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@x ¢ dx dt for: z = ³ = @³ @t + u ¢ d³ dx (5.36) The second term in this expression is a product of two values, which are both small because of the assumed small wave steepness. This product becomes even smaller (second order) and can be ignored, see …gure 5.7. This linearization provides the vertical velocity of the wave surface: dz dt = @³ @t for: z = ³ (5.37) The vertical velocity of a water particle in the free surface is then: @©w @z = @³ @t for: z = ³ (5.38) 5-10 CHAPTER 5. OCEAN SURFACE WAVES Figure 5.7: Kinematic Boundary Condition Analogous to equation 5.30 this condition is valid for z = 0 too, instead of for z = ³ only. A di¤erentiation of the free surface dynamic boundary condition (equation 5.30) with respect to t provides: @2©w @t2 + g @³ @t = 0 for z = 0 (5.39) or after re-arranging terms: @³ @t + 1 g ¢ @ 2©w @t2 = 0 for z = 0 (5.40) Together with equation 5.37 this delivers the free surface kinematic boundary condi- tion or the Cauchy-Poisson condition:¯¯¯¯ @z @t + 1 g ¢ @ 2©w @t2 = 0 for: z = 0 ¯¯¯¯ (5.41) Dispersion Relationship The information is now available to establish the relationship between ! and k (or equiva- lently T and ¸) referred to above. A substitution of the expression for the wave potential (equation 5.34) in equation 5.41 gives the dispersion relation for any arbitrary water depth h: j!2 = k g ¢ tanh khj (5.42) In many situations, ! or T will be know; one must determine k or ¸: Since k appears in a nonlinear way in 5.42, that equation will generally have to be solved iteratively. In deep water (tanhkh = 1), equation 5.42 degenerates to a quite simple form which can be used without di¢culty: !2 = k g (deep water) (5.43) and the deep water relation between T and ¸ becomes: T = r 2¼ g ¢ p ¸ or ¸ = g 2¼ ¢T 2 (deep water) (5.44) 5.2. REGULAR WAVES 5-11 Substitution of g = 9:81 m/s2 and ¼ yields: T t 0:80 ¢ p ¸ or ¸ t 1:56 ¢ T 2 (deep water) (5.45) Note that this regular wave relation cannot be used to describe the relation between the average wave length and the average wave period of an irregular sea. However in a more regular swell, this relation can be used with an accuracy of about 10 to 15 per cent. In shallow water, the dispersion relation is found by substituting tanhkh = kh in equa- tion 5.42; thus: ! = k ¢ p gh (shallow water) (5.46) and the shallow water relation between T and ¸ becomes: T = ¸p gh or ¸ = T ¢ p gh (shallow water) (5.47) Cauchy-Poisson Condition in Deep Water In deep water (short waves), the free surface kinematic boundary condition or Cauchy-Poisson condition is often given in another form in the literature. The space and time dependent wave potential, ©w(x; z; t), is divided in a space-dependent part, Áw(x; z), and a time-dependent part, 1 ¢ sin !t, by de…ning: ©w(x; z; t) = Áw(x; z) ¢ sin!t (5.48) Equation 5.41: @z @t + 1 g ¢ @ 2©w @t2 = 0 for: z = 0 can with equation 5.37 be written as: @³ @t + 1 g ¢ @ 2©w @t2 = 0 for: z = 0 (5.49) and using equation 5.38 results in: @©w @z + 1 g ¢ @ 2©w @t2 = 0 for: z = 0 (5.50) With the dispersion relation in deep water, !2 = k g; the free surface kinematic boundary condition or the Cauchy-Poisson condition becomes: @Áw @z ¡ k ¢ Áw = 0 for z = 0 (deep water) (5.51) 5.2.2 Phase Velocity With the dispersion relation (equation 5.42) the wave celerity (c = ¸=T = !=k) becomes:¯¯¯¯ c = r g k ¢ tanh kh ¯¯¯¯ (5.52) 5-12 CHAPTER 5. OCEAN SURFACE WAVES The phase velocity increases with the wave length (k = 2¼=¸); water waves display disper- sion in that longer waves move faster than shorter ones. As a result of this phenomena, sea sailors often interpret swell (long, relatively low wind-generated waves which have moved away from the storm that generated them) as a warning of an approaching storm. In deep water, the phase velocity is found by substituting tanhkh = 1 in equation 5.52; thus: c = r g k = g ! (deep water) (5.53) With some further substitution, one can get: c = r g k = r g ¢ ¸ 2¼ = r g 2¼ ¢ ¸ t 1:25 p ¸ t 1:56 ¢ T (deep water) (5.54) In shallow water, the phase velocity is found by substituting tanhkh = kh in equation 5.52; thus: c = p gh (shallow water) (5.55) The phase velocity is now independent of the wave period; these waves are not dispersive. This celerity, p gh, is called the critical velocity. This velocity is of importance when sailing with a ship at shallow water. Similar e¤ects occur with an airplane that exceeds the speed of sound. Generally, the forward ship speed will be limited to about 80% of the critical velocity, to avoid excessive still water resistance or squat (combined sinkage and trim). Indeed, only a ship which can plane on the water surface is able to move faster than the wave which it generates. 5.2.3 Water Particle Kinematics The kinematics of a water particle is found from the velocity components in the x- and z-directions, obtained from the velocity potential given in equation 5.34 and the dispersion relation given in equation 5.42. Velocities The resulting velocity components - in their most general form - can be expressed as: u = @©w @x = dx dt = ³a ¢ kg ! ¢ coshk (h+ z) coshkh ¢ cos (kx ¡ !t) w = @©w @z = dz dt = ³a ¢ kg ! ¢ sinhk (h+ z) coshkh ¢ sin (kx ¡ !t) (5.56) A substitution of: kg = !2 tanhkh derived from the dispersion relation in equation 5.42, provides: 5.2. REGULAR WAVES 5-13 ¯¯¯¯ u = ³a ¢ ! ¢ cosh k (h + z) sinhkh ¢ cos (kx¡ !t) ¯¯¯¯ ¯¯¯¯ w = ³a ¢ ! ¢ sinh k (h + z) sinkh ¢ sin (kx ¡ !t) ¯¯¯¯ (5.57) An example of a velocity …eld is given in …gure 5.8. Figure 5.8: Velocity Field in a Shallow Water Wave In deep water, the water particle velocities are given by: u = ³a! ¢ ekz ¢ cos (kx¡ !t) w = ³a! ¢ ekz ¢ sin (kx ¡ !t) (deep water) (5.58) An example of a velocity …eld is given in …gure 5.9. Figure 5.9: Velocity Field in a Deep Water Wave The circular outline velocity or orbital velocity in deep water (short waves) is: 5-14 CHAPTER 5. OCEAN SURFACE WAVES Vo = p u2 + w2 = ³a! ¢ ekz (deep water) (5.59) In shallow water, the water velocity components are: u = ³a! ¢ 1 kh ¢ cos (kx¡ !t) w = ³a! ¢ (1 + z h ) ¢ sin (kx¡ !t) (shallow water) (5.60) Displacements Because of the small steepness of the wave, x and z in the right hand side of these equations can be replaced by the coordinates of the mean position of the considered water particle: x1 and z1. Hence the distances x¡x1 and z ¡ z1 are so small that di¤erences in velocities resulting from the water motion position shifts can be neglected; they are of second order. Then, an integration of equations 5.57 over t yields the water displacements: ¯¯¯¯ x = ¡³a ¢ cosh k (h + z1) sinh kh ¢ sin (kx1 ¡ !t) +C1 ¯¯¯¯ ¯¯¯¯ z = +³a ¢ sinh k (h + z1) sinh kh ¢ cos(kx1 ¡ !t) + C2 ¯¯¯¯ (5.61) Trajectories It is obvious that the water particle carries out an oscillation in the x- and z-directions about a point (C1; C2). This point will hardly deviate from the situation in rest, so: C1 ¼ x1 and C2 ¼ z1. The trajectory of the water particle is found by an elimination of the time, t, by using: sin2 (kx1 ¡ !t) + cos2 (kx1 ¡ !t) = 1 (5.62) which provides: ¯¯¯¯ ¯¯¯ (x¡ x1)2³ ³a ¢ coshk(h+z1)sinhkh ´2 + (z ¡ z1)2³ ³a ¢ sinhk(h+z1)sinhkh ´2 = 1 ¯¯¯¯ ¯¯¯ (5.63) This equation shows that the trajectories of water particles are ellipses in the general case, as shown in …gure 5.10. The water motion obviously decreases as one moves deeper below the water surface. Half the vertical axis of the ellipse - the vertical water