4.11 Inverse Square Law Point Sources If the source of a wave is much smaller than the wavelength it emits, it can be represented by a 'point source'. It will tend to radiate energy equally in all directions, that is to say, with 'spherical symmetry'. Imagine a tiny spherical source which generates sound by rhythmically expanding and contracting. When it expands a pressure wave is transmitted outwards in all directions. The pressure pulse is followed by a rarefaction pulse as the sound source contracts. The resulting sound field (due to successive compressions and rarefactions of the surrounding fluid or gas) looks the same in all directions.

No energy is considered to have been lost! Inverse Square Law As one moves further from a source of spherical waves, the amplitude at your location gets less. This is because the intensity I of the wave decreases as it gradually spreads out over a greater area. The Intensity or Energy flux I is the Power P in the wave divided by the area A over which it is spread: I = P/A Assume that none of the wave power is absorbed (i.e. converted to heat) as it propagates from the source to you. This is the case for light sources radiating energy through free space – No energy is considered to have been lost!

If you are a distance r from the source, the area of the imaginary sphere over which the spherical wave is spread is  A = 4πr2. Substituting this into the previous equation gives I = P/4 π r2 where P is the Power of the source. As you move further from the source, r increases and the intensity of the wave decreases. Therefore the intensity decreases as an "inverse-square law" with distance r, that is like 1/r2. This physical law is explained by the figure below.

Try drawing this out on paper and measure the sides of the squares produced as a result. We should be able to measure this using 12V lamps and the dataloggers with light meters, moving them back gradually and logging the intensity of the light with distance in snapshot mode!

The inverse square law will apply to light sources Gamma sources Electric fields caused by a circular object Gravitational fields when viewed from a distance away from the body causing the field.