Superposition of linear plane progressive waves

1.

oblique plane waves: figure

Consider wave propagation at an angle $\theta$ to the x-axis

$$\begin{align}\eta = & A\cos ( {\overbrace {kx\cos \theta + kz\sin \theta }^{\ma... ...os \theta , k_z = k\sin \theta , k = \sqrt {k_x^2 + k_z^2 } \notag \end{align}$$

2.

Standing Waves:

$$\begin{align}\eta = & A\cos \left( {kx - \omega t} \right) + A\cos \left( { - k... ...osh k\left( {y + h} \right)}{\cosh kh}\cos kx \sin \omega t \notag \end{align}$$

$$\frac{\partial \eta }{\partial x}\sim \frac{\partial \phi }{\... ...kx = 0\mbox{ at }x = 0,\frac{n\pi }{k} = \frac{n\lambda }{2}$$

Therefore, $\left. {\frac{\partial \phi }{\partial x}} \right\vert _{x = 0} = 0$. To obtain an standing wave, it is necessary to have perfect reflection at the wall at $x = 0$. figure

$$A_I = A_R , R = \frac{A_R }{A_I } = 1 = \mbox{\ reflection coefficient}$$

3.

Oblique Standing Waves.

$$\begin{align}\eta _I = & A\cos \left( {kx\cos \theta + kz\sin \theta - \omega t}... ...+ kz\sin \left( {\pi - \theta } \right) - \omega t} \right) \notag \end{align}$$

Note: same A, R = 1.

$$\eta _T = \eta _I + \eta _R = 2A\underbrace {\cos \overbrace ... ...}^{k_z z - \omega t}}_{\mbox{\tiny {propagating wave in z}}}$$

and

$$\lambda _x = \frac{2\pi }{k\cos \theta }\mbox{ }V_{P_z } = \f... ...\sin \theta }\mbox{ }\lambda _z = \frac{2\pi }{k\sin \theta }$$

Check:

$$\frac{\partial \phi }{\partial \mbox{x}}\sim \frac{\partial \... ...\sin \left( {kx\cos \theta } \right) = 0 \mbox{\ on\ } x = 0$$

4.

Partial Reflection. figure

$$\begin{align}\eta _I = & A_I \cos \left( {kx - \omega t} \right) = A_I Re\left\{... ... {\mbox{R }e^{ - i\left( {kx + \omega t} \right)}} \right\} \notag \end{align}$$

$R$: Complex reflection coefficient

$$\begin{align}\mbox{R } = & \mbox{ }\left\vert \mbox{R} \right\vert e^{ - i\delta... ...{R} \right\vert\cos \left( {2kx + \delta } \right)} \right] \notag \end{align}$$
figure

At node,

$$\left\vert {\eta _T } \right\vert = \left\vert {\eta _T } \ri... ... = - 1 \mbox{ or } 2kx + \delta = \left( {2n + 1} \right)\pi$$

At antinode,

$$\left\vert {\eta _T } \right\vert = \left\vert {\eta _T } \ri... ...t( {2kx + \delta } \right) = 1\mbox{ or }2kx + \delta = 2n\pi$$

$$2kL = 2\pi \mbox{ so } L = \frac{\lambda }{2}$$

$$\left\vert R \right\vert = \frac{\left\vert {\eta _T } \right... ...\vert _{\min } } = \left\vert R \left( k \right)\right\vert$$

5.

Wave Group 2 waves, same amplitude A and direction, but $\omega$ and $k$ very close to each other.

$$\begin{align}\eta _1 = & \Re\left( {Ae^{i\left( {k_1 x - \omega _1 t} \right)}} ... ...( {k_{1,2} } \right)\mbox{\ and\ }V_{P_1 } \approx V_{P_2 } \notag \end{align}$$

$$\eta _T = \eta _1 + \eta _2 = \Re\left\{ {Ae^{i\left( {k_1 x ... ...2 - k_1 \mbox{\ and\ } \delta \omega = \omega _2 - \omega _1$$

$$\left. {\begin{array}{l} \left\vert {\eta _T } \right\vert ... ...t)t = 0 \mbox{\ then\ }V_g = \frac{\delta \omega }{\delta k}$$

In the limit,

$$\delta k,\delta \omega \to 0, V_g = \left. {\frac{d\omega }{dk}} \right\vert _{k_1 \approx k_2 \approx k},$$

and since

$$\omega ^2 = gk\tanh kh \Rightarrow V_g = \underbrace {\left( ... ...ce {\frac{1}{2}\left( {1 + \frac{2kh}{\sinh 2kh}} \right)}_n$$

$$\left. {\begin{array}{l} \mbox{(a) deep water }kh > > 1 \\ ... ...1}{2} < n < 1 \\ \end{array}} \right\} \mbox{ }V_g \le V_p$$

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