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Spherical Waves from a Point Source

Acoustic theory tells us that a point source produces a spherical wave in an ideal isotropic (uniform) medium such as air. Furthermore, the sound from any radiating surface can be computed as the sum of spherical wave contributions from each point on the surface (including any relevant reflections). The Huygens-Fresnel principle explains wave propagation itself as the superposition of spherical waves generated at each point along a wavefront (see, e.g., [352, p. 175]). Thus, all linear acoustic wave propagation can be seen as a superposition of spherical traveling waves.

To a good first approximation, wave energy is conserved as it propagates through the air. In a spherical pressure wave of radius $ r$ , the energy of the wavefront is spread out over the spherical surface area $ 4\pi r^2$ . Therefore, the energy per unit area of an expanding spherical pressure wave decreases as $ 1/r^2$ . This is called spherical spreading loss. It is also an example of an inverse square law which is found repeatedly in the physics of conserved quantities in three-dimensional space. Since energy is proportional to amplitude squared, an inverse square law for energy translates to a $ 1/r$ decay law for amplitude.

Figure 2.6: Geometry of wave propagation from a point source $ \mathbf {x}_1$ to a listening point $ \mathbf {x}_2$ .

The sound-pressure amplitude of a traveling wave is proportional to the square-root of its energy per unit area. Therefore, in a spherical traveling wave, acoustic amplitude is proportional to $ 1/r$ , where $ r$ is the radius of the sphere. In terms of Cartesian coordinates, the amplitude $ p(\mathbf{x}_2)$ at the point $ \mathbf{x}_2 =
(x_2,y_2,z_2)$ due to a point source located at $ \mathbf{x}_1 =
(x_1,y_1,z_1)$ is given by

$\displaystyle p(\mathbf{x}_2) = \frac{p_1}{r_{12}}

where $ p_1$ is defined as the pressure amplitude one radial unit from the point source located at $ \mathbf{x}=\mathbf{x}_1$ (i.e., $ p_1=p(\mathbf{x}_1+\mathbf{e})$ where $ \vert\vert\,\mathbf{e}\,\vert\vert =1$ ), and $ r_{12}$ denotes the distance from the point $ \mathbf {x}_1$ to $ \mathbf {x}_2$ :

$\displaystyle r_{12} \isdef \left\Vert\,\mathbf{x}_2 - \mathbf{x}_1\,\right\Vert
= \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}

This geometry is depicted for the 2D case in Fig.2.6.

In summary, every point of a radiating sound source emits spherical traveling waves in all directions which decay as $ 1/r$ , where $ r$ is the distance from the source. The amplitude-decay by $ 1/r$ can be considered a consequence of energy conservation for propagating waves. (The energy spreads out over the surface of an expanding sphere.) We often visualize such waves as ``rays'' emanating from the source, and we can simulate them as a delay line along with a $ 1/r$ scaling coefficient (see Fig.2.7). In contrast, since plane waves propagate with no decay at all, each ``ray'' can be considered lossless, and the simulation involves only a delay line with no scale factor, as shown in Fig.2.1 on page [*].

Figure 2.7: Point-to-point spherical wave simulator. In addition to propagation delay, there is attenuation by $ g=1/r$ .

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University