### 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.*,
[349, 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
, the energy of the wavefront is spread out over the spherical
surface area . Therefore, the energy per unit area of an
expanding spherical pressure wave decreases as . 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 decay law for amplitude.

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 , where is the radius of the sphere. In terms of Cartesian coordinates, the amplitude at the point due to a point source located at is given by

*i.e.*, where ), and denotes the distance from the point to :

In summary, every point of a radiating sound source emits spherical traveling waves in all directions which decay as , where is the distance from the source. The amplitude-decay by 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 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 .

**Next Section:**

Reflection of Spherical or Plane Waves

**Previous Section:**

Converting Propagation Distance to Delay Length