The intensity (power per unit area) of a spherical wave falls off as $1/4\pi r^2$.
My question: Is that equation correct? Does this mean the wave amplitude falls off as $1/2\pi^{1/2} r$ ?
I understand the $1/r$ and the $1/2$ but have not seen the $1/\pi^{1/2} $ used. The sources I have seen only say that the amplitude is inversely proportional to $r$.
See
http://resource.isvr.soton.ac.uk/spcg/tutorial/tutorial/Tutorial_files/Web-basics-pointsources.htm
and
The inverse square law $I=\frac{P}{4\pi r^2}$ is correct and its proof is astonishingly simple:
Let $P$ be the total power radiated from a point source. Let's assume this does not depend on time. When the wave is spherical then $P$ is constant on every sphere with radius $r$ around the source. Since the only source of $P$ is the source of the wave it is also clear that -due to energy conservation- $P$ does not depend on $r$. Since the surface area of a sphere with radius $r$ is $A=4\pi r^2$ we get $$ I=\frac{P}{A}=\frac{P}{4\pi r^2}\,. $$ Note that his law only holds in three dimensions.
