The superposition principle says that if $y_1(x,t)$ and $y_2(x,t)$ are both solutions to the wave equation then so is their linear combination

$$ y(x,t) = A_1\, y_1(x,t) + A_2 \, y_2(x,t) $$

Standing waves

Consider a taut string held fix at $x=0$ and $x=L$. We have a wave boundary such that $\mathcal T_c=0$ and $\mathcal R_c=-1$. The system supports the superposition of two waves of the same frequency and amplitude travelling in opposite directions. This is a standing wave.

$$ \begin{aligned} y_\mathrm{I}+ y_\mathrm{R} &= I e^{i(\omega t - k x)} - I e^{i(\omega t + k x)} \\ &= I e^{i \omega t} (e^{-ik x} - e^{i k x})\\ &= f(t) g(x) \end{aligned} $$

💥 Consider the incident wave as $y_\mathrm{I}(x, t)=\frac{A}{2} \sin (kx-\omega t)$ we get:

$$

\begin{aligned} y&=y_\mathrm{I}+y_\mathrm{R}=\underbrace{\frac{A}{2} \sin (kx -\omega t) + \frac{A}{2} \sin (kx +\omega t)}_{\sin(\alpha + \beta ) + \sin(\alpha - \beta) = 2 \sin \alpha \cos \beta} \\ y&= A \sin kx \cos \omega t. \end{aligned} $$

Discrete normal modes

if our string is fixed at $x=0$ and $x=L$ then

$$ \begin{aligned} y(0,t)&=0 \quad \Rightarrow \quad \sin(k\times 0)=0 \\ y(L,t)&=0 \quad \Rightarrow \quad \sin(kL)=0 \end{aligned} $$

Thus $k_n=\frac{n\pi}{L}$ where $n\in \Z$ thus $y$ becomes

$$ y_n= A_n \sin \left (\frac{n \pi x}{L}\right ) \cos ( \omega_n t) \quad n\in\Z $$

where $\omega_n = v\, k_n = \frac{v \, n \, \pi}{L}$, and $A_n$ is the amplitude of the n$^\text{th}$ mode.

standing.svg

A node is a point where $y=0$ and an anti-node is a point where the displacement $y$ is maximum

$$ \nu_n = \frac{v}{\lambda_n} = \frac{n v}{2 L} = \frac{n}{2L} \sqrt{\frac{T}{\mu}} $$

The superposition of normal modes

In general, the motion of the string will be a superposition of its normal modes given by