Acoustic waves are disturbances that propagate through a medium—such as air, water, or solids—by causing the particles of the medium to compress and expand.
The speed of an acoustic wave depends on the properties of the medium it travels through; for example, it travels at approximately 343 meters per second in air, and 1480 meters per second in water.
Acoustic waves encompass a broad range of phenomena, from audible sound to seismic waves and ultrasound, finding applications in diverse fields like acoustics, engineering, and medicine.
However, in solids, acoustic waves transmit in both longitudinal and transverse manners due to presence of shear moduli in such a state of matter.
The acoustic wave equation for sound pressure in one dimension is given by
where The wave equation for particle velocity has the same shape and is given by
where For lossy media, more intricate models need to be applied in order to take into account frequency-dependent attenuation and phase speed.
D'Alembert gave the general solution for the lossless wave equation.
This can be easily proven using the ideal gas law
As an acoustic wave propagates through the volume, adiabatic compression and decompression occurs.
As a sound wave propagates through a volume, the horizontal displacement of a particle
where 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
In general, the acoustic velocity c is given by the Newton-Laplace equation:
where 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
Note that sound waves in air are not polarized since they oscillate along the same direction as they move.
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.
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.
As pressure is maximum at the ends while velocity is zero, there is a 90 degrees phase difference between them.
An acoustic travelling wave can be reflected by a solid surface.
As a consequence, the local pressure in the near field is doubled, and the particle velocity becomes zero.
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.
When an acoustic wave propagates through a non-homogeneous medium, it will undergo diffraction at the impurities it encounters or at the interfaces between layers of different materials.
This is a phenomenon very similar to that of the refraction, absorption and transmission of light in Bragg mirrors.
[2] The acoustic absorption, reflection and transmission in multilayer materials can be calculated with the transfer-matrix method.