Question

Show that if dampening is ignored, the amlitude A of circular water waves decrese as the square root of the distance r from the source:

A â 1/âr

Answer 1

See proof below.

Step-by-step explanation:

For this case we need to show that
`A \propto (1)/(√(r))`

Let's assume that we have two concentric circles. We need to assume that we have the same energy passing through the two concentric circles.

We know that the intensity is defined as:

`I = (P)/(2 \pi r)`

We know that
`I =(P)/(A)` and
`P = (Energy)/(t)`, so then:

`I = (E)/(At)`

Where P represent the power and r the radius.

We know that the intensity is
`I \propto A^2` proportional to the amplitude square. So using transitivity we have that:

`I \propto A^2 \propto (P)/(2\pi r)`

Since we are assuming that P is a constant and if we take the square root we got that:

`A \propto (1)/(√(r))`