💼 Case: propagation on a stretched membrane

Consider a rectangular plate except along $x$ we move the boundary conditions effectively to infinity so that it is free to move along $\pm x$

• start with the wave equation

  $$ \frac{\partial^2 \phi}{\partial x^2}+\frac{\partial ^2 \phi}{\partial y^2}=\frac{1}{c^2} \frac{\partial^2 \phi}{\partial t^2} $$

  where $c$ is the speed of free waves

• We use separation of variable $\phi(x,y,t)=X(x) Y(y) T(t)$

  $$ \begin{aligned} YT \frac{\text d^2 X}{\text dx^2}+XT \frac{d^2 Y}{dy^2}&=\frac 1 {c^2} XY \frac{\text d^2 T}{\text dt^2} \\ \frac 1X \frac{\text d^2 X}{\text d x^2}+\frac 1Y\frac{\text d^2 Y}{\text dy}&= \frac 1 {c^2} \frac 1T \frac{d^2 T}{\text dt^2}-k^2 \\ \Rightarrow \quad \underbrace{\frac 1X \frac{\text d^2 X}{\text d x^2}}{\text{function of }x} &=\underbrace{-k^2 - \frac 1Y \frac{\text d^2 Y}{\text dy^2}}{\text{function of }y}=-k^2_x

  \end{aligned} $$

• Thus we get

  $$ \frac{\text d^2 X}{\text d x^2}=-k^2_x X \qquad \frac{d^2 Y}{dy^2}=-k^2_y Y \\ \frac 1{c^2}\frac{\text d^2 T}{\text d t^2}=-k^2 T $$

  where

  $$ k_x^2+k_y^2=k^2 $$

• We choose a solution of the form

  $$ Y=\sin k_y y \quad \text{with} \; k_y=n\pi/b $$

  but we adopt exponential solution along $x$ (due to the new boundary conditions) and this corresponds to traveling waves

  $$ X(x)=e^{ik_x x} $$

  and the time dependence is a sum of complex exponentials

  $$ T(t)=Ae^{i\omega t}+B^{-i\omega t} $$

• putting these together we have

  $$ \phi(x,y,t)=\sin\left ( \frac{n\pi y}{b} \right ) \left [ Ae^{i(k_x x+\omega t)}+Be^{i(k_xx-\omega t)} \right ] $$

  And the dispersion relation is

  $$ \frac{\omega^2}{c^2}=k^2_x+\left ( \frac{n\pi}{b}\right )^2 $$

• The general solution is a summation of the terms over $n$ and $k_x$

  $$ \begin{aligned} \phi(x,t)=\sum^\infin_{n=1}\frac{\text dk_x}{\sqrt{2\pi}}\sin\left ( \frac{n\pi y}{b} \right ) \left [ A(k_x)e^{i(k_x x+\omega t)}+B(k_x)e^{i(k_xx-\omega t)} \right ] \end{aligned} $$

  where the angular frequency $\omega$ satisfied the dispersion relation

• We can evaluate the phase and group velocity from this dispersion equation

  $$ \begin{aligned} v_p=\frac{\omega}{k_x}=c\sqrt{1+\frac{n^2\pi^2}{k_x^2b^2}} \\ v_g=\frac{\text d \omega}{\text d k_x}=\frac{c}{\sqrt{1+\frac{n^2\pi^2}{k_x^2 b^2}}}

  \end{aligned} $$

  We can relate them as

  $$ v_p v_g=c^2 $$

Typical dispersion curves

• We see that waves can propagate along the membrane if $\omega > c\frac{n\pi}b$
• below this angular frequency the waves are cut off and propagation is possible

For wavelengths

• if $\lambda =\frac {2\pi c}{\omega}>2b$ the wave will not propagate ( as $k_x$ will become imaginary and the wave will be exponentially damped)

in general $v_p>c$ and the group velocity is $v_g<c$.

• In the limit of $k_x\to \infin$ then $v_p\to c$ and $v_g \to c$
• In the limit of $k_x\to 0$ then $v_p \to \infin$ and $v_g \to 0$

Waves in a rectangular metal waveguide

💼 Case: the electric field for a Transverse Magnetic (TM) mode.

• The wave equation for the electric field along $y$ is

  $$ \frac{\partial ^2 E_y}{\partial x^2}+\frac{\partial^2 E_y}{\partial y^2}+\frac{\partial^2 E_y}{\partial z^2}=\frac 1{c^2}\frac{\partial^2 E_y}{\partial t^2} $$

  where $c$ is the velocity of light.

• We use separation of variables

  $$ E_y=X(x)Y(y)Z(z)T(t) $$

  and obtain

  $$ \begin{aligned} \frac{\text d^2 X}{\text d x^2}&=-k^2_x X \\ \frac 1 {c^2} \frac{\text d^2 T}{\text d t^2}&=-k^2 T \end{aligned} \quad \begin{aligned} \frac{\text d^2 Z}{\text dz^2}=-k^2_z Z \\ \frac{\text d^2 Y}{\text dy^2}=-k^2_y Y

  \end{aligned} $$

  here we also have

  $$ k_x^2+k_y^2+k_z^2=k^2=\omega^2/c^2 $$