Acoustic wave

Wave properties

An acoustic wave is a mechanical wave that transmits energy through the movements of atoms and molecules. Acoustic waves transmit through fluids in a longitudinal manner (movement of particles are parallel to the direction of propagation of the wave); in contrast to electromagnetic waves that transmit in transverse manner (movement of particles at a right angle to the direction of propagation of the wave). However, in solids, acoustic waves transmit in both longitudinal and transverse manners due to presence of shear moduli in such a state of matter.

Acoustic wave equation

The acoustic wave equation describes the propagation of sound waves. The acoustic wave equation for sound pressure in one dimension is given by $${\displaystyle \frac{{\mathrm{\partial }}^{2}p}{\mathrm{\partial }{x}^2}-\frac{1}{c^2}\frac{{\mathrm{\partial }}^{2}p}{\mathrm{\partial }{t}^2}=0}$$ where

• ${\displaystyle p}$ is sound pressure in Pa
• ${\displaystyle x}$ is position in the direction of propagation of the wave, in m
• ${\displaystyle c}$ is speed of sound in m/s
• ${\displaystyle t}$ is time in s

The wave equation for particle velocity has the same shape and is given by $${\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}-\frac{1}{c^2}\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^2}=0}$$ where

• ${\displaystyle u}$ is particle velocity in m/s

For lossy media, more intricate models need to be applied in order to take into account frequency-dependent attenuation and phase speed. Such models include acoustic wave equations that incorporate fractional derivative terms, see also the acoustic attenuation article.

D'Alembert gave the general solution for the lossless wave equation. For sound pressure, a solution would be $${\displaystyle p=R\cos (\omega t-kx)+(1-R)\cos (\omega t+kx)}$$ where

• ${\displaystyle \omega }$ is angular frequency in rad/s
• ${\displaystyle t}$ is time in s
• ${\displaystyle k}$ is wave number in rad·m⁻¹
• ${\displaystyle R}$ is a coefficient without unit

For ${\displaystyle R=1}$ the wave becomes a travelling wave moving rightwards, for ${\displaystyle R=0}$ the wave becomes a travelling wave moving leftwards. A standing wave can be obtained by ${\displaystyle R=0.5}$.

Phase

In a travelling wave pressure and particle velocity are in phase, which means the phase angle between the two quantities is zero.

This can be easily proven using the ideal gas law $${\displaystyle pV=nRT}$$ where

• ${\displaystyle p}$ is pressure in Pa
• ${\displaystyle V}$ is volume in m³
• ${\displaystyle n}$ is amount in mol
• ${\displaystyle R}$ is the universal gas constant with value $8.314472(15)\frac{\mathrm{J}}{\mathrm{m}\mathrm{o}\mathrm{l}\mathrm{K}}$

Consider a volume ${\displaystyle V}$. As an acoustic wave propagates through the volume, adiabatic compression and decompression occurs. For adiabatic change the following relation between volume ${\displaystyle V}$ of a parcel of fluid and pressure ${\displaystyle p}$ holds $${\displaystyle \frac{\mathrm{\partial }V}{V_m}=\frac{-1}{\gamma }\frac{\mathrm{\partial }p}{p_m}}$$ where ${\displaystyle \gamma }$ is the adiabatic index without unit and the subscript ${\displaystyle m}$ denotes the mean value of the respective variable.

As a sound wave propagates through a volume, the horizontal displacement of a particle ${\displaystyle \eta }$ occurs along the wave propagation direction. $${\displaystyle \frac{\mathrm{\partial }\eta }{V_m}A=\frac{\mathrm{\partial }V}{V_m}=\frac{-1}{\gamma }\frac{\mathrm{\partial }p}{p_m}}$$ where

• ${\displaystyle A}$ is cross-sectional area in m²

From this equation it can be seen that when pressure is at its maximum, particle displacement from average position reaches zero. As mentioned before, the oscillating pressure for a rightward traveling wave can be given by $${\displaystyle p=p_0\cos (\omega t-kx)}$$ Since displacement is maximum when pressure is zero there is a 90 degrees phase difference, so displacement is given by $${\displaystyle \eta ={\eta }_0\sin (\omega t-kx)}$$ Particle velocity is the first derivative of particle displacement: ${\displaystyle u=\mathrm{\partial }\eta /\mathrm{\partial }t}$. Differentiation of a sine gives a cosine again $${\displaystyle u=u_0\cos (\omega t-kx)}$$

During adiabatic change, temperature changes with pressure as well following $${\displaystyle \frac{\mathrm{\partial }T}{T_m}=\frac{\gamma -1}{\gamma }\frac{\mathrm{\partial }p}{p_m}}$$ This fact is exploited within the field of thermoacoustics.

Propagation speed

The propagation speed, or acoustic velocity, of acoustic waves is a function of the medium of propagation. In general, the acoustic velocity c is given by the Newton-Laplace equation: $${\displaystyle c=\sqrt{\frac{C}{\rho }}}$$ where

• C is a coefficient of stiffness, the bulk modulus (or the modulus of bulk elasticity for gas mediums),
• ${\displaystyle \rho }$ is the density in kg/m³

Thus the acoustic velocity increases with the stiffness (the resistance of an elastic body to deformation by an applied force) of the material, and decreases with the density. For general equations of state, if classical mechanics is used, the acoustic velocity ${\displaystyle c}$ is given by $${\displaystyle c^2=\frac{\mathrm{\partial }p}{\mathrm{\partial }\rho }}$$ with ${\displaystyle p}$ as the pressure and ${\displaystyle \rho }$ the density, where differentiation is taken with respect to adiabatic change.