[Acoustics / Theory] Plane Sound Waves

[Definition of plane waves]

Waves in which any acoustic variable at a given time is a constant on any plane perpendicular to the direction of propagation

[One-dimensional wave equation]

General solution:

[Harmonic plane progressive waves]

Harmonic plane wave propagating in x-direction:

where

 : angular frequency

 : angular wavenumber

Wavelength λ:

[Complex exponential representation of sound pressure]

Sound pressure in complex form:

where

: complex amplitude

Physical sound pressure is the real part of the complex pressure:

[Euler equation of motion in complex form]

 is a linear operator:

 and 

Euler equation of motion,

becomes

[Wave equation in complex form: Helmholtz equation]

or

[Complex notation of the solution]

From the equation (in the x-direction),

we get

This means that the particle velocity in a plane wave is proportional to the sound pressure with the ratio 1/ρc, which is the characteristic impedance of the medium.

[General solution to the 1-D Helmholtz equation]

Sum of

• a wave that travels in the positive x-direction (incident wave) and

• a wave that travels in the negative(opposite) x-direction (reflected wave)

Corresponding expression for the particle velocity:

[Relationship between the incident and the reflected wave]

From the boundary condition of zero particle velocity on a rigid surface,

gives

Thus

   

Similarly,

[Complex reflection factor]

Complex reflection factor R: Ratio of p_r to p_i,

R = 1: Perfect reflection with no phase shift