Standing Wave Equations

Standing Wave Equations

A standing wave forms when two waves of the same frequency and amplitude travel in opposite directions and interfere.
The resulting pattern does not travel along the medium.
Instead, it oscillates in place.

The standard equation for a standing wave is

$y(x, t) = 2A \sin(kx) \cos(\omega t)$y(x,t)=2Asin(kx)cos(ωt)

Here, $A$A is the amplitude of each travelling wave.
The wave number $k = \frac{2\pi}{\lambda}$k=λ2π​ and the angular frequency $\omega = 2\pi f$ω=2πf.

The term $2A \sin(kx)$2Asin(kx) gives the amplitude at position $x$x.
This amplitude changes along the string.
However, every point oscillates with the same frequency.

Nodes are points of zero displacement.
They occur where $\sin(kx) = 0$sin(kx)=0.
Therefore, nodes lie at $x = \frac{n\lambda}{2}$x=2nλ​, where $n = 0, 1, 2, \dots$n=0,1,2,….

Antinodes are points of maximum displacement.
They occur where $\sin(kx) = \pm 1$sin(kx)=±1.
Therefore, antinodes lie at $x = \frac{(2n+1)\lambda}{4}$x=4(2n+1)λ​.

On a string of length $L$L fixed at both ends, only certain wavelengths fit.
The allowed wavelengths are

$\lambda_n = \frac{2L}{n}, \quad n = 1, 2, 3, \dots$λn​=n2L​,n=1,2,3,…

The corresponding frequencies are

$f_n = n \frac{v}{2L}$fn​=n2Lv​

Here, $v$v is the speed of the waves on the string.
The value $n = 1$n=1 gives the fundamental frequency.
Higher values of $n$n give the overtones or harmonics.

In short, the standing-wave equation separates the space dependence from the time dependence.
This separation explains the fixed nodes and the oscillating antinodes.