Acoustic wave

Wave properties

Acoustic waves are longitudinal waves that exhibit phenomena like diffraction, reflection and interference. Sound waves however don't have any polarization since they oscillate along the same direction as they move.

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

∂ 2 p ∂ x 2 − 1 c 2 ∂ 2 p ∂ t 2 = 0

where

p is sound pressure in Pa x is particle displacement in m c is speed of sound in m/s t is time in s

The wave equation for particle velocity has the same shape and is given by

∂ 2 u ∂ x 2 − 1 c 2 ∂ 2 u ∂ t 2 = 0

where

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

p = R cos ⁡ ( ω t − k x ) + ( 1 − R ) cos ⁡ ( ω t + k x )

where

ω is angular frequency in rad/s t is time in s k is wave number in rad·m⁻¹ R is a coefficient without unit

For R = 1 the wave becomes a travelling wave moving rightwards, for R = 0 the wave becomes a travelling wave moving leftwards. A standing wave can be obtained by 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

  p V = n R T

where

p is pressure in Pa V is volume in m³ n is amount in mol R is the universal gas constant with value 8.314 472 ( 15 )   J m o l   K

Consider a volume V . As an acoustic wave propagates through the volume, adiabatic compression and decompression occurs. For adiabatic change the following relation between volume V of a parcel of fluid and pressure p holds

∂ V V m = − 1   γ ∂ p p m

where

γ is the adiabatic index without unit

As a sound wave propagates through a volume, displacement of particles x occurs along the wave propagation direction.

∂ x x m A = ∂ V V m = − 1   γ ∂ p p m

where

A is cross-sectional area in m²

From this equation it can be seen that when pressure is at its maximum, displacement reaches zero. As mentioned before, the oscillating pressure for a rightward travelling wave can be given by

p = p 0 c o s ( ω t − k x )

Since displacement is maximum when pressure is zero there is a 90 degrees phase difference, so displacement is given by

x = x 0 s i n ( ω t − k x )

Particle velocity is the first derivative of particle displacement. Differentiation of a sine gives a cosine again

u = u 0 c o s ( ω t − k x )

During adiabatic change, temperature changes with pressure as well following

∂ T T m = γ − 1   γ ∂ p p m

This fact is exploited within the field of thermoacoustics.

Propagation speed

The propagation speed of acoustic waves is given by the speed of sound. In general, the speed of sound c is given by the Newton-Laplace equation:

c = C ρ

where

C is a coefficient of stiffness, the bulk modulus (or the modulus of bulk elasticity for gas mediums), ρ is the density in kg/m³

Thus the speed of sound 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 speed of sound c is given by

c 2 = ∂ p ∂ ρ

where differentiation is taken with respect to adiabatic change.

where p is the pressure and ρ is the density

Interference

Interference is the addition of two or more waves that results in a new wave pattern. Interference of sound waves can be observed when two loudspeakers transmit the same signal. At certain locations constructive interference occurs, doubling the local sound pressure. And at other locations destructive interference occurs, causing a local sound pressure of zero pascals.

Standing wave

A standing wave is a special kind of wave that can occur in a resonator. In a resonator superposition of the incident and reflective wave occurs, causing a standing wave. Pressure and particle velocity are 90 degrees out of phase in a standing wave.

Consider a tube with two closed ends acting as a resonator. The resonator has normal modes at frequencies given by

f = N c 2 d N ∈ { 1 , 2 , 3 , … }

where

c is the speed of sound in m/s d is the length of the tube in m

At the ends particle velocity becomes zero since there can be no particle displacement. Pressure however doubles at the ends because of interference of the incident wave with the reflective wave. As pressure is maximum at the ends while velocity is zero, there is a 90 degrees phase difference between them.

Reflection

An acoustic travelling wave can be reflected by a solid surface. If a travelling wave is reflected, the reflected wave can interfere with the incident wave causing a standing wave in the near field. As a consequence, the local pressure in the near field is doubled, and the particle velocity becomes zero.

Attenuation causes the reflected wave to decrease in power as distance from the reflective material increases. As the power of the reflective wave decreases compared to the power of the incident wave, interference also decreases. And as interference decreases, so does the phase difference between sound pressure and particle velocity. At a large enough distance from the reflective material, there is no interference left anymore. At this distance one can speak of the far field.

The amount of reflection is given by the reflection coefficient which is the ratio of the reflected intensity over the incident intensity

R = I r e f l e c t e d I i n c i d e n t

Absorption

Acoustic waves can be absorbed. The amount of absorption is given by the absorption coefficient which is given by

α = 1 − R 2

where

α is the absorption coefficient without a unit R is the reflection coefficient without a unit

Often acoustic absorption of materials is given in decibels instead.

Measurement

Sound pressure can be measured directly using a microphone. Particle velocity can be measured directly using a particle velocity probe. It is also possible to measure the quantities indirectly using the opposite instrument. Sound intensity can be measured using different combinations:

Microphone and particle velocity probe (p-u probe)

Two microphones (p-p probe)

Two particle velocity probes (u-u probe)

The sound pressure is measured in pascal, the particle velocity in meters per second but not the sound intensity in watts. Often these quantities are measured as a level in decibels relative to a certain quantity.

The sound pressure level is given by

L p = 10 log 10 ⁡ ( p r m s 2 p r e f 2 )

where

p r m s is the root-mean square pressure in Pa p r e f is the reference value of 2*10⁻⁵ Pa

The particle velocity level is given by

L u = 10 log 10 ⁡ ( u r m s 2 u r e f 2 )

where

u r m s is the root-mean square particle velocity in m/s u r e f is the reference value of 5*10⁻⁸ m/s

The sound intensity level is given by

L I = 10 log 10 ⁡ I r m s I r e f

where

I r m s is the root-mean square sound intensity in W I r e f is the reference value of 1*10⁻¹² W