Superposition of Linear Plane Progressive Waves

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Superposition of Linear Plane Progressive Waves

Oblique Plane Waves

Consider wave propagation at an angle [math] \theta \, [/math] to the x-axis

[math] \eta = A \cos ( kx\cos\theta+kz\sin\theta-\omega{t}) = A \cos (k_xx+k_zz-\omega{t}) \, [/math]
[math] \phi = \frac{gA}{\omega} \frac{\cosh k (y+h)}{\cosh k h} \sin (kx\cos\theta+kz\sin\theta-\omega t) [/math]
[math] \omega = g k \tanh k h; \ k_x=k\cos\theta, k_z = k\sin\theta, \ k=\sqrt{k_x+k_z} [/math]

Standing Waves

[math] \eta = A \cos (kx-\omega t) + A \cos (-kx-\omega{t}) = 2 A \cos kx \cos \omega t \,[/math]
[math] \phi = - \frac{2 g A}{\omega} \frac{\cosh k (y+h)}{\cosh k h} \cos kx \sin \omega t [/math]
[math] \frac{\partial\eta}{\partial{x}} \sim \frac{\partial\phi}{\partial{x}} = \cdots \sin kx = 0 \, [/math] at [math] x=0, \ \frac{n\pi}{k} = \frac{n\lambda}{2} \, [/math]

Therefore, [math] \left. \frac{\partial\phi}{\partial{x}} \right|_x = 0 \, [/math]. To obtain a standing wave, it is necessary to have perfect reflection at the wall at [math] x=0 \, [/math].

Define the reflection coefficient as [math] R \equiv \frac{A_R}{A_I} (\leq 1) \, [/math].

[math] A_I = A_R \, [/math]
[math] R = \frac{A_R}{A_I} = 1 \, [/math]

Oblique Standing Waves

[math] \eta_I = A \cos ( k x \cos \theta + k z \sin \theta - \omega t ) \, [/math]
[math] \eta_R = A \cos ( k x \cos (\pi-\theta) + k z \sin (\pi-\theta) - \omega t ) \, [/math]
[math] \theta_R = \pi - \theta_I \, [/math]

Note: same [math] A, \ R = 1 \, [/math].

[math] \eta_T = \eta_I + \eta_R = 2 A \cos ( k x \cos \theta ) \cos ( k z \sin \theta - \omega t ) \ ,[/math]

and

[math] \lambda_x = \frac{2\pi}{k\cos\theta}; \ V_{P_x} = 0; \ \lambda_z = \frac{2\pi}{k\sin\theta}; \ V_{P_z} = \frac{\omega}{k\sin\theta} [/math]

Check:

[math] \frac{\partial\phi}{\partial{x}} \sim \frac{\partial\eta}{\partial{x}} \sim \cdots \sin (kx\cos\theta) = 0 \, [/math] on [math] x=0 \, [/math]

Partial Reflection

[math] \eta_I = A_I \cos ( k x - \omega t ) = A_I R_e \left\{ e^{i \ kx - \omega t} \right\} [/math]
[math] \eta_R = A_R \cos ( k x + \omega t + \delta ) = A_I R_e \left\{ e^{-i \ kx \omega t} \right\} [/math]

[math] R \, [/math]: Complex reflection coefficient

[math] R = |R| e^{-i\delta}, |R| = \frac{A_R}{A_I} \, [/math]
[math] \eta_T = \eta_I + \eta_R = A_I R_e \left\{ e^{i\ kx-\omega t} \left( 1 + R e^{-ikx} \right) \right\} [/math]
[math] |\eta_T| = A_I \left[ 1 + |R| + 2 |R| \cos ( 2 k x + \delta ) \right] \, [/math]

At node,

[math] |\eta_T| = |\eta_T| = A_I ( 1 - |R| ) \, [/math] at [math] \cos (2 k x + \delta) = -1 \, [/math] or [math] 2 k x + \delta = ( 2 n + 1 ) \pi \, [/math]

At antinode,

[math] |\eta_T| = |\eta_T| = A_I ( 1 + |R| ) \, [/math] at [math] \cos (2 k x + \delta) = 1 \, [/math] or [math] 2 k x + \delta = 2 n \pi \, [/math]
[math] 2 k L = 2 \pi \, [/math] so [math] L = \frac{\lambda}{2} \, [/math]
[math] |R| = \frac{|\eta_T|-|\eta_T|}{|\eta_T|+|\eta_T|} = |R(k)| \, [/math]

Wave Group

2 waves, same amplitude [math] A \, [/math] and direction, but [math] \omega \, [/math] and [math] k \, [/math] very close to each other.

[math] \eta = \Re \left( A e^{i k_1 x - \omega_1 t } \right) \, [/math]
[math] \eta = \Re \left( A e^{i k_2 x - \omega_2 t } \right) \, [/math]
[math] \omega, = \omega, ( k , ) \, [/math] and [math] V_{P_1} \approx V_{P_2} \, [/math]
[math] \eta_T = \eta + \eta = \Re \left\{ A e^{i\ k_1x-\omega_1t} \left[ 1 + e^{i\ \delta kx - \delta\omega t} \right] \right\} \, [/math] with [math] \delta k = k - k \, [/math] and [math] \delta \omega = \omega - \omega \, [/math]
[math] \begin{Bmatrix} |\eta_T| = 2 |A| \ \mbox{when} \ \delta k x - \delta \omega t = 2n \pi \\ |\eta_T| = 0 \ \mbox{when} \ \delta k x - \delta \omega t = (2n+1) \pi \end{Bmatrix} x_g = V_g t, \ \delta k V_g t =0 \ \mbox{when} \ V_g = \frac{\delta\omega}{\delta k} [/math]

In the limit,

[math] \delta k, \delta\omega \to 0, \ \left. V_g = \frac{d\omega}{dk} \right|_{k_1\approx k_2\approx k} , [/math]

and since

[math] \omega = g k \tanh k h \Rightarrow \, [/math]
[math] V_g = \underbrace{\left( \frac{\omega}{k} \right)}_{V_P} \underbrace{\frac{1}{2} \left( 1+\frac{2kh}{\sinh 2kh} \right)}_n [/math]

[math] \begin{Bmatrix} & (a) \ \mbox{deep water} \ kh \gg 1 & n = \frac{V_g}{V_P} = -1 \\ & (b) \ \mbox{shallow water} \ kh \ll 1 & n=\frac{V_g}{V_P}=1 \ \mbox{no dispersion} \\ & (c) \ \mbox{intermediate depth} & -1 \lt n \lt 1 \end{Bmatrix} V_g \leq V_P [/math]

Wave Energy -Energy Associated with Wave Motion.

For a single plane progressive wave:

align="center" ! Energy per unit surface area of wave
[math] \bullet [/math] Potential energy PE [math] \bullet [/math] Kinetic energy KE
PE without wave [math] = \int_{-h} \rho g y dy = - - \rho g h \, [/math]
PE with wave [math] \int_{-h}^\eta \rho g y dy = - \rho g ( \eta - h ) \, [/math]
[math] PR_{wave} = - \rho g \eta = - \rho g A \cos ( kx - \omega t) \, [/math]
[math] KE_{wave} = \int_{-h}^\eta dy - \rho ( u + v ) [/math]
Deep water [math] = \cdots = - \rho g A \ [/math] to leading order
Finite depth [math] = \cdots \, [/math]
Average energy over one period or one wavelength
[math] \overline{PE}_{wave} = - \rho g A \, [/math] [math] \overline{KE}_{wave} = - \rho g A \, [/math] at any [math] h \, [/math]
  • Total wave energy in deep water:

[math] E = PE + KE = - \rho g A \left[ \cos ( k x - \omega t ) + - \right] \, [/math]

  • Average wave energy [math] E \, [/math] (over 1 period or 1 wavelength) for any water depth:

[math] \overline{E} = - \rho g A \left[ \overline{PE} + \overline{KE} \right] = - \rho g A = E_S , \, [/math]
[math] E_S \equiv \, [/math] Specific Energy: total average wave energy per unit surface area.

  • Linear waves: [math] \overline{PE} = \overline{KE} = \frac{1}{2} E_S \, [/math] (equipartition).
  • Nonlinear waves: [math] \overline{PE} \gt \overline{PE} \, [/math].

Energy Propagation - Group Velocity

Consider a fixed control volume [math] V \, [/math] to the right of 'screen' [math] S \, [/math]. Conservation of energy:

[math] \underbrace{\frac{dW}{dt}} \, [/math] [math] = \, [/math] [math] \underbrace{\frac{dE}{dt}} \, [/math] [math] = \, [/math] [math] \underbrace{\Im} \, [/math]
rate of work done on [math] S \, [/math] rate of change of energy in [math] V \, [/math] energy flux left to right

where

[math] \Im = \int_{-h}^\eta pu dy \ \, [/math] with [math] \ p = - \rho \left( \frac{d\phi}{dt} + gy \right) \ [/math] and [math] \ u = \frac{\partial\phi}{\partial x} \, [/math]
[math] \overline{\Im} = \underbrace{\left( -\rho g A \right)}_{\overline{E}} \underbrace{\underbrace{\frac{\omega}{k}}_{V_P} \underbrace{\left[-\left(1+\frac{kh}{kh}\right)\right]}_n}_{V_g} = \overline{E} (n V_P) = \overline{E} V_g [/math]

e.g. [math] A = 3m, \ T = 10\mbox{sec} \rightarrow \overline{\Im} = 400KW/m \, [/math]

Equation of Energy Conservation

[math] \left( \overline{\Im} - \overline{\Im} \right) \Delta t = \Delta \overline{E} \Delta x \, [/math]
[math] \overline{\Im} = \overline{\Im} + \left. \frac{\partial\overline{\Im}}{\partial{x}} \right| \Delta x + \cdots \, [/math]
[math] \frac{\partial\overline{E}}{\partial{t}} + \frac{\partial\overline{\Im}}{\partial{x}} = 0 \, [/math], but [math] \overline{\Im} = V_g \overline{E} \, [/math]
[math] \frac{\partial\bar{E}}{\partial{t}} + \frac{\partial}{\partial{x}} \left( V_g \overline{E} \right) = 0 \, [/math]

1. [math] \frac{\partial\overline{E}}{\partial{t}}=0, \ V_g \overline{E} = \ \, [/math] constant in [math] x \, [/math] for any [math] h(x) \, [/math].

2. [math] V_g = \, [/math] constant (i.e., constant depth, [math] \delta k \ll k )\, [/math]

[math] \left( \frac{\partial}{\partial t} + V_g \frac{\partial}{\partial{x}} \right) \bar{E} = 0, \ [/math] so [math] \ \overline{E} = \overline{E} (x-V_g t) \ [/math] or [math] \ A = A ( x - V_g t ) \, [/math]

i.e., wave packet moves at [math] V_g \, [/math].

Steady Ship Waves, Wave Resistance

  • Ship wave resistance drag [math] D_w \, [/math]
Rate of work done = rate of energy increase
[math] D_w U + \overline{\Im} = \frac{d}{dt} (\overline{E}L) = \overline{E}U \, [/math]
[math] D_w = \frac{1}{U} ( \overline{E} U - \overline{E} U /2 ) = - \overline{E} = - \rho g A \ \Rightarrow \ D_w \propto A [/math]
  • Amplitude of generated waves

The amplitude [math] A \, [/math] depends on [math] U \, [/math] and the ship geometry. Let [math] \ell \equiv \, [/math] effective length.

To approximate the wave amplitude [math] A \, [/math] superimpose a bow wave ([math] \eta_b \, [/math]) and a stern wave ([math] \eta_s \, [/math]).

[math] \eta_b = a \cos (kx) \ \, [/math] and [math] \ \eta_S = - a \cos (k ( x+ \ell )) \, [/math]
[math] \eta_T = \eta_b + \eta_S \, [/math]
[math] A = | \eta_T | = 2 a \left|\sin (-k\ell)\right| \ \leftarrow \ [/math] envelope amplitude
[math] D_w = - \rho g A = \rho g a \sin ( -k \ell ) \ \Rightarrow \ D_w = \rho g a \sin \left( - \frac{g\ell}{U^2} \right) \, [/math]
  • Wavelength of generated waves To obtain the wave length, observe that the phase speed of the waves must equal [math] U \, [/math]. For deep water, we therefore have
[math] V_p = U \ \Rightarrow \ \frac{\omega}{k} = U \ \begin{matrix} \mbox{deep} \\ \longrightarrow \\ \mbox{water} \end{matrix} \sqrt{\frac{g}{k}} = U, \ [/math] or [math] \lambda = 2 \pi \frac{U}{g} [/math]
  • Summary Steady ship waves in deep water.
[math] U = \, [/math] ship speed
[math] V_p = \sqrt{\frac{g}{k}} = U; \ [/math] so [math] \ k = \frac{g}{U} \ \, [/math] and [math] \ \lambda = 2 \pi \frac{U}{g} \, [/math]
[math] L = \, [/math] ship length, [math] \ \ell \sim L \, [/math]
[math] D_w = \rho g a \sin \left( - \frac{g\ell}{U^2} \right) \cong \rho g a \sin \left( \frac{1}{2F_{rL}} \right) \cong \rho g \sin \left( \frac{1}{2F_{rL}} \right) [/math]



This article is based on the MIT open course notes and the original article can be found here.

Marine Hydrodynamics