Vibration and waves are very common and interesting subject of physics. Almost all the information are obtained by human with the help of waves. (eg. sound wave for speak/hear, light wave for read/see).

Waves in free space

A simple case of sound wave in free space is the 1D sound wave in an infinite 1D media. Sound wave is specifically refers to the longitudinal wave in a media, which has parallel displacement with the wave vector ($\vec{\xi} \parallel \vec{k}$). Besides sound wave (longitudinal wave), there also exists transverse waves in the media, which has perpendicular perturbation to the wave vector ($\vec{\xi} \perp \vec{k}$). The wave equation has the form:

$\frac{\partial^2 X}{\partial x^2} = \frac{1}{C_s^2}\frac{\partial^2 X}{\partial t^2}$

Where $C_s = \sqrt{\frac{d\rho}{dP}} = \frac{\gamma k_B T}{m}$ with $\frac{dP}{d\rho} = \frac{\gamma P}{\rho}$, where $\gamma$ is adiabatic coefficient, $k_B$ is Bothzman constant, $m$ is density of a air molecure, $\rho$ is media density, $P$ is media pressure, $X = X(x, t)$ is the displacement of perturbation in the media. Actually, this wave equation applies for many kinds of waves, such as: sound wave, transverse wave, electromagnetic wave.

Solution of wave equation in 1D free space

Suppose the displacement has the form of a function like this: $X(x, t) =f(kx-\omega t)$. We put this test solution in the wave equation and get:

$f'' \omega^2 = \frac{1}{C_s^2}f'' k^2 \Rightarrow C_s = \frac{k}{\omega}$

This proves that $f(kx - \omega t)$ is a solution to the above wave equation. Notices that this also means the displacement of the longitudinal/transverse wave could have any different shape. When I was a young kid, I play with a long line by creating different kinds of waves on it. Many different kind of perturbations could all travel in this line to the other side.

This means the wave that travel in free space could have any possible shape at any possible frequency.

Waves in confined space

For waves in free space, their perturbation could have any shape, and their frequency could also take any value. But for waves confined in some limited space, their perturbation as well as frequency could take some isollated values.
Becasue, the solution of the wave equation (differential equation) is determined by the boundry condition.

Standing wave under 1D confinement

Suppose we have a wave in x direction only with the wave equation as: $\frac{\partial^2 \chi}{\partial x^2} = \frac{1}{C_s^2}\frac{\partial^2 \chi}{\partial t^2}$ and a fixed boundry condition: $\chi(0) = \chi(L) = 0$. Soppose the general solution of the wave equation can be expressed as the fourier series: $\chi(x, t) = \sum_n e^{i(kx -\omega_n t)}$. When consider the reflection wave: $\chi(-x, t)$, the general solution is a combination of the two waves. Then the total solution should be like this:

$\chi = \sum_n e^{i(kx - \omega_n t)} -e^{i(-kx -\omega_n t)} = \sum_n e^{i\omega_n t}(2i sin(kx))$

Then we consider the boundary condition: $\chi(0)=\chi(L)=0$, which means $sin(kx)=0$. Mathematically, it means: $kx=n\pi\Rightarrow k=\frac{n\pi}{L}$, where $n=0,1,2,3,...$. Thus the frequency of the confined wave could only take the isolated values as: $\omega = k C_s = \frac{n\pi}{L}C_s$. This is the solution of the so called “1D standing wave”.