Question

How are a wave's energy and amplitude related?

Answer 1

`IpropA^2`

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Intensity, `I`, is the amount of wave energy arriving at a point per unit area per unit time.
`I=E/(At)=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.

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Extending `IpropA^2`

If `IpropA^2` then `I/A^2=k`

So `I_1/A_1^2=I_2/A_2^2` and `I_1/I_2=(A_1/A_2)^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 `Iprop1/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: `IpropA^2prop1/d^2` so `Aprop1/d`.