The energy density of sound waves is given by:
(164) |
For this reason the energy carried by sound waves is best measured by
intensity: the energy per unit time per unit area perpendicular to
the direction of wave propagation. Imagine a box with sides given by
(perpendicular to the direction of the wave's propagation)
and (in the direction of the wave's propagation. All the
energy in this box crosses through in time . That
is:
(165) |
(166) |
However, consider a spherical wave. For a spherical wave, the intensity
looks something like:
(167) |
(168) |
This makes sense from the point of view of energy conservation and symmetry. If a source emits a power , that energy has to cross each successive spherical surface that surrounds the source. Those surfaces have an area that varies like . A surface at has 4 times the area of one at , but the same total power has to go through both surfaces. Consequently, the intensity at the surface has to be the intensity at the surface.
It is important to remember this argument, simple as it is. Think back to Newton's law of gravitation. Remember that gravitational field diminishes as with the distance from the source. Electrostatic field also diminishes as . There seems to be a shared connection between symmetric propagation and spherical geometry; this will form the basis for Gauss's Law in electrostatics and much beautiful math.