6. Spherical Waves
Spherical waves in general are discussed in the section on wave spectra. Here we consider only the most elementary spherical wave. Compared to the plane wave solution, a spherical wave is much closer to waves encountered in the real world. An outgoing spherical wave solution to equations (26) and (27) is
This is the wave produced by an elementary point source. Power flow in this case is inversely proportional to r2. Except for this, when r is much greater than the wavelength λ=2π/k, the behavior is very similar to a plane wave. This is called "far-field behavior". When r is comparable to, or smaller than, λ the behavior is more complex, even for this simple source. For a real source of finite dimension, the behavior in the "near field" is even more complex.
This wave propagates isotropically - that is, the wave amplitude is spherically symmetric. Another interesting elementary source is a Huygens' wavelet, which provides a useful way of understanding wave propagation. It is not an isotropic source; a description of a Huygens' wavelet can be found here.
The ratio of the magnitudes of pressure and velocity for this elementary wave is the blue curve shown here [36 kb] as a function of r. It is seen that when r is greater than a wavelength the ratio is already close to the ratio for a plane wave in free space. For small values of r the ratio is very small. The relative phase (not shown) is close to 90 degrees for r=.01 wavelengths, and about 10 degrees at r=1 wavelength. At r=10 wavelengths the phase is about 1 degree, so at this point the behavior is very close to a plane wave.
The fact that the magnitude ratio is small when r is a small fraction of a wavelength means that for a given pressure the velocity is much larger than for a plane wave. This means that it takes a lot of velocity to create much sound pressure, and is the reason that loudspeaker response drops off at low frequencies. Conversely, it also implies that a small (with respect to a wavelength) piston mounted in a baffle, or a small diameter tube in free space, will respond primarily to the pressure of an incident plane wave, and relatively weakly to the velocity. The coupling to a tube computed in the previous section is shown by the red curve, confirming this expectation.
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