5: Waves

The wave equation

The general form of the wave equation is: $\Box u=0$, or:

$$\nabla^2u-\frac{1}{v^2}\frac{\partial^2 u}{\partial t^2}=\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}+\frac{\partial^2 u}{\partial z^2}-\frac{1}{v^2}\frac{\partial^2 u}{\partial t^2}=0$$

where $u$ is the disturbance and $v$ the propagation velocity. In general $v=f\lambda$ holds. By definition $k\lambda=2\pi$ and $\omega=2\pi f$.

In principle, there are two types of waves:

1. Longitudinal waves: for these  $\vec{k}\parallel\vec{v}\parallel\vec{u}$ holds.
2. Transversal waves: for these $\vec{k}\parallel\vec{v}\perp\vec{u}$. holds

The phase velocity is given by $v_{\rm ph}=\omega/k$. The group velocity is given by:

$$v_{\rm g}=\frac{d\omega}{dk}=v_{\rm ph}+k\frac{dv_{\rm ph}}{dk}= v_{\rm ph}\left(1-\frac{k}{n}\frac{dn}{dk}\right)$$

where $n$ is the refractive index of the medium. If $v_{\rm ph}$ does not depend on $\omega$ then: $v_{\rm ph}=v_{\rm g}$. In a dispersive medium it is possible that $v_{\rm g}>v_{\rm ph}$ or $v_{\rm g}<v_{\rm ph}$, and $v_{\rm g}\cdot v_{\rm f}=c^2$. If one wants to transfer information with a wave, e.g. by modulation of an EM wave, the information travels with the velocity at which a change in the electromagnetic field propagates. This velocity is often almost equal to the group velocity.

For some media, the propagation velocity follows from:

• Pressure waves in a liquid or gas: $v=\sqrt{\kappa/\varrho}$, where $\kappa$ is the modulus of compression.
• Further for pressure waves in a gas: $v=\sqrt{\gamma p/\varrho}=\sqrt{\gamma RT/M}$.
• Pressure waves in a thin solid bar with diameter $<<\lambda$: $v=\sqrt{E/\varrho}$
• Waves in a string: $v=\sqrt{F_{\rm span}l/m}$
• Surface waves on a liquid: $\displaystyle v=\sqrt{\left(\frac{g\lambda}{2\pi}+ \frac{2\pi\gamma}{\varrho\lambda}\right)\tanh\left(\frac{2\pi h}{\lambda}\right)}$
  where $h$ is the depth of the liquid and $\gamma$ the surface tension. If $h\ll\lambda$ then $v\approx\sqrt{gh}$ holds.

Solutions of the wave equation

Plane waves

In $n$ dimensions a harmonic plane wave is defined by:

$$u(\vec{x},t)=2^n\hat{u}\cos(\omega t)\sum_{i=1}^n\sin(k_ix_i)$$

The equation for a harmonic traveling plane wave is: $u(\vec{x},t)=\hat{u}\cos(\vec{k}\cdot\vec{x}\pm\omega t+\varphi)$

If waves reflect at the end of a spring this will result in a change in phase. A fixed end imposes a phase change of $\pi/2$ to the reflected wave, with boundary condition $u(l)=0$. A loose end yields no change in the phase of the reflected wave, with boundary condition $(\partial u/\partial x)_l=0$.

If an observer is moving w.r.t. the wave with a velocity $v_{\rm obs}$, they will observe a change in frequency: the Doppler effect. This is given by: $\displaystyle\frac{f}{f_0}=\frac{v_{\rm f}-v_{\rm obs}}{v_{\rm f}}$.

Spherical waves

When the situation is spherically symmetric, the homogeneous wave equation is given by:

$$\frac{1}{v^2}\frac{\partial^2 (ru)}{\partial t^2}-\frac{\partial^2 (ru)}{\partial r^2}=0$$

with a general solution:

$$u(r,t)=C_1\frac{f(r-vt)}{r}+C_2\frac{g(r+vt)}{r}$$

Cylindrical waves

When the situation has a cylindrical symmetry, the homogeneous wave equation becomes:

$$\frac{1}{v^2}\frac{\partial^2 u}{\partial t^2}-\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial u}{\partial r}\right)=0$$

This is a Bessel equation, with solutions that can be written as Hankel functions. For sufficient large values of $r$ these are approximated by:

$$u(r,t)=\frac{\hat{u}}{\sqrt{r}}\cos(k(r\pm vt))$$

The general solution in one dimension

Starting from the equation:

$$\frac{\partial^2 u(x,t)}{\partial t^2}=\sum_{m=0}^{N}\left(b_m\frac{\partial ^m}{\partial x^m}\right)u(x,t)$$

where $b_m \in  \mathbb{R}$. Substituting $u(x,t)=A{\rm e}^{i(kx-\omega t)}$ gives two solutions $\omega_j=\omega_j(k)$ as dispersion relations. The general solution is given by:

$$u(x,t)=\int\limits_{-\infty}^{\infty}\left(a(k){\rm e}^{i(kx-\omega_1(k)t)}+ b(k){\rm e}^{i(kx-\omega_2(k)t)}\right)dk$$

Because in general the frequencies $\omega_j$ are non-linear in $k$ there is dispersion and the solution cannot be written any more as a sum of functions depending only on $x\pm vt$: the wave front transforms.

The stationary phase method

Usually the Fourier integrals of the previous section cannot be calculated exactly. If $\omega_j(k) \in \mathbb{R}$ the stationary phase method can be applied. Assuming that $a(k)$ is only a slowly varying function of $k$, one can state that the parts of the $k$-axis where the phase of $kx-\omega(k)t$ changes rapidly will give no net contribution to the integral because the exponent oscillates rapidly there. The only areas contributing significantly to the integral are areas with a stationary phase, determined by $\displaystyle\frac{d}{dk}(kx-\omega(k)t)=0$. Now the following approximation is possible:

$$\int\limits_{-\infty}^\infty a(k){\rm e}^{i(kx-\omega(k)t)}dk\approx \sum_{i=1}^{N}\sqrt{\frac{2\pi}{\frac{d^2\omega(k_i)}{dk_i^2}}} \exp\left[-i\mbox{$\frac{1}{4}$}\pi+i(k_ix-\omega(k_i)t)\right]$$

Green functions for the initial-value problem

This method is preferable if the solutions deviate strongly from the stationary solutions, like point-like excitations. Starting with the wave equation in one dimension, with $\nabla^2=\partial^2/\partial x^2$  if $Q(x,x',t)$ is the solution with initial values $Q(x,x',0)=\delta(x-x')$ and $\displaystyle\frac{\partial Q(x,x',0)}{\partial t}=0$, and $P(x,x',t)$ the solution with initial values $P(x,x',0)=0$ and $\displaystyle\frac{\partial P(x,x',0)}{\partial t}=\delta(x-x')$, then the solution of the wave equation with arbitrary initial conditions $f(x)=u(x,0)$ and $\displaystyle g(x)=\frac{\partial u(x,0)}{\partial t}$ is given by:

$$u(x,t)=\int\limits_{-\infty}^\infty f(x')Q(x,x',t)dx'+ \int\limits_{-\infty}^\infty g(x')P(x,x',t)dx'$$

$P$ and $Q$ are called the propagators. They are defined by:

$$\begin{aligned} Q(x,x',t)&=&\frac{1}{2} [\delta(x-x'-vt)+\delta(x-x'+vt)]\\ P(x,x',t)&=& \left\{\begin{array}{ll} \displaystyle \frac{1}{2v}&~~~\mbox{if}~~|x-x'|<vt\\ 0&~~~\mbox{if}~~|x-x'|>vt \end{array}\right.\end{aligned}$$

Further the relation: $\displaystyle Q(x,x',t)=\frac{\partial P(x,x',t)}{\partial t}$ holds.

Waveguides and resonating cavities

The boundary conditions for a perfect conductor can be derived from Maxwell's equations. If $\vec{n}$ is a unit vector $\perp$ the surface, pointing from 1 to 2, and $\vec{K}$ is a surface current density, then:

$$\begin{array}{ll} \vec{n}\cdot(\vec{D}_2-\vec{D}_1)=\sigma~~&~~\vec{n}\times(\vec{E}_2-\vec{E}_1)=0\\ \vec{n}\cdot(\vec{B}_2-\vec{B}_1)=0~~&~~\vec{n}\times(\vec{H}_2-\vec{H}_1)=\vec{K} \end{array}$$

In a waveguide because of the cylindrical symmetry: $\vec{E}(\vec{x},t)=\vec{\cal E}(x,y){e}^{i(kz-\omega t)}$ and $\vec{B}(\vec{x},t)=\vec{\cal{B}}(x,y){e}^{i(kz-\omega t)}$ holds. From this one can now deduce that, if ${\cal B}_z$ and ${\cal E}_z$ are not $\equiv0$:

$$\begin{array}{ll} \displaystyle {\cal B}_x=\frac{i}{\varepsilon\mu\omega^2-k^2}\left(k\frac{\partial {\cal B}_z}{\partial x}-\varepsilon\mu\omega\frac{\partial {\cal E}_z}{\partial y}\right)~~&~~ \displaystyle {\cal B}_y=\frac{i}{\varepsilon\mu\omega^2-k^2}\left(k\frac{\partial {\cal B}_z}{\partial y}+\varepsilon\mu\omega\frac{\partial {\cal E}_z}{\partial x}\right)\\ \displaystyle {\cal E}_x=\frac{i}{\varepsilon\mu\omega^2-k^2}\left(k\frac{\partial {\cal E}_z}{\partial x}+\varepsilon\mu\omega\frac{\partial {\cal B}_z}{\partial y}\right)~~&~~ \displaystyle {\cal E}_y=\frac{i}{\varepsilon\mu\omega^2-k^2}\left(k\frac{\partial {\cal E}_z}{\partial y}-\varepsilon\mu\omega\frac{\partial {\cal B}_z}{\partial x}\right) \end{array}$$

Now one can distinguish between three cases:

1. $B_z\equiv0$: the Transverse Magnetic Modes (TM). Boundary condition: ${\cal E}_z|_{\rm surf}=0$.
2. $E_z\equiv0$: the Transverse Electric Modes (TM). Boundary condition: $\frac{\displaystyle {\partial \cal B}_z}{\partial n}   \bigg\rvert _{surf}=0$.
   For the TE and TM modes this results in an eigenvalue problem for ${\cal E}_z$ resp. ${\cal B}_z$ with boundary conditions: $$\left(\frac{\partial^2 }{\partial x^2}+\frac{\partial^2 }{\partial y^2}\right)\psi=-\gamma^2\psi~~\mbox{with eigenvalues}~~ \gamma^2:=\varepsilon\mu\omega^2-k^2$$
   This has a discrete solution $\psi_\ell$ with eigenvalue $\gamma_\ell^2$: $k=\sqrt{\varepsilon\mu\omega^2-\gamma_\ell^2}$. For $\omega<\omega_\ell$, $k$ is imaginary and the wave is damped. Therefore, $\omega_\ell$ is called the cut-off frequency. In rectangular conductors the following expression can be found for the cut-off frequency for modes TE$_{m,n}$ or TM$_{m,n}$:
   $$\lambda_\ell=\frac{2}{\sqrt{(m/a)^2+(n/b)^2}}$$
3. $E_z$ and $B_z$ are zero everywhere for the Transversal Eectro-Magnetic Modes (TEM). Then: $k=\pm\omega\sqrt{\varepsilon\mu}$ and $v_{\rm f}=v_{\rm g}$, just as if there were no waveguide. Further $k \in \mathbb{R}$, so no cut-off frequency exists.

In a rectangular, three dimensional resonating cavity with edges $a$, $b$ and $c$ the possible wave numbers are given by: $\displaystyle k_x=\frac{n_1\pi}{a}~,~~k_y=\frac{n_2\pi}{b}~,~~k_z=\frac{n_3\pi}{c}$ This results in the possible frequencies $f=vk/2\pi$ in the cavity:

$$f=\frac{v}{2}\sqrt{\frac{n_x^2}{a^2}+\frac{n_y^2}{b^2}+\frac{n_z^2}{c^2}}$$

For a cubic cavity, with $a=b=c$, the possible number of oscillating modes $N_{\rm L}$ for longitudinal waves is given by:

$$N_{\rm L}=\frac{4\pi a^3f^3}{3v^3}$$

Because transverse waves have two possible polarizations $N_{\rm T}=2N_{\rm L}$ holds for them.

Non-linear wave equations

The Van der Pol equation is given by:

$$\frac{d^2x}{dt^2}-\varepsilon\omega_0(1-\beta x^2)\frac{dx}{dt}+\omega_0^2x=0$$

$\beta x^2$ can be ignored for very small values of the amplitude. Substitution of $x\sim{\rm e}^{i\omega t}$ gives: $\omega= \frac{1}{2} \omega_0(i\varepsilon\pm2\sqrt{1-\frac{1}{2}\varepsilon^2})$. The lowest-order instabilities grow as $\frac{1}{2} \varepsilon\omega_0$. While $x$ is growing, the 2nd term becomes larger and which limits the growth. Oscillations on a time scale $\sim\omega_0^{-1}$ can exist. If $x$ is expanded as $x=x^{(0)}+\varepsilon x^{(1)}+\varepsilon^2x^{(2)}+\cdots$ and this is substituted one obtains, additional periodic, secular terms $\sim\varepsilon t$. If it is assumed that there exist timescales $\tau_n$, $0\leq\tau\leq N$ with $\partial\tau_n/\partial t=\varepsilon^n$ and if the secular terms are put to 0 one obtains:

$$\frac{d}{dt}\left\{\frac{1}{2}\left(\frac{dx}{dt}\right)^2+\frac{1}{2} \omega_0^2x^2\right\}= \varepsilon\omega_0(1-\beta x^2)\left(\frac{dx}{dt}\right)^2$$

This is an energy equation. Energy is conserved if the left-hand side is 0. If $x^2>1/\beta$, the right-hand side changes sign and an increase in energy changes into a decrease of energy. This mechanism limits the growth of oscillations.

The Korteweg-De Vries equation is given by:

$$\frac{\partial u}{\partial t}+\frac{\partial u}{\partial x}-\underbrace{au\frac{\partial u}{\partial x}}_{\rm non-lin}+ \underbrace{b^2\frac{\partial ^3u}{\partial x^3}}_{\rm dispersive}=0$$

This equation is for example a model for ion-acoustic waves in a plasma. For this equation, soliton solutions of the following form exist:

$$u(x-ct)=\frac{-d}{\cosh^2(e(x-ct))}$$ with $c=1+\frac{1}{3}ad$ and $e^2=ad/(12b^2)$.