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Sound Wavefronts

Plane waves

Plane waves are the form of wave propagation in one dimension. The wavefronts of plane waves are planes perpendicular to the propagation direction.

Plane waves are generated by a plane source. The surface area of wavefront does not vary with the distance away from the sound source. Since there is no geometrical divergence along the propagation path, both the sound intensity and sound pressure remain unchanged. 

and

where

A_x is the surface area at distance x and is a constant
I_x is the sound intensity at distance x
p_x is the sound pressure at distance x

Therefore, both the sound intensity level and the sound pressure level remains constant with distance away from the source.

Plane waves propagation can be found in musical pipe instrument, e.g. flutes, clarinets, or noise in ducts.

Cylindrical Waves

Cylindrical waves are the form of wave propagation in two dimension. The wavefronts of cylindrical waves are coaxial cylinders with axis of symmetry normal to the propagation direction.

Cylindrical waves are generated by a line source. The surface area of wavefront increases with the distance away from the sound source. Since there is geometrical divergence along the propagaton path, both the sound intensity and sound pressure  decreases with distance away from the source.

and

where

2πrh is the surface area at distance r
I_r is the sound intensity at distance r
p_r is the sound intensity at distance r
2π(2r)h is the surface area at distance 2r
I_2r is the sound intensity at distance 2r
p_2r is the sound intensity at distance 2r

Therefore, both the sound intensity level in term of I and sound pressure level in term of p²  decrease by 3dB as the distance of receiver away from the source is increased by double. 

and
and

where

L_Ir is the sound intensity level at distance r
L_p2r  is the sound pressure level at distance r
I_r is the sound intensity at distance r
I_o is the sound intensity reference
p_r is the sound intensity at distance r
p_o is the sound pressure reference
L_I2r  is the sound intensity level at distance 2r
L_p2r  is the sound pressure level at distance 2r
I_2r is the sound intensity at distance 2r
p_2r is the sound intensity at distance 2r

Cylindrical waves propagation can be found in sound radiation from a circular pipes or duct, or noise field in an open space radiated by a lane of closely spaced vehicles.

Spherical Waves

Spherical waves are the form of wave propagation in three dimension. The wavefronts of spherical waves are concentric spheres with center at the center of propagation in all directions.

Spherical waves are generated by a point source. The surface area of wavefront increases with the distance away from the sound source. Since there is geometrical divergence along the propagation path, both the sound intensity and sound pressure level decreases with distance away from the source.

and

where

4πr² is the surface area at distance r
I_r is the sound intensity at distance r
p_r is the sound intensity at distance r
4π(2r)² is the surface area at distance 2r
I_2r is the sound intensity at distance 2r
p_2r is the sound intensity at distance 2r

Therefore, both the sound intensity level in term of I and sound pressure level in term of p² decrease by 6dB as the distance of receiver away from the source is increased by double. 

and
and

where

L_Ir is the sound intensity level at distance r
L_p2r  is the sound pressure level at distance r
I_r is the sound intensity at distance r
I_o is the sound intensity reference
p_r is the sound intensity at distance r
p_o is the sound pressure reference
L_I2r  is the sound intensity level at distance 2r
L_p2r  is the sound pressure level at distance 2r
I_2r is the sound intensity at distance 2r
p_2r is the sound intensity at distance 2r

Spherical waves propagation in practical can be found by approximating a sound source with source dimension and radiated wavelength much smaller than the distance away from the receiver.