Superposition of Linear Plane Progressive Waves

Superposition of Linear Plane Progressive Waves

Oblique Plane Waves

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

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

Therefore, [math]\displaystyle{ \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]\displaystyle{ x=0 \, }[/math].

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

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

Oblique Standing Waves

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

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

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

and

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

Partial Reflection

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

[math]\displaystyle{ R \, }[/math]: Complex reflection coefficient

[math]\displaystyle{ R = |R| e^{-i\delta}, |R| = \frac{A_R}{A_I} \, }[/math] [math]\displaystyle{ \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]\displaystyle{ |\eta_T| = A_I \left[ 1 + |R| + 2 |R| \cos ( 2 k x + \delta ) \right] \, }[/math]

At node,

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

At antinode,

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

Wave Group

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

[math]\displaystyle{ \eta = \Re \left( A e^{i k_1 x - \omega_1 t } \right) \, }[/math] [math]\displaystyle{ \eta = \Re \left( A e^{i k_2 x - \omega_2 t } \right) \, }[/math] [math]\displaystyle{ \omega, = \omega, ( k , ) \, }[/math] and [math]\displaystyle{ V_{P_1} \approx V_{P_2} \, }[/math] [math]\displaystyle{ \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]\displaystyle{ \delta k = k - k \, }[/math] and [math]\displaystyle{ \delta \omega = \omega - \omega \, }[/math] [math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \omega = g k \tanh k h \Rightarrow \, }[/math] [math]\displaystyle{ V_g = \underbrace{\left( \frac{\omega}{k} \right)}_{V_P} \underbrace{\frac{1}{2} \left( 1+\frac{2kh}{\sinh 2kh} \right)}_n }[/math]

[math]\displaystyle{ \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]\displaystyle{ \bullet }[/math] Potential energy PE	[math]\displaystyle{ \bullet }[/math] Kinetic energy KE
PE without wave [math]\displaystyle{ = \int_{-h} \rho g y dy = - - \rho g h \, }[/math] PE with wave [math]\displaystyle{ \int_{-h}^\eta \rho g y dy = - \rho g ( \eta - h ) \, }[/math] [math]\displaystyle{ PR_{wave} = - \rho g \eta = - \rho g A \cos ( kx - \omega t) \, }[/math]	[math]\displaystyle{ KE_{wave} = \int_{-h}^\eta dy - \rho ( u + v ) }[/math] Deep water [math]\displaystyle{ = \cdots = - \rho g A \ }[/math] to leading order Finite depth [math]\displaystyle{ = \cdots \, }[/math]
Average energy over one period or one wavelength
[math]\displaystyle{ \overline{PE}_{wave} = - \rho g A \, }[/math]	[math]\displaystyle{ \overline{KE}_{wave} = - \rho g A \, }[/math] at any [math]\displaystyle{ h \, }[/math]

• Total wave energy in deep water:

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

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

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

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

• Nonlinear waves: [math]\displaystyle{ \overline{PE} \gt \overline{PE} \, }[/math].

Energy Propagation - Group Velocity

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

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

where

[math]\displaystyle{ \Im = \int_{-h}^\eta pu dy \ \, }[/math] with [math]\displaystyle{ \ p = - \rho \left( \frac{d\phi}{dt} + gy \right) \ }[/math] and [math]\displaystyle{ \ u = \frac{\partial\phi}{\partial x} \, }[/math] [math]\displaystyle{ \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]\displaystyle{ A = 3m, \ T = 10\mbox{sec} \rightarrow \overline{\Im} = 400KW/m \, }[/math]

Equation of Energy Conservation

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

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

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

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

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

Steady Ship Waves, Wave Resistance

• Ship wave resistance drag [math]\displaystyle{ D_w \, }[/math]

Rate of work done = rate of energy increase [math]\displaystyle{ D_w U + \overline{\Im} = \frac{d}{dt} (\overline{E}L) = \overline{E}U \, }[/math] [math]\displaystyle{ 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]\displaystyle{ A \, }[/math] depends on [math]\displaystyle{ U \, }[/math] and the ship geometry. Let [math]\displaystyle{ \ell \equiv \, }[/math] effective length.

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

[math]\displaystyle{ \eta_b = a \cos (kx) \ \, }[/math] \ \eta_S = - a \cos (k ( x+ \ell )) \, </math> [math]\displaystyle{ \eta_T = \eta_b + \eta_S \, }[/math] [math]\displaystyle{ A = | \eta_T | = 2 a \left|\sin (-k\ell)\right| \ \leftarrow \ }[/math] envelope amplitude [math]\displaystyle{ D_w = - \rho g A = \rho g a \sin ( -k \ell ) \ \Righarrow \ 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]\displaystyle{ U \, }[/math]. For deep water, we therefore have

[math]\displaystyle{ V_p = U \ \Righarrow \ \frac{\omega}{k} = U \ \begin{matrix} \mbox{deep} \\ \rightarrow \\ \mbox{water} \end{matrix} \sqrt{\frac{g}{k}} = U, \ }[/math] or [math]\displaystyle{ \lambda = 2 \pi \frac{U}{g} }[/math]

• Summary Steady ship waves in deep water.

[math]\displaystyle{ U = \, }[/math] ship speed [math]\displaystyle{ V_p = \sqrt{\frac{g}{k}} = U; \ }[/math] so [math]\displaystyle{ \ k = \frac{g}{U} \ \, }[/math] and [math]\displaystyle{ \ \lambda = 2 \pi \frac{U}{g} \, }[/math] [math]\displaystyle{ L = \, }[/math] ship length, [math]\displaystyle{ \ \ell \sim L \, }[/math] [math]\displaystyle{ 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]