Environmental Fluid Mechanics - Chapter 8 pot

8 Surface Water Waves 8.1 INTRODUCTION Mechanical systems may vibrate if their displacement from a state of equi- librium is subject to a restoring force. Some examples of such vibrations are represented by water hammer in pipes, sound waves, surface gravity waves, surface capillary waves, and internal waves. The restoring force in these partic- ular cases may be the pipe elasticity, fluid compressibility, gravity, surface tension, or Coriolis force in cases of rotating fluids. In most cases wave motions decay by viscous shear stresses unless there is a constant supply 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 formation and propagation. For example, waves carry mechanical energy, which can lead to the destruction of maritime structures. With regard to environmental issues, wave energy leads to mixing in the water column and also along the bottom, causing movement of sediments along coastlines, affecting various current patterns in a water body and aiding the transport of solutes and floating materials in the environment. The present chapter discusses basic features and properties of waves in such environments, focusing on surface waves. These waves, on a horizontal water surface of a marine or lake environment, are confined to two-dimensional propagation in the horizontal plane and are subject to a vertical external gravity restoring force. An initial presentation of the wave equation is given, along with a discussion of its application to several specific issues of environmental fluid mechanics. Special attention also is given to the development and prop- agation of waves in open channels and rivers. Internal waves are presented in Sec. 13.2. Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. 8.2 THE WAVE EQUATION Neglecting viscous and Coriolis forces, the general governing equations for fluid motion are the equations of motion and mass conservation as developed in Chap. 2,   ∂ E V ∂t C E VÐr E V  Drp CgZ 8.2.1 ∂ ∂t C E VÐr C rÐ E V D 0 8.2.2 where V is the velocity, p is pressure, t is time, g is gravitational accelera- tion,  is fluid density, and Z is the elevation above an arbitrary datum. These equations are considered to apply to a flow field that represents a small devi- ation from an initial state of fluid at rest and uniform fluid density,  0 . Thus velocities, pressure variations from hydrostatic values, and density variations all are assumed to be small, so that any products of these quantities become even smaller (i.e., in nondimensional terms, these quantities are less than one). Equations (8.2.1) and (8.2.2) may then be linearized by neglecting all terms involving products of these small quantities, resulting in  0 ∂ E V ∂t Drp C 0 gZ 8.2.3 ∂ ∂t D 0 rÐ E V8.2.4 The velocity vector may comprise two parts. One part is rotational and is associated with the vorticity. An equation for vorticity can be obtained by taking the curl of Eq. (8.2.3). Then, since the curl of the right-hand side of that 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. Other properties 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 depen- dent. This part can be considered as originating from a potential function . Therefore we may write (see Sec. 4.2) E V Dr8.2.5 This expression implies that the velocity of interest for wave propagation stems from a potential function. According to linear wave theory, any steady rotational velocity field does not affect this velocity. Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. In the usual case of waves developing on a homogeneous water environ- ment, we consider a fluid with constant density and apply a coordinate system in which z is the vertical upward coordinate, with z D 0 corresponding to the elevation 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 ∂ ∂t C p  C gz D 0 8.2.6 If the density of the fluid is not constant, then combining Eqs. (8.2.5) and (8.2.4) produces ∂ ∂t D 0 r 2 8.2.7 If there is no change in the fluid density, as with surface water waves, then the left-hand side of this last equation vanishes and the potential function satisfies Laplace’s equation. To summarize the results so far, Eqs. (8.2.1)–(8.2.7) indicate that with regard to small amplitude wave motion, the equations of motion can be linearized and the velocity of interest originates from a potential function. However, the phenomenon of surface water waves is associated with the prop- agation of surface disturbances. The equations representing the free surface of the water, as shown below, also can be linearized to represent the boundary condition of the water free surface by a linear differential equation with regard to the potential function. Simple wave motions of small amplitude are described by the wave equation, ∂ 2 Á ∂t 2 D c 2 r 2 Á8.2.8 where Á is the displacement of the free surface and c is the wave velocity. This is a linear hyperbolic differential equation in a two-dimensional space. For the analysis and calculation of many wave phenomena, it is sufficient to consider a one-dimensional space. For this case, Eq. (8.2.8) collapses to ∂ 2 Á ∂t 2 D c 2 ∂ 2 Á ∂x 2 8.2.9 where it has been assumed that the wave propagates along the x-direction. The general form of the solution of Eq. (8.2.9) is given by Á D fx  ct C Fx C ct 8.2.10 where f and F represent arbitrary functions. Specific forms of these functions are discussed in the following sections of this chapter. Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. 8.3 GRAVITY SURFACE WAVES Surface water waves develop at the water free surface, which is the interface between the water and air phases. In general, this interface is considered as a discontinuity in the overall distribution of density in the domain. The state of stable equilibrium of the system is represented by water occupying the lowest portions of the domain. Disturbances to the state of equilibrium are represented by surface gravity waves. Such waves propagate only in the horizontal direc- tion, while the restoring gravity force acts in the vertical direction. Therefore there is no preferred horizontal direction of the disturbance propagation, and the waves are isotropic (they may move equally in any horizontal direction). However, waves of different wavelengths penetrate to different depths into the water 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 coordinate system as in Sec. 8.2, using z as the vertical upward coordinate, with its origin at the water free surface. We also drop the subscript 0 for , with the understanding that water density is constant (not stratified). The undisturbed absolute pressure, p 0 is distributed hydrostatically, p 0 D p a  gz 8.3.1 where p a is the atmospheric pressure. The pressure disturbance, p e , originating from the wave disturbance, is defined by p e D p  p 0 8.3.2 The linearized equation of motion, from Eq. (8.2.3), is given by  ∂ E V ∂t Drp e 8.3.3 where p a 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 to Laplace’s equation, r 2  D 0 8.3.4 Laplace’s equation cannot describe wave propagation in a fluid that is completely bounded by stationary surfaces, but it can describe wave propaga- tion by the employment of the boundary condition at the original free water surface. At the original free water surface, the pressure value is associated with the wave displacement, namely the disturbed free surface elevation, Á, according to p e D gÁ 8.3.5 Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. where Á D Áx, y, t 8.3.6 Equation (8.3.5), along with Eqs. (8.3.1) and (8.3.2), indicates that at the disturbed 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 the velocity, which is associated with the wave propagation, p e D ∂ ∂t 8.3.7 Therefore at the disturbed free water surface, Eqs. (8.3.5)–(8.3.7) yield the surface boundary condition, ∂ ∂t     zDÁ DgÁ 8.3.8 In practice, this represents a very complicated boundary condition, since the value of Á is not known (in fact, it is part of the desired solution). However, according to linear theory, we may consider that Á is a small quantity. There- fore Eq. (8.3.8) is approximated by ∂ ∂t     zD0 DgÁ 8.3.9 According to the mean value theorem, the difference between the values of the left-hand sides of Eqs. (8.3.8) and (8.3.9) is equal to the product of the disturbance Á with the derivative of the left-hand-side expression with respect to z, evaluated at a point intermediate to the disturbed and undisturbed water surfaces. 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 Á is equal to the vertical fluid velocity at the surface, namely, ∂Á ∂t C E VÐrÁ D ∂ ∂z     zDÁ 8.3.10 According to linear theory, this expression is simplified by neglecting the advective rate of change of Á, since it is a product of two small quantities. Furthermore, the right-hand side of Eq. (8.3.10) is evaluated for z D 0, instead of z D Á. Therefore ∂Á ∂t D ∂ ∂z     zD0 8.3.11 Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. By differentiating Eq. (8.3.9) with respect to t and applying Eq. (8.3.11), we obtain ∂ 2  ∂t 2 C g ∂ ∂z D 0atz D 0 8.3.12 The wave propagation in the domain is then fully determined by solving the Laplace equation (8.3.4), subject to the boundary condition given by Eq. (8.3.12). 8.4 SINUSOIDAL SURFACE WAVES ON DEEP WATER 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, Á, as a function of time and position. An alternative function to describe wave movement is F D F 8.4.1 where  D ωt  kx, ω is the angular velocity,orradian frequency,andk is the wave number. The parameters ω and k are connected with the wave velocity, c,as c D ω k 8.4.2 In addition, the wave number is related to the wave length, , and the angular velocity is related to the wave period, t p ,as  D 2 k ; t p D 2 ω 8.4.3 By combining Eqs. (8.4.2) and (8.4.3), it is seen that the wave propagates a distance of a single wavelength during one period, namely,  D ct p 8.4.4 The function F of Eq. (8.4.1) has a constant value for a constant value of the variable . Therefore this equation represents constant values of the function F, moving in the positive x-direction. Such a function may refer to waves propagating in that direction. Differentiating Eq. (8.4.1) twice with respect to t and twice with respect to x gives ∂ 2 F ∂t 2 D ω 2 F 00 ∂ 2 F ∂x 2 D k 2 F 00 8.4.5 Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. where the double prime represents differentiation with respect to the variable . Using the wave velocity from Eq. (8.4.2) then gives ∂ 2 F ∂t 2 D c 2 ∂ 2 F ∂x 2 8.4.6 This is the same wave equation as was obtained earlier in Eq. (8.2.9). The potential function (Eq. 8.2.5), which represents motions in the entire water domain subject to surface water waves, usually consists of a product of a function similar to that given by Eq. (8.4.1) and another function, which describes an attenuation of the fluid motion with the water depth. In the case of periodic surface waves, F is usually specified as a sine or cosine function. Therefore the potential function can be represented using a sine, a cosine, or the real part of a complex function, such as  D fz sinωt  kx D fz sin  ω  t  x c  8.4.7a  D fz cosωt  kx D fz cos  ω  t  x c  8.4.7b  D fz exp[iωt kx] D fz exp  iω  t  x c  8.4.7c where fz is a function that represents the variation in the vertical direc- tion. Since we know that the potential function is governed by the Laplace equation (8.3.5), it follows that f 00 z k 2 fz D 0 8.4.8 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 should vanish. Therefore a solution of Eq. (8.4.8) that is consistent with a vanishing value of  with increasing depth (where z !1)isgivenby fz D  0 e kz 8.4.9 where  0 is a constant equal to the value of f at the surface, and z D 0. Combining Eqs. (8.4.7a) and (8.4.9), we obtain ∂ ∂t Dω 0 e kz sinωt kc ∂ ∂z D k ∂ 2  ∂t 2 Dω 2 8.4.10 Then, using the first part of this result with Eq. (8.3.10), a solution for the surface displacement is obtained as Á D ω g  0 sinωt kx D a sinωt  kx a D ω g  0 8.4.11 Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. This is the relationship between the wave amplitude a and the maximum amplitude of the potential function. By introducing Eq. (8.4.10) into Eq. (8.3.13), a relationship between the frequency and the wave number for gravity waves on a deep-water environ- ment is found as ω 2 D gk 8.4.12 This is known as a dispersion relationship, giving the dependence of the wave propagation on the wavelength (or wave number). Using the defini- tion for wave speed (Eq. 8.4.2), this last result shows that waves of different wavelengths propagate with different velocities: c D ω k D  g k D  g 2 8.4.13 Considering that g D 9.81 m/s 2 , we apply Eqs. (8.4.13) and (8.4.3) to obtain c D 1.25 p t p D 0.80 p 8.4.14 where  is measured in meters, t p is measured in seconds, and c in m/s. Then, considering that the range of typical wavelengths for surface water waves is between 1 m and 100 m, ranges of typical wave velocity and period are 1.25 m/s Ä c Ä 12.5m/s 0.8 s Ä t p Ä 8.0 s8.4.15 It should be noted that although Eq. (8.4.15) represents typical values, extreme cases may exist, with wavelengths as low as 0.1 m or as large as 1000 m. Near the sea shore the wavelength is generally much less, and the waves should be described with alternative theories, since the deep-water assumption is no longer valid. 8.4.3 Pathlines of the Fluid Particles Velocity components in a wavy flow field are obtained by differentiating the potential function. For example, taking Eq. (8.4.7b), the velocity components are found as u D ∂ ∂x D k 0 e kz sinωt  kx 8.4.16 and w D ∂ ∂z D k 0 e kz cosωt kx 8.4.17 Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. Thus both velocity components vary sinusoidally with time and have the same amplitude, which decays exponentially with depth. However, it should be noted 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) to describe the potential function). Equations (8.4.16) and (8.4.17) also indicate that at a fixed position, the velocity vector maintains a constant absolute value and 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 that the deviations of a fluid particle from an initial position x 0 ,z 0  are relatively small, then the differential equations of the fluid particle pathline are given approximately by dx dt D u ¾ D k 0 e kz 0 sinωt  kx 0 8.4.18 and dz dt D w ¾ D k 0 e kz 0 cosωt kx 0 8.4.19 Direct integration of these results gives an approximation for the instantaneous position of the fluid particle, x D k ω  0 e kz 0 cosωt  kx 0  C C 1 8.4.20a z D k ω  0 e kz 0 sinωt kx 0  C C 2 8.4.20b where C 1 and C 2 are constants for each particular fluid particle. By eliminating time from Eqs. (8.4.20a) and (8.4.20b), we obtain x  C 1  2 C z  C 2  2 D  k ω  0 e kz 0  2 8.4.21 Figure 8.1 Schematic description of pathlines and instantaneous positions of fluid particles subject to motion due to sinusoidal surface wave on deep water. Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. This result shows that the fluid particles move in circular pathlines, which also is evident from the previous conclusion that the velocity components have equal amplitudes (Eqs. 8.4.16 and 8.4.17). The radius of the pathline decays exponentially with water depth and does not depend on the horizontal coordinate. According to Eqs. (8.4.20) and (8.4.21), the phase of the fluid particle location does not depend on the z coordinate. Figure 8.1 provides a schematic description of various pathlines of different fluid particles and their instantaneous positions. 8.4.4 The Shape of the Streamlines The shape of the streamlines is calculated by considering the following rela- tionships between the stream function and the real parts of the velocity compo- nents given by Eqs. (8.4.15) and (8.4.16): ∂ ∂z D u D k 0 e kz sinωt kx 8.4.22 ∂ ∂x Dw Dk 0 e kz cosωt kx 8.4.23 Direct integration of these expressions yields  D  0 e kz sinωt  kx C C8.4.24 where C is an arbitrary constant. This shows that streamlines, like the prop- agating surface waves, are sinusoidal and the amplitude of the streamlines decays exponentially with the water depth. 8.4.5 The Wave Energy The excess energy in surface water waves is divided between kinetic and potential energy. The excess potential wave energy incorporated in a surface area of unit width, with length equal to one wavelength, and where the water depth, h, is large but finite, is WE p D   0   Á h gz dz   0 h gz dz  dx D   0  1 2 gÁ 2  h 2  C 1 2 gh 2  dx D 1 2 g   0 Á 2 dx 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 Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. [...]... earlier as Eq (7.2.21), Sf D 0 8. 7.16 y By introducing Eq (8. 7.2) into Eq (8. 7.13), it can be shown that 2c ∂c ∂V ∂V CV C D g S0 ∂x ∂x ∂t 8. 7.17 Sf Also, introducing Eq (8. 7.2) into Eq (8. 7.6), we obtain 2V ∂c ∂V ∂c Cc C2 D0 ∂x ∂x ∂t 8. 7. 18 Adding Eqs (8. 7.17) and (8. 7. 18) results in ∂ V C 2c ∂ V C 2c C VCc D g S0 ∂t ∂x Copyright 2001 by Marcel Dekker, Inc All Rights Reserved Sf 8. 7.19 and subtracting them... for the calculation of Ek is at the water surface Therefore Eq (8. 4.27) can be replaced with WEk D 0 ∂ 1  2 ∂z dx 8. 4. 28 zD0 This result is further modified by introducing an expression for the velocity potential From Eq (8. 4.10), D 1 ∂ k ∂z 8. 4.29 Substituting Eqs (8. 4.29) and (8. 3.12) into Eq (8. 4. 28) , we obtain WEk D 0 1 2k ∂Á ∂t 2 dx 8. 4.30 The appearance of the wave number k in this result is... ∂ f0 D ∂z f  D [k tanh kh ] 8. 5.6 zD0 Where tanh x D sinh x / cosh x is the hyperbolic tangent function Equations (8. 3.12) and (8. 5.6) then give ω2 D gk tanh kh 8. 5.7 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 cD ω D k g tanh kh k 8. 5 .8 In order to illustrate the effect... C V ∂V 2c 8. 7.20 D g S0 Sf ∂t ∂x Equations (8. 7.19) and (8. 7.20) are two first-order hyperbolic partial differential equations We may refer to the x –t plane and identify the characteristics of each of these equations The family of characteristics of Eqs (8. 7.19) and (8. 7.20) are given, respectively, by c dx dx DVCc DV c 8. 7.21 dt dt If the terms on the right-hand sides of Eqs (8. 7.19) and (8. 7.20) are... we obtain ∂k ∂ω C D0 ∂t ∂x 8. 6.7 We consider that ω is a function of k, as implied by Eq (8. 4.2), namely, ωDωk 8. 6 .8 Introducing this relationship into Eq (8. 6.7) gives ∂k ∂k CU D0 ∂t ∂x 8. 6.9 where U is the group velocity, defined as UDU k D dω dk 8. 6.10 Equation (8. 6.9) indicates that k is constant along paths in the x –t plane, which satisfy the condition that x Ut D const 8. 6.11 In the case of waves... ∂ QV C ∂t ∂x 8. 7 .8 Combining Eqs (8. 7.7) and (8. 7 .8) yields ∂ ∂Q C QV ∂t ∂x D A ∂h ∂x 0P 8. 7.9 where h is the elevation of the water surface, given by hDzCy Copyright 2001 by Marcel Dekker, Inc All Rights Reserved 8. 7.10 For a wide rectangular channel, P is approximately equal to the width, and Eq (8. 7.9) may be divided by the channel width to obtain ∂ yV ∂ C ∂t ∂x yV2 C g y2 2 D yS0 0 8. 7.11 where... rearranging Eq (8. 4.12) and substituting into Eq (8. 4.11), we obtain ÁD k 0 sin ωt ω kx D a sin ωt 8. 4.31 kx Thus ∂Á D k0 cos ωt ∂t kx D ωa cos ωt kx 8. 4.32 This result is then substituted into Eqs (8. 4.25) and (8. 4.30), also using Eqs (8. 4.31) and (8. 4.12) After dividing by the wave length, , to obtain the total wave energy per unit area of the water surface, the total energy is E D Ep C Ek D 1 2 ga2 8. 4.33... given by Eq (8. 4.25) The expression for the kinetic energy of the wave also is identical to that of waves 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 same relationship typical of waves on deep water, namely Eq (8. 4.33) 8. 6 THE GROUP... Rights Reserved 8. 7.24a 8. 7.24b 8. 7.24c 8. 7.24d Figure 8. 11 Characteristic curves for two types of moving observers Since AB is a characteristic of the first type, we also find VA C 2cA D VB C 2cB 8. 7.24e The various expressions of Eq (8. 7.24) can be satisfied only if VA D VB cA D cB 8. 7.25 This indicates that the curve AB is a straight line, and therefore all characteristic curves of Fig 8. 11 are straight... bed, dz dx S0 D 8. 7.12 We subtract the continuity result, Eq (8. 7.6), from Eq (8. 7.11) and divide the result by the water depth and the specific weight of the water, to obtain 1 g ∂V ∂t C ∂ ∂x V2 CyCz 2g D 0 8. 7.13 y This also can be represented in another form, as ∂H 1 ∂V 0 C C D0 ∂x g ∂t y 8. 7.14 where H is the water head, HDhC V2 2g 8. 7.15 The last term on the left-hand side of Eq (8. 7.14) represents . There- fore Eq. (8. 3 .8) is approximated by ∂ ∂t     zD0 DgÁ 8. 3.9 According to the mean value theorem, the difference between the values of the left-hand sides of Eqs. (8. 3 .8) and (8. 3.9). governed by the Laplace equation (8. 3.5), it follows that f 00 z k 2 fz D 0 8. 4 .8 8. 4.2 The Potential Function for Deep Water Waves The general solution of Eq. (8. 4 .8) can be obtained by a linear. kx 8. 5.18a ∂ ∂x Dw D aω coshkh sinh[kz C h]cosωt kx 8. 5.18b By direct integration of Eq. (8. 5. 18) , the stream function is found as  D aω k coshkh sinh[kz C h]sinωt kx 8. 5.19 This