Copyright © Michael Richmond.
This work is licensed under a Creative Commons License.

#
The energy and power carried by a wave

As a wave passes through a medium, it transfers
energy to that medium.
For example, when a water wave enters
a placid, still pool of water (low energy)
it turns that pool into a choppy mess of ripples and waves (high energy).

Is there some way to quantify the energy transferred
by the wave to the water?

Let's start with a very simplified situation:
a wave approaches a chunk of stuff of mass **M**
which is
currently at rest (brown = low energy):

After the wave passes through the chunk,
the material has been excited by the wave,
and has lots of energy (green = high energy):

How much energy has the wave given to the particles
of this chunk?
Well, the **kinetic energy** of all the particles
depends on their motion:

Q: Can you fill in the value of "v_{y}" here?

Now, at any given time, some of the particles will be
moving rapidly, some slowly, some not at all:
that's what the "sine" function means.
But if we average over all the particles, we can find
the average KE.

Okay, that gives us the average kinetic energy of the particles
in this chunk of material.
But they will also have **potential energy**, as they move
away from their equilibrium positions.
It turns out that the average potential energy is exactly equal
in size to the average kinetic energy,
for objects in SHM.
Therefore,

And so

####
The power transmitted by a wave

Note that the energy deposited in this chunk of material
depends upon the total mass of the particles in the chunk.
It is often useful to describe things in slightly
different terms:
in terms of **power**, which is the energy transmitted
per unit time.

Look at the chunk. It has a mass **M** and a length **d**.

We can define its **linear mass density μ** as the
mass per unit length:

Now, the distance that the wave travels in some time interval **t**
is simply the wave's speed **v** multiplied by the interval.
So, the amount of mass disturbed by the wave during that time
interval must be

The total energy of this disturbed mass is simply

Q: Can you write an equation for the POWER transmitted to the
chunk by the wave?

Pretty simple, isn't it?

Note that the power depends strongly on
both the amplitude of the wave,
and on the frequency of the wave.

Copyright © Michael Richmond.
This work is licensed under a Creative Commons License.