Acoustic wave
Encyclopedia
Acoustic waves are a type of longitudinal waves that propagate by means of adiabatic compression and decompression. Longitudinal waves are waves that have the same direction of vibration as their direction of travel. Important quantities for describing acoustic waves are sound pressure
, particle velocity
, particle displacement
and sound intensity
. Acoustic waves travel with the speed of sound
which depends on the medium they're passing through.
, reflection
and interference
. Sound waves however don't have any polarization since they oscillate along the same direction as they move.
describes the propagation of sound waves. The acoustic wave equation for sound pressure
in one dimension
is given by
where
is sound pressure
in Pa
is particle displacement
in m
is speed of sound
in m/s
is time
in s
The wave equation for particle velocity
has the same shape and is given by
where
is particle velocity
in m/s
D'Alembert's gave the general solution for the wave equation. For sound pressure, a solution would be
where
is angular frequency
in rad/s
is time in s
is wave number in rad·m−1
is a coefficient without unit
For
the wave becomes a travelling wave moving rightwards, for 
the wave becomes a travelling wave moving leftwards. A standing wave
can be obtained by
.
, which means the phase angle between the two quantities is zero.
This can be easily proven using the ideal gas law
where
is pressure
in Pa
is volume in m3
is amount in mol
is the universal gas constant with value 
Consider a volume
. As an acoustic wave propagates through the volume, adiabatic compression and decompression occurs. For adiabatic change the following relation between volume 
of a parcel of fluid and pressure 
holds
where
is the adiabatic index without unit
As a sound wave propagates through a volume, displacement of particles
occurs along the wave propagation direction.
where
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
Since displacement is maximum when pressure is zero there is a 90 degrees phase difference, so displacement is given by
Particle velocity is the first derivative of particle displacement. Differentiation of a sine gives a cosine again
During adiabatic change, temperature changes with pressure as well following
This fact is exploited within the field of thermoacoustics
.
. In general, the speed of sound c is given by the Newton-Laplace equation:
where
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
is given by
where differentiation is taken with respect to adiabatic change.
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.
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
where
is the speed of sound in m/s
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.
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
where
is the absorption coefficient without a unit
is the reflection coefficient
without a unit
Often acoustic absorption
of materials is given in decibels instead.
. 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:
The sound pressure is measured in pascal, the particle velocity in meters per second and 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
where
is the root-mean square pressure in Pa
is the reference value of 2*10-5 Pa
The particle velocity level is given by
where
is the root-mean square particle velocity in m/s
is the reference value of 5*10-8 m/s
The sound intensity level is given by
where
is the root-mean square sound intensity in W
is the reference value of 1*10-12 W
Sound pressure
Sound pressure or acoustic pressure is the local pressure deviation from the ambient atmospheric pressure caused by a sound wave. Sound pressure can be measured using a microphone in air and a hydrophone in water...
, particle velocity
Particle velocity
Particle velocity is the velocity v of a particle in a medium as it transmits a wave. In many cases this is a longitudinal wave of pressure as with sound, but it can also be a transverse wave as with the vibration of a taut string....
, particle displacement
Particle displacement
Particle displacement or particle amplitude is a measurement of distance of the movement of a particle from its equilibrium position in a medium as it transmits a wave....
and sound intensity
Sound intensity
Sound intensity or acoustic intensity is defined as the sound power Pac per unit area A. The usual context is the noise measurement of sound intensity in the air at a listener's location.-Acoustic intensity:...
. Acoustic waves travel with the speed of sound
Speed of sound
The speed of sound is the distance travelled during a unit of time by a sound wave propagating through an elastic medium. In dry air at , the speed of sound is . This is , or about one kilometer in three seconds or approximately one mile in five seconds....
which depends on the medium they're passing through.
Wave properties
Acoustic waves are longitudinal waves that exhibit phenomena like diffractionDiffraction
Diffraction refers to various phenomena which occur when a wave encounters an obstacle. Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1665...
, reflection
Reflection
Reflection or reflection may refer to:-Science:* Mirror image, a reflection in a mirror or in water* Reflection nebula, a nebula that is extended and has no boundaries* Reflection , a common wave phenomenon...
and interference
Interference
In physics, interference is a phenomenon in which two waves superpose to form a resultant wave of greater or lower amplitude. Interference usually refers to the interaction of waves that are correlated or coherent with each other, either because they come from the same source or because they have...
. Sound waves however don't have any polarization since they oscillate along the same direction as they move.
Acoustic wave equation
The acoustic wave equationAcoustic wave equation
In physics, the acoustic wave equation governs the propagation of acoustic waves through a material medium. The form of the equation is a second order partial differential equation. The equation describes the evolution of acoustic pressure p or particle velocity u as a function of position r and...
describes the propagation of sound waves. The acoustic wave equation for sound pressure
Sound pressure
Sound pressure or acoustic pressure is the local pressure deviation from the ambient atmospheric pressure caused by a sound wave. Sound pressure can be measured using a microphone in air and a hydrophone in water...
in one dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
is given by
where
is sound pressureSound pressure
Sound pressure or acoustic pressure is the local pressure deviation from the ambient atmospheric pressure caused by a sound wave. Sound pressure can be measured using a microphone in air and a hydrophone in water...
in Pa
Pascal (unit)
The pascal is the SI derived unit of pressure, internal pressure, stress, Young's modulus and tensile strength, named after the French mathematician, physicist, inventor, writer, and philosopher Blaise Pascal. It is a measure of force per unit area, defined as one newton per square metre...
is particle displacementParticle displacement
Particle displacement or particle amplitude is a measurement of distance of the movement of a particle from its equilibrium position in a medium as it transmits a wave....
in m
M
M is the thirteenth letter of the basic modern Latin alphabet.-History:The letter M is derived from the Phoenician Mem, via the Greek Mu . Semitic Mem probably originally pictured water...
is speed of soundSpeed of sound
The speed of sound is the distance travelled during a unit of time by a sound wave propagating through an elastic medium. In dry air at , the speed of sound is . This is , or about one kilometer in three seconds or approximately one mile in five seconds....
in m/s
is timeTime
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....
in s
S
S is the nineteenth letter in the ISO basic Latin alphabet.-History: Semitic Šîn represented a voiceless postalveolar fricative . Greek did not have this sound, so the Greek sigma came to represent...
The wave equation for particle velocity
Particle velocity
Particle velocity is the velocity v of a particle in a medium as it transmits a wave. In many cases this is a longitudinal wave of pressure as with sound, but it can also be a transverse wave as with the vibration of a taut string....
has the same shape and is given by
where
is particle velocityParticle velocity
Particle velocity is the velocity v of a particle in a medium as it transmits a wave. In many cases this is a longitudinal wave of pressure as with sound, but it can also be a transverse wave as with the vibration of a taut string....
in m/s
D'Alembert's gave the general solution for the wave equation. For sound pressure, a solution would be
where
is angular frequencyAngular frequency
In physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
in rad/s
is time in sS
S is the nineteenth letter in the ISO basic Latin alphabet.-History: Semitic Šîn represented a voiceless postalveolar fricative . Greek did not have this sound, so the Greek sigma came to represent...
is wave number in rad·m−1 is a coefficient without unitFor
the wave becomes a travelling wave moving rightwards, for the wave becomes a travelling wave moving leftwards. A standing waveStanding wave
In physics, a standing wave – also known as a stationary wave – is a wave that remains in a constant position.This phenomenon can occur because the medium is moving in the opposite direction to the wave, or it can arise in a stationary medium as a result of interference between two waves traveling...
can be obtained by
.Phase
In a travelling wave pressure and particle velocity are in phasePhase
-In physics:*Phase , a physically distinctive form of a substance, such as the solid, liquid, and gaseous states of ordinary matter**Phase transition is the transformation of a thermodynamic system from one phase to another*Phase...
, which means the phase angle between the two quantities is zero.
This can be easily proven using the ideal gas law
Ideal gas law
The ideal gas law is the equation of state of a hypothetical ideal gas. It is a good approximation to the behavior of many gases under many conditions, although it has several limitations. It was first stated by Émile Clapeyron in 1834 as a combination of Boyle's law and Charles's law...
where
is pressurePressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...
in Pa
Pascal (unit)
The pascal is the SI derived unit of pressure, internal pressure, stress, Young's modulus and tensile strength, named after the French mathematician, physicist, inventor, writer, and philosopher Blaise Pascal. It is a measure of force per unit area, defined as one newton per square metre...
is volume in m3 is amount in molMol
-Places:* City Municipality of Ljubljana, known after the acronym MOL in Slovene language * Märkisch-Oderland, a rural district of Brandenburg, Germany* Mol, Belgium, a municipality in Belgium* Mol, Serbia, a town in Serbia...
is the universal gas constant with value Consider a volume
. As an acoustic wave propagates through the volume, adiabatic compression and decompression occurs. For adiabatic change the following relation between volume of a parcel of fluid and pressure holds
where
is the adiabatic index without unitAs a sound wave propagates through a volume, displacement of particles
occurs along the wave propagation direction.
where
is cross-sectional area in m2From 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
Since displacement is maximum when pressure is zero there is a 90 degrees phase difference, so displacement is given by
Particle velocity is the first derivative of particle displacement. Differentiation of a sine gives a cosine again
During adiabatic change, temperature changes with pressure as well following
This fact is exploited within the field of thermoacoustics
Thermoacoustics
Thermoacoustics is about the interaction between thermodynamic and acoustic phenomena. Thermoacoustics is a relatively new field of science and engineering. Few devices based on this principle have been made thus far...
.
Propagation speed
The propagation speed of acoustic waves is given by the speed of soundSpeed of sound
The speed of sound is the distance travelled during a unit of time by a sound wave propagating through an elastic medium. In dry air at , the speed of sound is . This is , or about one kilometer in three seconds or approximately one mile in five seconds....
. In general, the speed of sound c is given by the Newton-Laplace equation:
where
- C is a coefficient of stiffnessElastic modulusAn elastic modulus, or modulus of elasticity, is the mathematical description of an object or substance's tendency to be deformed elastically when a force is applied to it...
, the bulk modulusBulk modulusThe bulk modulus of a substance measures the substance's resistance to uniform compression. It is defined as the pressure increase needed to decrease the volume by a factor of 1/e...
(or the modulus of bulk elasticity for gas mediums),
is the densityDensityThe mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
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
is given bywhere differentiation is taken with respect to adiabatic change.
- where
is the pressure and 
is the density
Interference
InterferenceInterference
In physics, interference is a phenomenon in which two waves superpose to form a resultant wave of greater or lower amplitude. Interference usually refers to the interaction of waves that are correlated or coherent with each other, either because they come from the same source or because they have...
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 waveStanding wave
In physics, a standing wave – also known as a stationary wave – is a wave that remains in a constant position.This phenomenon can occur because the medium is moving in the opposite direction to the wave, or it can arise in a stationary medium as a result of interference between two waves traveling...
is a special kind of wave that can occur in a resonator
Resonance
In physics, resonance is the tendency of a system to oscillate at a greater amplitude at some frequencies than at others. These are known as the system's resonant frequencies...
. In a resonator superposition
Superposition
Superposition can refer to:* The superposition principle in physics, mathematics, and engineering, describes the overlapping of waves. Particular applications include :** Quantum superposition, in quantum physics** Superposition theorem, in electronics....
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
where
is the speed of sound in m/s is the length of the tube in mM
M is the thirteenth letter of the basic modern Latin alphabet.-History:The letter M is derived from the Phoenician Mem, via the Greek Mu . Semitic Mem probably originally pictured water...
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 reflectedReflection (physics)
Reflection is the change in direction of a wavefront at an interface between two differentmedia so that the wavefront returns into the medium from which it originated. Common examples include the reflection of light, sound and water waves...
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
Near field
Near field may refer to:*Near-field , an algebraic structure*Near and far field, parts of an electromagnetic field*Near field communication, a set of short-range wireless technologies, typically requiring a distance of 4 cm or less...
. 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
Absorption
Acoustic waves can be absorped. The amount of absorption is given by the absorption coefficient which is given bywhere
is the absorption coefficient without a unit is the reflection coefficientReflection coefficient
The reflection coefficient is used in physics and electrical engineering when wave propagation in a medium containing discontinuities is considered. A reflection coefficient describes either the amplitude or the intensity of a reflected wave relative to an incident wave...
without a unit
Often acoustic absorption
Absorption (acoustics)
Acoustic absorption is that property of any material that changes the acoustic energy of sound waves into another form, often heat, which it to some extent retains, as opposed to that sound energy that material reflects or conducts. Acoustic absorption is represented by the symbol A in calculations...
of materials is given in decibels instead.
Measurement
Sound pressure can be measured directly using a microphoneMicrophone
A microphone is an acoustic-to-electric transducer or sensor that converts sound into an electrical signal. In 1877, Emile Berliner invented the first microphone used as a telephone voice transmitter...
. Particle velocity can be measured directly using a particle velocity probe
Particle velocity probe
A Particle velocity probe is a probe capable of measuring the acoustic particle velocity.The only particle velocity probe to date has been produced by Microflown Technologies and is called the Microflown....
. 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 and 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
where
is the root-mean square pressure in Pa is the reference value of 2*10-5 PaThe particle velocity level is given by
where
is the root-mean square particle velocity in m/s is the reference value of 5*10-8 m/sThe sound intensity level is given by
where
is the root-mean square sound intensity in W is the reference value of 1*10-12 WSee also
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