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: