Consider again our little element of string at position .
We have shown that (for fixed ) the mass element will execute
SHM as a function of time . Therefore there is an
effective LINEAR RESTORING FORCE in the direction
acting on the mass element :
.
But for a simple traveling wave we have^{14.8}
so
,
giving
.
In other words, the effective spring constant
for an element of string long is
where I have used the unconventional notation
for the effective spring constant to avoid confusing it
with the wavenumber , which is something
completely different. Applying our knowledge of the
potential energy stored in a stretched spring,
,
we have the elastic potential energy stored in the
string per unit length,
or,
plugging in ,
What about kinetic energy? From SHM we expect
the energy to be shared between potential and kinetic energy
as each mass element oscillates through its period.
Well, the kinetic energy of our little element of string
is just
.
Again
and now we must evaluate .
Working from
we have
,
from which we can write
The total energy density is of course the sum of these two:
Note that the net energy density (potential plus kinetic) is constant in time and space for such a uniform traveling wave. It just switches back and forth between potential and kinetic energy twice every cycle. Since the average of either or is 1/2, the energy density is on average shared equally between kinetic and potential energy.
If we want to know the energy per unit time (power )
transported past a certain point by the wave, we just
multiply by to get
Note that while the wave does not transport any mass down the string (all physical motion is transverse) it does transport energy. This is an ubiquitous property of waves, lucky for us!