# How are a wave's energy and amplitude related?

Jun 6, 2014

$I \propto {A}^{2}$

Intensity, $I$, is the amount of wave energy arriving at a point per unit area per unit time.
$I = \frac{E}{A t} = \frac{P}{A}$

For a point source of a wave the energy is distributed across an area equal to the surface area of a sphere with a radius equal to the distance from the receiving location to the point source:
I=P/(4πd^2)

$P$ is the power at the source.
$d$ is the distance from the source.

Extending $I \propto {A}^{2}$

If $I \propto {A}^{2}$ then $\frac{I}{A} ^ 2 = k$

So ${I}_{1} / {A}_{1}^{2} = {I}_{2} / {A}_{2}^{2}$ and ${I}_{1} / {I}_{2} = {\left({A}_{1} / {A}_{2}\right)}^{2}$

That is useful for calculating intensity or amplitude ratios given enough information to determine the other ratio.

A more general relationship that you can see from the intensity equation above is $I \propto \frac{1}{d} ^ 2$. So wave intensity falls as you increase the distance from the source.

You can then also see the relationship between amplitude and distance: $I \propto {A}^{2} \propto \frac{1}{d} ^ 2$ so $A \propto \frac{1}{d}$.