AcademySound
Academy
Sound as a Wave
Level 1 - Physics topic page in Sound.
Principle
Sound is a longitudinal pressure disturbance that travels through a material medium.
Notation
\(\xi(x,t)\)
air-particle displacement
\(\mathrm{m}\)
\(\Delta p(x,t)\)
pressure variation from equilibrium
\(\mathrm{Pa}\)
\(\rho_0\)
equilibrium density
\(\mathrm{kg\,m^{-3}}\)
\(v\)
sound speed
\(\mathrm{m\,s^{-1}}\)
\(k\)
wavenumber
\(\mathrm{rad\,m^{-1}}\)
\(\omega\)
angular frequency
\(\mathrm{rad\,s^{-1}}\)
Method
Sound in air is not a sideways displacement wave. Air parcels oscillate back and forth along the direction that the pressure pattern travels.
Longitudinal motion
\[\vec \xi \parallel \vec v\]
The particle displacement is along the propagation direction.
Pressure split
\[p(x,t)=p_0+\Delta p(x,t)\]
Sinusoidal displacement
\[\xi(x,t)=\xi_0\cos(kx-\omega t)\]
Pressure phase
\[\Delta p(x,t)=\Delta p_{\max}\sin(kx-\omega t)\]
Pressure is largest where neighboring parcels are closest together.
A compression has positive pressure variation. A rarefaction has negative pressure variation.
Rules
These are the compact sound-wave relations.
Pressure split
\[p=p_0+\Delta p\]
Wave speed
\[v=f\lambda=\frac{\omega}{k}\]
Longitudinal sound
\[\vec \xi\parallel \vec v\]
Examples
Question
A sound wave has frequency
\[680\,\mathrm{Hz}\]
and speed \[340\,\mathrm{m\,s^{-1}}\]
Find its wavelength.Answer
Use
\[v=f\lambda\]
\[\lambda=\frac{v}{f}=\frac{340}{680}=0.50\,\mathrm{m}\]
Checks
- Sound needs a material medium; it is not an electromagnetic wave.
- The air parcels oscillate locally rather than traveling with the sound across the room.
- Pressure variation \(\Delta p\) is measured relative to the equilibrium pressure.
- Compression and rarefaction describe pressure, not permanent motion of air.