. WAVES 

Figure 4.6: The difference between beach surf (upper panel) and a tsunami (lowerpanel). Unlike surf in which a flooding crest is negated by the following trough, atsunami behaves as a bore, with a single rush of water. [Courtesy Department of Earth and Space Sciences, University of Washington, Seattle]

amplitude as they propagate into shallower waters, so that what may begin as aninnocuous 1-m wave in the middle of the ocean, which a ship hardly notices, canturn into a catastrophic multi-meter surge on the beach. Because shallow-water

waves propagate at the speed √ gH , they slow down as they encounter shallowerwaters and their energy1, density increases, raising their amplitude by the timethey reach the shore. Deep-sea swell turning into beach surf behaves similarly, withone major exception. Swell is characterized by a periodic motion so that the nexttrough swallows the water spilled by the previous crest, but a tsunami behaves asa bore, with a single rush of water. The rising water level has no place for retreat.

Examples of disastrous tsunamis are those that occurred in the eastern PacificOcean on 22 May 1960, in the Indian Ocean on 26 December 2004, and in the westernPacific Ocean on 11 March 2011. Tsunamis are relatively easy to forecast using acomputer model. The key to an effective warning system is the early detectionof the originating earthquake, to track the rapid propagation of the tsunami frompoint of origin to the coastline on time to issue a warning before the high wavestrikes. A vital precaution for coastal communities is the preservation of coral reefs

to act as damping ground for tsunamis (Fernando et al., 2005; Marris, 2005).

1To be exact, the conserved property is the   wave action , which is the energy density dividedby the frequency (Lighthill, 1978). But since the frequency of shallow-water waves is proportionalto wavenumber and thus inversely proportional to wavelength, the conserved quantity is energydensity times the wavelength.

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4.1. SURFACE GRAVITY WAVES    83

Figure 4.7: Aerial view of the water east of Whiskey Island off the coast of Louisiana(USA) showing refration of long-crested surface waves upon approaching a slopingbottom at an oblique angle. The portion of the wave in deeper water travels faster

and gradually catches up with the part in shallower water. The net effect is atendency toward alignment of wave crests with the shoreline. [Photograph by JohnLivzey]

4.1.7 Wave refraction in shallow water

A common observation at the seashore is that approaching waves have crest linesnearly parallel to the shore. This may appear somewhat puzzling given that oceanwaves can propagate in any direction and thus approach the shore from any angle.The explanation lies in the phenomenon known as  wave refraction .

When a wave approaches a sloping beach obliquely (Figure 4.7), the crest line

spans various depths, the part closer to shore travels over a shallow bottom while theouter portion travels over deeper water. As the wave speed is

√ gH , this outer por-

tion travels faster making the wave bend and aligning the wave crest progressivelywith the shoreline.

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84   CHAPTER 4. WAVES 

4.2 Internal Gravity Waves

A stably stratified fluid system can support internal waves. These are gravity wavespropagating on the density stratification not unlike surface waves propagating atthe air-water interface. The main differences are that internal waves are generallyslower because they rely on a weaker density difference and can propagate bothvertically and horizontally.

Under summertime heating, lakes become thermally stratified, that is, the watertemperature increases gradually from the bottom, where the water is coldest, tothe surface, where it is warmest. And, since thermal expansion causes density todecrease with increasing temperature, the water density varies in the opposite way,with buoyant waters floating on top of denser waters. The stratified water column isthus gravitationally stable in summer. (In winter, cooling causes the water columnis top heavy and unstable. The result is convection.) Similarly, whenever a warmerair mass lies above a layer of colder air, a gravitationally stable stratification occurs

in the atmosphere. A common occurrence is nighttime stratification caused by theair near the ground cooling due to radiative loss to space once the sun has set.

Internal gravity waves are generated whenever a source of energy displaces fluidvertically in the presence of density stratification. A prototypical example in thesea is a tide passing over an irregular bottom, such as a sill at the entrance of anembayment or fjord. In the atmosphere, convective storms radiate energy outwardthrough the stratified surroundings via internal waves, and mountain waves areinternal waves generated by wind passing over a mountain range when stratificationis simultaneously present. For additional examples, the reader is referred to thebooks by Roberts (1975) and Sutherland (2010).

Although temperature variations create changes in density, the actual densitydifferences remain modest, and a linear relation between density and temperaturemay be invoked (see Section 2.5):

ρ   =   ρ0[1  −   α(T  − T 0)],   (4.34)

in which  ρ0  is the density at the reference temperature  T 0  and  α  the coefficient of thermal expansion. The smallness of the density difference permits the Boussinesqapproximation, namely the substitution of the reference density  ρ0   everywhere inthe equations, except in connection with gravity where density variations matterbecause they create a buoyancy force.

4.2.1 Theory

To simplify the problem we restrict our attention to the two-dimensional case, thatis, considering motions confined to a single vertical plane across the system, with

coordinates x  running horizontally and  z   extending vertically from bottom to top.The variables of the problem are the velocity components   u   and   w, in the   x

and  z   directions, respectively, the pressure  p  and the temperature  T , all of whichdepend on  x,   z   and time   t. The four equations are the continuity equation, thetwo momentum equations, in the horizontal and vertical directions, and the energyequation.

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4.2. INTERNAL GRAVITY WAVES    85

Figure 4.8: Cloud formations showing evidence of internal waves.

Under the 2D assumption, the continuity equation (3.5) reduces to:

∂u

∂x  +

  ∂w

∂z  = 0.   (4.35)

For momentum, we invoke the additional assumption that wave motions are weak

in order to linearize the equations. This is in addition to assuming that   ρ   maybe replaced by  ρ0  everywhere, except in connection with gravity. We also neglectfriction in order to avoid the complication of wave damping. Equations (3.10) and(3.12), together with (4.34) provide:

∂u

∂t  =   −   1

ρ0

∂p

∂x  (4.36)

∂w

∂t  =   −   1

ρ0

∂p

∂z  −   g[1− α(T  − T 0)] (4.37)

For the fourth and last equation, we decompose the temperature into a compo-nent  T̄  that represents the existing thermal stratification and an additional compo-

nent  T ′ that is a small perturbation due to the wave:

T (x,z,t) =   T̄ (z) +   T ′(x,z,t).   (4.38)

The perturbation T ′ is caused by the wave-induced vertical displacement of the basictemperature gradient, with a positive/negative  T ′ created by a downward/upward

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4.2. INTERNAL GRAVITY WAVES    87

Figure 4.9: Vertical displace-ment of a parcel in a stablystratified fluid. At its new po-sition, the parcel experiences anet force, equal to its weight mi-nus buoyancy, directed towardthe parcel’s place of origin, andthe parcel executes oscillationsof period 2π/N .

Under the Boussinesq approximation, variations in density are small, so that

we may replace  ρ(z) on the left by the reference density  ρ0  while, on the right, wemay use a Taylor expansion to express the density at the new level [ρ(z  +  h) ≃ρ(z) + h(dρ/dz)], to obtain, after division by  V  ,

d2h

dt2  =

  g

ρ0

dz  h,   (4.43)

or

d2h

dt2  +   N 2 h   = 0,   (4.44)

after defining

N 2 =   −   gρ0

dρdz

  .   (4.45)

Should the density stratification be attributed to a temperature gradient, definition(4.45) is equivalent to (4.42).

If  N 2 >  0, Equation (4.44) admits periodic solutions cos(N t) and sin(N t) forh(t), indicating that the displaced parcel executes vertical oscillations at frequencyN . Physically, when density decreases upward, the parcel once displaced upward isheavier than its surroundings, feels a downward recalling force, falls down, acquiresa vertical velocity, overshoots its original position, becomes lighter than the ambientfluid, rises again and oscillates in this manner until friction eventually brings it torest.

On the contrary, should   N 2 be negative, the particle does not return toward

its original position but its displacement h(t) grows exponentially as exp(√ −N 2 t).Physically, the fluid is top heavy and gravitationally unstable.

Typical values of the frequency  N  of stable stratifications correspond to periods2π/N   of a few seconds (atmosphere) or minutes (lakes). This is usually shortcompared to the period of internal waves, which implies  ω < N .

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88   CHAPTER 4. WAVES 

The set of equations is now reduced to (4.35) and (4.41) for the two unknownsu  and  w . Since these are linear and the coefficients are time independent, we seek

a trigonometric solution of the type

u   =   U  sin(kxx + kzz − ωt)

w   =   W  sin(kxx + kzz − ωt),

with angular frequency  ω  and wavenumbers  kx   and  kz   in the horizontal and ver-tical directions, respectively. The equations relating the amplitudes U   and  W   areobtained by substitution in the preceding equations:

kxU   +   kzW    = 0 (4.46)

kzω2U   +   kx   (N 2

−ω2)W    = 0,   (4.47)

which form a two-by-two system of equations for the the velocity amplitudes  U  andW .

The system of equations for  U   and  W  admits non-zero solutions only if 

ω2 =   N 2  k2x

k2x + k2z.   (4.48)

This is the dispersion relation for internal gravity waves. It admits two frequenciesfor every wavenumber pair (kx,  kz):

ω   =   ±N 

   k2x

k2x + k2z,   (4.49)

from which it is clear that the absolute value of the wave’s frequency   ω   must besmaller than the stratification frequency  N   of the fluid. Should a periodic pertur-bation be imposed on a stratified fluid at an angular frequency higher than  N , thefluid would be unable to respond in the form of wave radiation, and the energywould have to be dissipated locally by means of turbulence.

4.2.2 Internal seiches in a rectangular basin

Long internal waves, with wavelengths comparable to the length of the lake, arecapable of reflecting back and forth between the extremities of the basin withoutappreciable damping, and the result is a standing internal wave, called an  internal 

seiche . On occasions, under favorable wind forcing, an internal seiche can assume a

very large amplitude. Figure 4.10 shows an example for Cayuga Lake, USA. Suchdramatic oscillation in the basin evokes the concept of  resonance .

For a rectangular basin, that is, for an unrealistic lake with a flat horizontalbottom and perfectly vertical sides, the mathematical problem is separable and canbe solved by modal decomposition. The boundary conditions require impermeabilityall around the domain, i.e., is u  = 0 at x = 0, L in the horizontal direction and w  = 0

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4.2. INTERNAL GRAVITY WAVES    89

Figure 4.10: Temperature versus depth and time in Cayuga Lake (New York State,USA) during 11–26 September 2001. The swing in temperature values about everytwo days at depths of 12–20 meters are indicative of large vertical oscillations of thethermal stratification, generated by a basin-wide resonant internal wave (internalseiche). [Figure courtesy of Prof. Edwin A. Cowen, Cornell University]

at  z  = 0, H  in the vertical direction. Here, L  is the length of the lake and  H  is itsuniform depth. We thus seek a solution of the type:

u   =   U   sinmπx

L

cos

nπz

cos(ωt) (4.50)

w   =   W   cosmπx

L

sinnπz

cos(ωt) (4.51)

which meets all boundary conditions as long as  m  and  n   are integers. Substitutionin Equations (4.35)–(4.41) demands:

m  U 

L  +   n

  W 

H   = 0 (4.52)

nω2   U 

H   +   m(N 2 − ω2)

  W 

L  = 0.   (4.53)

This set of equations implies:

ω2 =   N 2  m2H 2

m2H 2 + n2L2  .   (4.54)

In other words, a wave solution exists only if the frequency  ω  takes one among aset of discrete values (because  m  = 1, 2,...  and  n  = 1, 2,...). Physically, this means

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90   CHAPTER 4. WAVES 

Figure 4.11: Schematic of the horizontal and vertical motions in the internal seicheof lowest frequency in a uniformly stratified lake with flat bottom.

that for a given lake of length   L, depth   H   and stratification frequency  N , therecorresponds a discrete set of oscillation frequencies. Alternatively, to an externallyimposed frequency of oscillation (due to tides or wind), there exists a discrete setof stratification frequencies  N  that cause the basin to resonate.

The gravest mode, that corresponding to   m   =   n   = 1 (Figure 4.11), has afrequency given by:

ω   =  N H √ 

H 2 + L2.

Since lakes are typically much longer than they are deep (L ≫   H ), this can beapproximated to:

ω  ≃   N H L

  =   H L

 αg   dT̄ 

dz  .   (4.55)

Of particular interest in environmental problems is the value of the horizontalvelocity along the bottom. Indeed, resonance can generate large oscillations ac-companied by strong bottom currents capable of eroding sediments and depositingthem elsewhere. According to (4.50), the maximum horizontal velocity at the bot-tom is umax  = |U |, which is related to the maximum vertical velocity along the sidewmax  = |W |  by umax   = (nL/mH )wmax, according to (4.52). In turn, the verticalvelocity is the time derivative of the vertical displacement. If the range of verticalexcursion (= twice the amplitude) along the side is ∆z, then  wmax  =  ω ∆z/2 andumax   = (nLω∆z/2mH ). For the gravest mode, the maximum horizontal velocityalong the bottom is thus

umax   =  N ∆z

2  ,   (4.56)

and it occurs in the middle of the lake. It is left to the reader to show that themaximum bottom velocity of all other modes (m  and  n  arbitrary) is the same butoccurs at other locations along the bottom of the basin.

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4.3. MOUNTAIN WAVES    91

Figure 4.12: Mountain wave over the Presidential range in New Hampshire mani-fested by undulating clouds. [Photo credit: AccuWeather.com]

If the basin is not rectangular, the situation is complicated, and the analysislies beyond the scope of this book. For details, see Maas and Lam (1995) andCushman-Roisin (2004). Data-model comparisons were performed for internal se-iches in Upper Mystic Lake in Massachusetts (Fricker and Nepf, 2000) and LakeGeneva in Switzerland (Lemmin  et al., 2005), which both showed that the modesexcited by wind events tend to be the lowest modes and that the structure of theseseiche modes is highly sensitive to both bathymetry and stratification. There is atendency toward a narrowing of wave energy into beams with a resulting ampli-fication of velocities along the beams and where they reflect on the bottom andsurface.

4.3 Mountain Waves

Mountain waves, also called   lee waves , are atmospheric internal waves generatedby wind passing over a mountain ridge (Figure 4.12). Such waves can occasionallybe accompanied by sudden and localized strong winds that can topple trees andstructures (Gill, 1982, Section 8.8).

As long as the wave remains of moderate amplitude to permit a descriptionby linear dynamics, a stratified flow over a mountain, hill or land topography of any shape can be represented by means of a Fourier decomposition into sinusoidalwaves. When this is done, each wave component has a given horizontal wavenumbercomponent kx. Because the topography is fixed in space, so, too, is the wave, whichmeans that its phase propagation speed  c  =  ω/kx   must be equal and opposite tothe mean wind velocity U , and this sets the wave frequency ω  to

ω   =   −   U kx.   (4.57)

At the same time, the dispersion relation Equation (4.48) must hold, from which

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92   CHAPTER 4. WAVES 

we can determine the vertical wavenumber component that the wave is required tohave:

kz   =   ±   kx

 N 2

ω2 − 1 =   ±

 N 2

U 2 − k2x   .   (4.58)

Clearly, two possibilities arise depending on the sign of the expression under thesquare root. Either the expression is positive in which case   kz   is real, or it isnegative and causing   kz   to be imaginary. In the first case, which is that of thelonger horizontal wavelengths, the wave is undulating in the vertical (sine wavein  z), and wave energy generated by the topography is propagating upward. Bycontrast in the second case, corresponding to shorter horizontal wavelengths, thewave is evanescent in the vertical (exponential decay in  z), and the correspondingenergy is trapped near the ground.

The Fourier decomposition of an actual topography will necessarily include many

wavelengths. While the longer wavelengths may fall in the first regime, the shorterwavelengths will fall into the second regime, and the overall wave structure can bequite complex. Figure 4.13 illustrates the case of a wave generated by an idealized,bell-shaped mountain,

h(x) =  hmax1 + (x/L)2

  ,   (4.59)

which involves waves of both types.In stratified waters, internal waves are frequently generated by a current (tidal

or not) passing over bumpy topography. The difference with mountain waves in theatmosphere is the presence of an upper surface against which upward propagatingwaves can reflect as downward propagating waves. This situation is very commonin fjords where stratification created by fresher river water overlying saltier seawa-ter is made to oscillate vertically by tidal currents over the entrance sill (Baines,1982; Cushman-Roisin and Svendsen, 1983). Along the bottom, the buoyancy forceresponsible for internal-wave generation is proportional to:

1

h(x)

dh

dx  (4.60)

and is thus greater where the bottom slope (dh/dx) occurs closer to the surface.The reflection of internal waves against the surface may cause significant hor-

izontal currents, and the spatially periodic structure of these creates a pattern of alternating convergence and divergence, leading to the appearance of visible lines,called  slicks , resulting from the gathering of small floating debris or bubbles alongconvergence lines (Klemas, 2012).

Likewise, reflection of the same internal waves along the bottom may influ-ence the erosion/sedimentation pattern. This is particularly pronounced where theoblique propagation direction of the wave happens to match the bottom slope, caus-ing a   critical reflection  (Cacchione and Southard, 1974; Puig et al., 2004).

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4.3. MOUNTAIN WAVES    93

Figure 4.13: Structure of a wave generated by a uniform flow (U  = 10 m/s, flowingfrom left to right) of a uniformly stratified fluid (N   = 0.01/s) over a bell-shapedmountain profile with length scale  L  = 1 km. The upper panel depicts the verti-cal displacements (upward in blue and downward in red), the zeros of which areoutlined with dashed lines. In a moist atmosphere, the zones of updraft may be as-sociated with cloud formation. The lower panel shows the ground-level distributionof pressure and wind. [From Queney, 1948, with colors added]

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94   CHAPTER 4. WAVES 

4.4 Inertia-Gravity Waves

Inertia-gravity waves are long surface waves modified by the effect of the earth’srotation. The theory proceeds as in Section 4.1.3 except for the inclusion of theCoriolis terms in the momentum equations and the assumption of hydrostatic bal-ance, both attributed to the long length scale:

∂u

∂t  −   f v   =   −   1

ρ0

∂p

∂x  (4.61)

∂v

∂t  +   f u   =   −   1

ρ0

∂p

∂y  (4.62)

0 =   −   1

ρ0

∂p

∂z  −   g   (4.63)

∂u∂x

  +   ∂v∂y

  +   ∂w∂z

  = 0.   (4.64)

The hydrostatic balance (4.63) provides the structure of the pressure field:

 p(x,y,z,t) =   pa   +   ρ0g(H  + a− z),   (4.65)

where a(x,y,t) is the surface elevation, expressed as a wave form:

a(x,y,t) =   A sin(kxx + kyy − ωt).   (4.66)

Since for long waves, the vertical velocity varies linearly over depth [see (4.18) inthe limit of small  k z  and  kH ], is zero at the bottom [w(z  = 0) = 0] and fluctuateswith the surface displacement on top [w(z  =  H ) =  ∂a/∂t], its expression is

w(x,y,z,t) =   zH 

∂a∂t

  =   −   ωA   zH 

  cos(kxx + kyy − ωt).   (4.67)

With the elimination of pressure p and vertical velocity w  in terms of the surfacedisplacement a, Equations (4.61)–(4.62)–(4.64) become

∂u

∂t  −   f v   =   −  g

  ∂a

∂x  (4.68)

∂v

∂t  +   f u   =   −  g

  ∂a

∂y  (4.69)

∂u

∂x  +

  ∂v

∂y  +

  1

∂a

∂t  = 0.   (4.70)

It can be shown with some algebra that the wave form (4.66) for  a  and similartrigonometric functions for the velocity components  u  and  v  lead to two solutionsfor the frequency ω. The first solution is simply  ω  = 0, which corresponds to a stateof equilibrium, called the geostrophic balance . This state is of profound importancein meteorology and oceanography but is not pursued here, as our present concernis wave motion.

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4.5. ENERGY PROPAGATION    95

The second solution is

ω2 =   f 2 +   gH k2,   (4.71)

where   k   is the magnitude of the wavenumber (k2 =   k2x +  k2y). These waves areisotropic because their frequency depends only on the magnitude of the wavenemu-ber and not its direction, and are said to be   superinertial  because their frequencyexceeds the  inertial frequency  f . These waves are also dispersive as their propaga-tion speed

c   =  ω

k  =   ±

 gH  +

 f 2λ2

4π2  ,   (4.72)

is wavelength dependent, except in the limit of short wavelengths, so short thatthe Coriolis effect becomes negligible (f λ ≪ 2π

√ gH ). In this limit, we recover the

shallow-water gravity waves of Section 4.1.5. For additional information on inertia-

gravity waves, the reader is referred to Section 9.3 of Cushman-Roisin and Beckers(2011).

4.5 Energy Propagation

An interesting aspect of wave propagation is that the energy carried by waves doesnot always travel in the same direction or at the same speed as crests and troughs.In a wave of a single wavelength, the energy density is uniformly distributed becauseof the pattern repetition every wavelength. But, in a wave consisting of multiplecomponents, different wavelengths dominate at different locations, and the energy

distribution is non uniform. See for example the ripple pattern of Figure 4.4 inwhich the local wavelength increases from center to rim and energy is radiatedoutward.

In a multi-component wave, called a   wave group, the trigonometric functionsin(kx−ωt) is no longer applicable and needs to be generalized to sin α, where thephase  α(x, t) is a more complicated function of space and time. Nonetheless, onecan define a local wavenumber  k  and a local frequency  ω  as

k   =  ∂α

∂x  , ω   =   −   ∂α

∂t  ,   (4.73)

to which correspond a local wavelength  λ   = 2π/k  and a local period   T   = 2π/ω.The uniqueness of the function   α   in Equations (4.73) implies a relation betweenwavenumber and frequency:

∂k

∂t  +

  ∂ω

∂x  = 0.   (4.74)

Since the dispersion relation [such as (4.16) or (4.48) or (4.71)] prescribes arelation between frequency and wavenumber, it follows that ω  =  ω(k) and ∂ω/∂x =(dω/dk)(∂k/∂x), leading to:

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96   CHAPTER 4. WAVES 

∂k

∂t  +   cg

∂k

∂x  = 0,   (4.75)

where the coefficient  cg  stands for  dω/dk  and is a function of  k   only. In terms of the local wavelength, we have

∂λ

∂t  +   cg

∂λ

∂x  = 0,   (4.76)

with cg  now taken as a function of  λ. This equation is a transport equation statingthat the quantity  λ   travels unchanged at speed  cg. Each wavelength travels at itsown speed   cg(λ). Ahead and behind this wave are waves of the group with otherwavelengths. Thus, the energy associated with wavelength  λ  travels at the speed

cg   =  dω

dk

  (4.77)

and not at the propagation speed   c   =   ω/k   of local crests and troughs. For thisreason, a distinction is to be made between the two propagation speed:   c  is calledthe  phase speed , and  cg   the  group velocity .

That the difference between the two speeds can be quite important is exemplifiedby deep-water waves. For these waves indeed, the frequency  ω , given by Equation(4.23), is clearly a function of wavenumber  k , with

c   =  ω

k  =

 g

k  (4.78)

cg   =  dω

dk

  =  1

g

k

  =  c

2

  ,   (4.79)

indicating that the energy travels at half the speed of crests and troughs. But, howis that possible? Close examination of the evolution of a ripple pattern (of Figure4.4, for example) reveals that crests vanish by destructive interference and newones form by constructive interference. An observer tracking by eye the progressof one crest will suddenly lose sight of it and be led to believe that the eyes weresomehow confused between the crest that disappears and the following one, only todiscover that this one, too, disappears! And, while crests disappear at the head of the formation, new ones keep appearing behind. In other words, individual crestsovertake the group, and the group proceeds at a slower pace than individual crests.In the case of deep-water gravity waves, the group travels half as fast as crestswithin in.

In two or three dimensions, there is a wavenumber in each spatial direction,and one forms the wavenumber vector  k  with components (kx,  ky,  kz). The groupvelocity, too, becomes a vector cg   formed with the derivatives of  ω  with respect tothe wavenumber components (∂ω/∂kx,  ∂ω/∂ky,  ∂ω/∂kz), that is,

cg   =    ∇k  ω.   (4.80)

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4.6. NONLINEAR EFFECTS    97

Figure 4.14: Structure of a ray

of internal gravity waves. Whilethe ray radiates energy towardthe top right, crest and troughlines are parallel to the ray andtravel down to the right. Thus,in an internal wave, phase andenergy travel at a right angle of each other.

Internal gravity waves are peculiar in the sense that their energy propagatesvertically in the opposite direction of their crests and troughs. Indeed, according

to (4.49), the vertical phase speed is

cz   =  ω

kz=   ±   N kx

kkz(4.81)

while their vertical group velocity is

cgz   =  ∂ω

∂kz=   ∓  N kxkz

k3  ,   (4.82)

where k2 = k2x+ k2z . In contrast, the horizontal wave speed and group velocity sharethe same sign, and it can be shown (Problem 4.7) that the group velocity vector isaligned with the crest and trough lines, implying that phase and energy propagateat a right angle of each other (Figure 4.14)

For more on group velocity, the reader is referred to Lighthill (1978, Section

3.6), Kundu (1990, Chapter 7, Section 9) or Cushman-Roisin and Beckers (2011,Appendix B).

4.6 Nonlinear Effects

The preceding wave theories were all predicated on the assumption of a weak am-plitude in order to justify linear dynamics and thus permit a relatively easy math-ematical treatment. But as anyone who has ever contemplated surf on a beach, itis clear that waves can acquire finite amplitudes, depart in shape from simple sinu-soidal curves, sometimes to the point of tipping over and rolling into surf (Figure

4.15). Needless to say, nonlinear wave theory has received considerable attention,at the cost of major mathematical difficulties.

For surface gravity waves, a long-wave, finite-amplitude theory can be developedin the case when the wave amplitude  a  is on the order of  H 3/λ2 in which  H   is theresting water depth and  λ   the wavelength. This leads to the following for  a(x, t)first obtained by J. Boussinesq in 1872 (see Long, 1964):

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98   CHAPTER 4. WAVES 

Figure 4.15: Rolling wave uponapproaching a beach.

∂ 2a

∂t2  −   gH 

  ∂ 2a

∂x2  =   gH 

 3

∂a

∂x

2+

 3a

∂ 2a

∂x2 +

 H 2

3

∂ 4a

∂x4

.   (4.83)

Among other solutions to this equation, one is the so-called solitary wave  which hasa single crest and travels without changing shape over time, at a speed dependent onits amplitude and a little greater than the linear-wave speed

 √ gH . This particular

solution also possesses the property of being “a highly favoured type of disturbance”(Long, 1964) in the sense that it is often the shape ultimately assumed by a weaklynonlinear wave after a certain period of adjustment. This situation arises after

reaching a balance between two opposite tendencies. Because the highest point of adisturbance moves faster than others, there is a tendency for wave steepening. Thisis represented by the two nonlinear terms on the right of Equation (4.83). At thesame time, because longer waves propagate faster than shorter ones, according to thedispersion effect represented by the last, linear term of (4.83), the waveform spreadsand tends to flatten. The combination of wave steepening and wave flattening givesrise to an equilibrated wave shape, the solitary wave.

The preceding equation and its solutions, however, fail to apply to breakingwaves so often seen on beaches, such as the one caught by photograph for Figure4.15. The modeling of breaking (or rolling) waves can only be done with a non-hydrostatic numerical model, and the interested reader is referred to the books byMader (2004) and Lin (2008).

Finite-amplitude internal waves, too, have been studied (Turner, 1973, Section3.1). As their surface cousins, internal waves possess solitary forms in which abalance is struck between nonlinear steepening and linear dispersion.

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4.6. NONLINEAR EFFECTS    99

Problems

4-1.   After a few glasses of whiskey, a retired captain recounts a very stormy nightin the South Pacific when the waves were so fierce that his 45-m long shipwith its bow and stern spanning the distance from trough to crest was rockedback and forth in less than three seconds. Can this possibly be true?

4-2.   In which two limits do surface gravity waves have their wavelength propor-tional to a power of their period (i.e.   λT n)? What are then the values of theexponent n?

4-3.  Swell is a surface wave generated in the open ocean by a wind storm. Considera swell of 15-m wavelength in the middle of the Atlantic Ocean heading forMiami Beach. As the swell encounters shallower waters along its path, its

nature changes from deep-water wave to intermediate-water wave and ulti-mately to shallow-water wave. At which respective depths do these changesoccur?

If the change from deep-water to intermediate-water wave occurs after 1200km of travel, how old is the swell by then? Finally, at which depth will theswell propagate twice as slowly as it did in the deep ocean? [ Hint : While theswell undergoes transformation, its frequency  ω  remains unchanged.]

4-4.   By taking the limit   kH  →   0 in Equations (4.17) and (4.18), show that thehorizontal velocity   u   associated with a shallow-water wave is nearly depthindependent. What is the vertical structure of the accompanying verticalvelocity  w?

4-5.  At very short wavelengths (millimeters to centimeters), the restoring force of water waves is surface tension, and the waves are called capillary waves  (Figure4.16). At slightly longer wavelengths, surface water waves are intermediatebetween pure capillary and pure gravity waves, and the theory leads to thefollowing dispersion relation:

ω   = 

k  (g + γk2) tanh(kH )  ,   (4.84)

in which  γ   is a parameter related to the surface tension, a joint property of the two fluids in contact at the surface. Because the gravitational accelerationg  = 9.81 m/s2 and the surface tension parameter  γ  = 7.38× 10−5 m3/s2 forwater-air contact, the   γk2 correction becomes important only at relativelyshort wavelengths.

(a) For which wavelength  λ   does the surface-tension correction term becomeequal to the gravitational term?

(b) If the water depth   H  = 3 m, should the waves at the preceding wave-length be considered as deep-water waves, intermediate waves or shallow-waterwaves?

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100   CHAPTER 4. WAVES 

Figure 4.16: Capillary wavesriding on gravity waves.

(c) Simplify the dispersion relation accordingly.

(d) Derive the expressions for the phase speed   c   and the group velocity   cgfrom this simplified dispersion relation, and then form the ratio   cg/c  of thetwo. Plot this ratio as a function of wavelength and explore the two limitsof only surface tension (pure capillary waves) and only gravity (pure gravitywaves).

(e) What is a main difference in energy propagation between capillary andgravity waves?

(f) After throwing a stone in a pond and while watching the waves radiateoutward, where should you see the capillary waves, near the center or at theouter edge?

4-6.  What are the periods of the first three seiche modes of Lake Erie, which is388-km long and 18.9-m deep in average?

4-7.   What is the magnitude of the oscillatory bottom current if a wind storm overLake Erie (see previous problem) generates the second seiche mode with atrough-to-crest height of 60 cm?

4-8.  In a 28-m deep lake during summer, the temperature varies gradually from ahigh of 25◦C at the surface to a low of 17◦C at the bottom. This stratificationsupports internal waves, which are manifested on the surface by a patternof distorsions (slicks) of small wind-induced waves. Observations reveal arepeating slick pattern of 163 meters along the main axis of the lake andtraveling horizontally with a speed of 22.6 cm/s.

What is the corresponding vertical wavelength of the internal waves? Compare

this to the water depth. Is there any relation that strikes you that could havea significance?

4-9.   Norwegian fjords are former glacier valleys now flooded by seawater. Thereare therefore long and narrow. The freshwater discharged by lateral rivers pa-tially mixes into the seawater, creating a vertical stratification with decreasing

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4.6. NONLINEAR EFFECTS    101

salinity upward. At the open end, sea tides force motions at fixed frequen-cies, which inside the fjord generate internal waves at tidal frequencies, called

internal tides.

Consider Skjomen Fjord near Narvik in northern Norway, with length  L  of 25 km, average depth  H  of about 110 m, stratification frequency  N  of about2.0 ×  10−3 s−1, and in which the semi-diurnal tide generates motions witha 12.42-hour period. What are the horizontal wavelengths of the first twovertical modes (those with half vertical wavelengths respectively equal to thedepth and half the depth)? Can these waves fit in the length of the fjord?

4-10.   Derive dispersion relation (4.71) for inertia-gravity waves. Are the velocitycomponents u  and  v   in phase, out of phase, or in quadrature (phase shift of 0◦, 180◦ or ±90◦)?

4-11.  Show that the energy of internal gravity waves travels in a direction parallel

to the lines of crests and troughs.


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