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
