# Standing waves of a string

Why standing waves cannot be form from every frequency but only a particular frequency? I understand standing waves as a pattern that will be formed from one wave interfering with another, but i dont understand why it should be in a particular frequency.

• When you want to satisfy the boundary conditions of the wave equation, it forces you to have those frequencies. – Sayan Mandal Sep 22 '17 at 11:35
• There's a minimum frequency determined by the speed of the wave and the boundaries (the distance between the fixed points or nodes.) You can then form only integer multiples of that fundamental frequency. Voila! Quantization. – DWin Sep 23 '17 at 2:14

With no constraints you can form standing waves at any frequency.

If you introduce constraints these limit the frequencies at which standing waves are formed.
For example if you insist that at two positions there must be a node then the separation of those two positions $d$ must be equal to an integer number $n$ of half wavelengths $n\frac \lambda 2$.

Since the speed of a wave $c$ is related to the frequency $f$ and the wavelength $f\lambda =c$ the allowed frequencies are given by the equation $f= n \frac{c}{2d}$.
You will see that your choice of $d$ determines the frequencies at which standing waves are formed.

The waves interfere at any frequency. But to have constructive interference,their phase difference must have some specific values. As the phase difference depends on the boundary conditions and frequency, for a given set of boundaries there are some frequencies for the condition of perfectly constructive interference. If the frequency is a little off you still have interference but the amplitude of the resultant wave is lower. Some of the energy remain in a travelling wave component.