AcademySound
Academy
Standing Sound Waves
Level 1 - Physics topic page in Sound.
Principle
Standing sound waves are fixed pressure and displacement patterns in an air column.
Notation
\(L\)
air-column length
\(\mathrm{m}\)
\(\lambda_n\)
allowed wavelength
\(\mathrm{m}\)
\(f_n\)
allowed frequency
\(\mathrm{Hz}\)
\(v\)
sound speed
\(\mathrm{m\,s^{-1}}\)
\(n\)
mode number
1
Method
At an open end, the pressure variation is approximately zero. At a closed end, air displacement is approximately zero.
Open-open geometry
\[L=n\frac{\lambda_n}{2}\]
Both ends are pressure nodes.
Open-open frequencies
\[f_n=\frac{nv}{2L}\]
Open-closed geometry
\[L=(2n-1)\frac{\lambda_n}{4}\]
One pressure node and one pressure antinode fit in the fundamental.
Open-closed frequencies
\[f_n=\frac{(2n-1)v}{4L}\]
Open-open pipes support all integer harmonics. Open-closed pipes support only odd harmonics in the ideal model.
Rules
These are the compact air-column results.
Open-open wavelengths
\[\lambda_n=\frac{2L}{n}\]
Open-open frequencies
\[f_n=\frac{nv}{2L}\]
Open-closed wavelengths
\[\lambda_n=\frac{4L}{2n-1}\]
Open-closed frequencies
\[f_n=\frac{(2n-1)v}{4L}\]
Examples
Question
An open-open pipe has length
\[0.85\,\mathrm{m}\]
Use \[v=340\,\mathrm{m\,s^{-1}}\]
to find its fundamental frequency.Answer
\[f_1=\frac{v}{2L}=\frac{340}{2(0.85)}=200\,\mathrm{Hz}\]
Checks
- Open end means pressure node, not displacement node.
- Closed end means displacement node and pressure antinode.
- Open-open pipes have harmonics \(f_n=nf_1\).
- Open-closed pipes have odd harmonics \(f_1,3f_1,5f_1,\ldots\).