Environmental Fluid Mechanics - Chapter 8 pot

This shows that streamlines, like the agating surface waves, are sinusoidal and the amplitude of the streamlinesdecays exponentially with the water depth.. The kinetic energy due to the

equi-of energy to maintain the motion.

Water quality issues of the marine environment, as well as of lakes,channels, and rivers, are often closely related to topics of wave formationand propagation For example, waves carry mechanical energy, which canlead to the destruction of maritime structures With regard to environmentalissues, wave energy leads to mixing in the water column and also along thebottom, causing movement of sediments along coastlines, affecting variouscurrent patterns in a water body and aiding the transport of solutes and floatingmaterials in the environment

The present chapter discusses basic features and properties of waves insuch environments, focusing on surface waves These waves, on a horizontalwater surface of a marine or lake environment, are confined to two-dimensionalpropagation in the horizontal plane and are subject to a vertical external gravityrestoring force An initial presentation of the wave equation is given, alongwith a discussion of its application to several specific issues of environmentalfluid mechanics Special attention also is given to the development and prop-agation of waves in open channels and rivers Internal waves are presented inSec 13.2

8.2 THE WAVE EQUATION

Neglecting viscous and Coriolis forces, the general governing equations forfluid motion are the equations of motion and mass conservation as developed

The velocity vector may comprise two parts One part is rotational and

is associated with the vorticity An equation for vorticity can be obtained bytaking the curl of Eq (8.2.3) Then, since the curl of the right-hand side ofthat equation vanishes, it is seen that the vorticity cannot be a time dependent

variable Therefore it is noted that linear wave theory (based on the linearized

equations of motion) neglects movements of vortex lines with the fluid Otherproperties may propagate in the domain

In addition to the rotational part of the velocity, which is independent

of time, the velocity incorporates another, irrotational part that is time dent This part can be considered as originating from a potential function .Therefore we may write (see Sec 4.2)

depen-E

This expression implies that the velocity of interest for wave propagationstems from a potential function According to linear wave theory, any steadyrotational velocity field does not affect this velocity

In the usual case of waves developing on a homogeneous water ment, we consider a fluid with constant density and apply a coordinate system

environ-in which z is the vertical upward coordenviron-inate, with z D 0 correspondenviron-ing to theelevation of the free surface, where the value of p vanishes For such a case,introducing Eq (8.2.5) into Eq (8.2.3) and integrating both sides results in

to the potential function

Simple wave motions of small amplitude are described by the wave equation,

8.3 GRAVITY SURFACE WAVES

Surface water waves develop at the water free surface, which is the interfacebetween the water and air phases In general, this interface is considered as adiscontinuity in the overall distribution of density in the domain The state ofstable equilibrium of the system is represented by water occupying the lowestportions of the domain Disturbances to the state of equilibrium are represented

by surface gravity waves Such waves propagate only in the horizontal tion, while the restoring gravity force acts in the vertical direction Thereforethere is no preferred horizontal direction of the disturbance propagation, and

direc-the waves are isotropic (direc-they may move equally in any horizontal direction).

However, waves of different wavelengths penetrate to different depths into thewater phase This phenomenon has an implication with regard to the inertia

of the fluid particles that are directly affected by the waves Therefore waves

of different wavelengths have different wave speed The dependence of the

wave speed on the wavelength causes dispersion of the waves.

To develop a solution of the wave equation, we adopt a coordinatesystem as in Sec 8.2, using z as the vertical upward coordinate, with itsorigin at the water free surface We also drop the subscript 0 for
, with theunderstanding that water density is constant (not stratified) The undisturbedabsolute pressure, p0 is distributed hydrostatically,

where pa has been assumed to be constant, so that its gradient is zero Due

to the incompressibility of the fluid, the continuity Eq (8.2.7) collapses toLaplace’s equation,

Laplace’s equation cannot describe wave propagation in a fluid that iscompletely bounded by stationary surfaces, but it can describe wave propaga-tion by the employment of the boundary condition at the original free watersurface At the original free water surface, the pressure value is associatedwith the wave displacement, namely the disturbed free surface elevation, ,according to

Equation (8.3.5), along with Eqs (8.3.1) and (8.3.2), indicates that at thedisturbed free water surface, the pressure is identical to the atmospheric pres-sure

According to Eqs (8.2.6) and (8.3.3) for the irrotational part of thevelocity, which is associated with the wave propagation,

to z, evaluated at a point intermediate to the disturbed and undisturbed watersurfaces This gives a means of checking the degree to which Eq (8.3.9)provides a good approximation to Eq (8.3.8)

The rate of change of  is equal to the vertical fluid velocity at thesurface, namely,

By differentiating Eq (8.3.9) with respect to t and applying Eq (8.3.11),

8.4.1 The General Wave Propagation Equation

Recall that a general form of the solution for the wave equation was

Eq (8.2.10), which gives the surface displacement, , as a function of timeand position An alternative function to describe wave movement is

to waves propagating in that direction Differentiating Eq (8.4.1) twice withrespect to t and twice with respect to x gives

∂2F

∂t2 D ω2F00 ∂

2F

where the double prime represents differentiation with respect to the variable

 Using the wave velocity from Eq (8.4.2) then gives

of a function similar to that given by Eq (8.4.1) and another function, whichdescribes an attenuation of the fluid motion with the water depth In the case ofperiodic surface waves, F is usually specified as a sine or cosine function.Therefore the potential function can be represented using a sine, a cosine, orthe real part of a complex function, such as

 D fzsinωt
kx D fz sin

ω



t
xc



8.4.7a

 D fzcosωt
kx D fz cos

ω



t
xc



8.4.7b

 D fzexp[iωt
kx] D fz exp

iω



t
xc



8.4.7c

where fz is a function that represents the variation in the vertical tion Since we know that the potential function is governed by the Laplaceequation (8.3.5), it follows that

8.4.2 The Potential Function for Deep Water Waves

The general solution of Eq (8.4.8) can be obtained by a linear combination of

an exponentially decaying term and an exponentially growing term However,

if the water depth is very large, then the exponential growing term shouldvanish Therefore a solution of Eq (8.4.8) that is consistent with a vanishingvalue of  with increasing depth (where z !
1) is given by

This is the relationship between the wave amplitude a and the maximumamplitude of the potential function.

By introducing Eq (8.4.10) into Eq (8.3.13), a relationship between thefrequency and the wave number for gravity waves on a deep-water environ-ment is found as

This is known as a dispersion relationship, giving the dependence of the

wave propagation on the wavelength (or wave number) Using the tion for wave speed (Eq 8.4.2), this last result shows that waves of differentwavelengths propagate with different velocities:

defini-c D ω

k D

g

k D

g

It should be noted that although Eq (8.4.15) represents typical values, extremecases may exist, with wavelengths as low as 0.1 m or as large as 1000 m Nearthe sea shore the wavelength is generally much less, and the waves should

be described with alternative theories, since the deep-water assumption is nolonger valid

8.4.3 Pathlines of the Fluid Particles

Velocity components in a wavy flow field are obtained by differentiating thepotential function For example, taking Eq (8.4.7b), the velocity componentsare found as

u D ∂

∂x D k0ekzsinωt
kx 8.4.16and

w D∂

∂z D k0ekzcosωt
kx 8.4.17

Thus both velocity components vary sinusoidally with time and have the sameamplitude, which decays exponentially with depth However, it should benoted that at a fixed position, the horizontal velocity lags the vertical velocity

by 90°(this result holds when using either of the expressions of Eq (8.4.7) todescribe the potential function) Equations (8.4.16) and (8.4.17) also indicatethat at a fixed position, the velocity vector maintains a constant absolute valueand rotates in the clockwise direction

Based on this oscillating velocity field, it is seen that over a long period

of time, there is no net movement of a fluid particle If it is assumed thatthe deviations of a fluid particle from an initial position x0, z0are relativelysmall, then the differential equations of the fluid particle pathline are givenapproximately by

dx

dt D u ¾D k0ekz0sinωt
kx0 8.4.18and

dz

dt D w ¾D k0ekz0cosωt
kx0 8.4.19Direct integration of these results gives an approximation for the instantaneousposition of the fluid particle,

where C1 and C2 are constants for each particular fluid particle

By eliminating time from Eqs (8.4.20a) and (8.4.20b), we obtain

x
C12C z
C22 D

k

This result shows that the fluid particles move in circular pathlines, whichalso is evident from the previous conclusion that the velocity componentshave equal amplitudes (Eqs 8.4.16 and 8.4.17) The radius of the pathlinedecays exponentially with water depth and does not depend on the horizontalcoordinate According to Eqs (8.4.20) and (8.4.21), the phase of the fluidparticle location does not depend on the z coordinate.Figure 8.1 provides aschematic description of various pathlines of different fluid particles and theirinstantaneous positions.

8.4.4 The Shape of the Streamlines

The shape of the streamlines is calculated by considering the following tionships between the stream function and the real parts of the velocity compo-nents given by Eqs (8.4.15) and (8.4.16):

rela-∂

∂z D u D k0ekzsinωt
kx 8.4.22

∂

∂x D
w D
k0ekzcosωt
kx 8.4.23Direct integration of these expressions yields

 D 0ekzsinωt
kx C C 8.4.24

where C is an arbitrary constant This shows that streamlines, like the agating surface waves, are sinusoidal and the amplitude of the streamlinesdecays exponentially with the water depth

prop-8.4.5 The Wave Energy

The excess energy in surface water waves is divided between kinetic andpotential energy The excess potential wave energy incorporated in a surfacearea of unit width, with length equal to one wavelength, and where the waterdepth, h, is large but finite, is

1

2dx 8.4.25

It should be noted that wave displacements above the level z D 0, as well

as below z D 0, carry positive potential energy The raised free surface adds

potential energy by adding fluid above the level z D 0, while the depressedfree surface adds potential energy by the removal of fluid from below the level

z D0

The kinetic energy due to the wave motion can be obtained by a metric integral performed over a volume of a prism incorporating the entirewater depth and a water surface area of unit width and a single wavelength

volu-in the longitudvolu-inal direction,

WEkD 1

2
r

where dU is an elementary volume Considering that Laplace’s equation, given

by Eq (8.3.4), is satisfied, and applying the divergence theorem, Eq (8.4.26)

WEkD 

0

12

By rearranging Eq (8.4.12) and substituting into Eq (8.4.11), we obtain

 D k

ω0sinωt
kx D a sinωt
kx 8.4.31Thus

∂

∂t D k0cosωt
kx D ωa cosωt
kx 8.4.32This result is then substituted into Eqs (8.4.25) and (8.4.30), also usingEqs (8.4.31) and (8.4.12) After dividing by the wave length, , to obtainthe total wave energy per unit area of the water surface, the total energy is

E D EpC EkD 1

Note that this result refers to energies per unit area, whereas previous equationsrefer to the energies per an area of unit width and a single wavelength in length.Equation (8.4.33) indicates that although the amount of potential andkinetic energies per unit surface area of the water varies from point to point,the total sum of potential and kinetic energies per unit water surface is constantover the entire water surface

WATER DEPTH 8.5.1 The Potential Function of Shallow Water Waves

Waves in deep water were considered in the previous section In practice, deepwater is defined when depth is greater than the wavelength,  If the waterdepth is uniform, but smaller than , then the water environment is considered

to be shallow and the solution of the differential Eq (8.4.8) should satisfy thecondition of zero normal velocity at the bottom, or

where 0 is a constant Then Eq (8.5.2) becomes

which is the dispersion relation for shallow-water waves.

8.5.2 The Wave Velocity of Propagation

Wave velocity is given by Eq (8.4.2), which, when used with Eq (8.5.7),gives

c D ω

k D

g

In order to illustrate the effect of the wavelength on the velocity of wavepropagation in a water layer of constant depth,Fig 8.2demonstrates severalcurves showing the variation of the hyperbolic functions Using the relation-ships of Eq (8.4.2), an expression relating wave velocity of propagation tothe wavelength is obtained:

c D

g

Figure 8.2 Graphs of cosh(x), sinh(x), and tanh(x).

Using this relation in Eq (8.5.10) gives

c D

For waves on deep water, the value of the tanh term of Eq (8.5.9) is imately unity (see Eq 8.4.12).Figure 8.3 shows the variability of the wavevelocity as a function of the relationship between the wavelength and the

approx-Figure 8.3 The wave speed versus the wavelength for water environment of constant depth.

depth of the water environment This figure provides some guidance about thepossible definition for deep and shallow water, as well as for long waves.Using Eqs (8.4.2), (8.5.7), and (8.5.9), it can be shown thatcω

g D tanh

ωhc



8.5.12

This result is useful for the evaluation of the variability of the wave velocity

of propagation due to gradual decrease of the water depth, for a constantfrequency This is shown in Fig 8.4, where Eq (8.5.12) is applied to depictthe variability of the wave velocity versus water depth, for constant wavefrequency Sinusoidal waves approaching the coastline pass through water ofgradually decreasing depth, while their frequency is kept unchanged Thereforethe number of wave crests reaching the beach per unit time is equal to thenumber approaching the coastline.Figure 8.4 shows how the wave speed ofsuch waves gradually decreases with the water depth In addition, wavelengthdecreases and, in fact, is much more significant than the decrease of the wavevelocity

In general, the original wave crests approach the coastline with someorientation angle Due to the decrease of the water depth, such wave crests tend

to align with the coastline, and their orientation angle decreases Figure 8.5illustrates this phenomenon, which is associated with the decrease of the wavevelocity in the region of shallow water In other words, as a wave approachesthe shore at some angle, the portion of the wave closest to the shorelineexperiences a decrease in velocity sooner than portions further away This

Figure 8.4 Wave speed as a function of water depth.

Figure 8.5 Alignment of wave crest approaching the coastline.

causes the wave crest to effectively change its alignment with respect to theshoreline

8.5.3 Pathlines of the Fluid Particles

Following the same approach as in Sec 8.4.3, expressions for the velocitycomponents of fluid particles are obtained by differentiating the potentialfunction Eq (8.4.7b), using Eq (8.5.4) for the depth variation, giving

u D ∂

∂x D fk0cosh[kz C h]g sinωt
kx 8.5.13a

w D∂

∂z D fk0sinh[kz C h]g cosωt
kx 8.5.13bUsing Eq (8.3.12), w is found from Eq (8.5.13) for z D 0, and the resultingexpression is integrated over time to obtain the relationship between the poten-tial function amplitude and the wave amplitude,

u D dx

dt D aωcoshkhcosh[kz C h] sinωt
kx 8.5.15a

w Ddz

dt D aωcoshkhsinh[kz C h] cosωt
kx 8.5.15a

In contrast to the situation for waves on deep water, in the case ofwater of finite depth, the amplitude of the fluid particle horizontal velocity isdifferent from that in the vertical direction As shown inFig 8.2, for waves

on finite water depth the amplitude of the horizontal velocity is larger thanthat of the vertical velocity, but they become almost identical for large waterdepth

Using the same approach as was used for deep water waves (i.e., grating Eq 8.5.15 over time), the following expressions for the pathlines ofthe fluid particles can be obtained:

inte-x D
acosh[kz0C h]

coshkh cosωt
kx0 C C1 8.5.16a

z D asinh[kz0C h]

coshkh sinωt
kx0 C C2 8.5.16bwhere C1 and C2 are constants connected with the initial location of the fluidparticle

We eliminate the time-dependent expression from the two parts of

Eq (8.5.16) to obtain the expression for the curve representing the particlepathline,

Figure 8.6 Pathlines of fluid particles in a sinusoidal wave on water with finite depth.

This indicates that particle pathlines are ellipses, as shown in Fig 8.6 Themajor and minor axes of the elliptical pathlines are represented by the terms ofthe denominators of Eq (8.5.17) In deeper water the pathlines tend to be morecircular since, except near the bottom, the cosh and sinh terms of Eq (8.5.17)become practically identical, as was the result found in Eq (8.4.21) In veryshallow water, the pathlines are flattened, as the minor axis of the ellipticalpathlines decreases

With regard to the calculation of the streamlines, we follow a similarprocedure as before, to find

coshkhsinh[kz C h] cosωt
kx 8.5.18b

By direct integration of Eq (8.5.18), the stream function is found as

 D aω

kcoshkhsinh[kz C h] sinωt
kx 8.5.19This result indicates that the streamlines, namely lines of constant value of ,have the shape of a sinusoidal wave whose amplitude decays with depth assinh At the bottom of the water environment z D
h, so it is represented bythe streamline with  D 0

The excess potential energy of the waves on water of uniform finitedepth takes the same form as for waves on deep water, given by Eq (8.4.25).The expression for the kinetic energy of the wave also is identical to that ofwaves on deep water, given by Eqs (8.4.28)–(8.4.30), with Eq (8.5.6) used

to specify the vertical gradient of the potential function (i.e., in Eq 8.4.29).However, the total energy per unit area of the water surface has the samerelationship typical of waves on deep water, namely Eq (8.4.33)

Previous sections of the present chapter refer to sinusoidal waves on deepwater, as well as water of finite depth A variety of properties of surface

water waves were introduced and analyzed while assuming that the surfacewater waves are sinusoidal It should be noted that the initial reference tosinusoidal surface water waves is justified for two main reasons: (1) surfacewaves are most commonly observed to be roughly sinusoidal, and (2) waves of

more complicated shape can be analyzed by Fourier analysis, in which waves

are considered as a linear combination of different sinusoidal disturbances.Due to the linearity of Laplace’s equation and the linear boundary condition

at the water surface as given by Eq (8.3.12), a linear combination of variouspotential functions describing different sinusoidal waves can also be a potentialfunction for a more general wave field

When surface waves are represented as a linear combination ofsinusoidal waves, it is important to keep in mind the dispersive propertycharacterizing waves of different wavelengths As shown previously, waves

of different wavelengths have different wave speeds (refer to Eqs 8.4.13 and8.5.9) Therefore if a disturbance at the water surface is created at one location

of the water environment, then at a later time different sinusoidal components

of the water surface disturbance will be found at other locations, due to theoriginal disturbance For example, consider large disturbances created by astorm In general, this causes the boundary conditions to be complicated andnonlinear, so that at the time and place of the storm, linear wave theory maynot be appropriate However, storm waves are reduced to groups of smallersize waves, which obtain energy from the high waves due to wave dispersion

and are called swell These reduced size waves can be analyzed by use of

linear wave theory Consider a small group of waves whose wave speed is c.After a time interval, t, this group of waves will be found at a distance Ut

from the origin of the disturbance, where U is called the group velocity and is

defined below (Eq 8.6.10) In deep water, the group velocity is approximatelyequal to half of the wave speed Also, it can be shown that the wave energypropagates at the group velocity

Consider that the wavelength gradually varies from one wave to the next

in the group of waves A local phase, ˛, can be defined, and the value of ˛ atevery wave crest can be expressed as an even multiple of , namely,

where n is an integer The value of n increases by one for each successivewave crest that passes a particular point Between the wave crests, the value

of ˛ varies smoothly In the wave troughs, the value of ˛ is an odd multiple

of The rate of change of ˛ with time is given in radians per second by

∂˛

The phase also is a function of the longitudinal distance between sive wave crests At a given time, the value of ˛ decreases with x, at a rate

succes-equal to the wavenumber k in radians per meter,

∂˛

Equations (8.6.2) and (8.6.3) imply that the water surface displacement,

, can be represented as a wave of slowly variable amplitude 1, by either

 D 1x, texp[i˛x, t] 8.6.4aor

Around a particular location x0, and particular time t0, the value of ˛ can beexpressed by a Taylor series, from which the following linear combination isconsidered:

˛x, t D ˛x0, t0
k0x
x0 C ωt
t0 8.6.5Then, differentiating Eq (8.6.2) with respect to x and Eq (8.6.3) with respect

to t and subtracting one from the other, we obtain

g

k D 12

ω

k D 1

to specify the vertical gradient of the potential function (i.e., in Eq 8. 4.29).However, the total energy per... 8. 5.18b

By direct integration of Eq (8. 5. 18) , the stream function is found as

 D aω

kcoshkhsinh[kz C h] sinωt
kx 8. 5.19This... k0cosωt
kx D ωa cosωt
kx 8. 4.32This result is then substituted into Eqs (8. 4.25) and (8. 4.30), also usingEqs (8. 4.31) and (8. 4.12) After dividing by the wave length, ,