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# Energy Density

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 have14.8    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  ,

 (14.22)

-- that is, the potential energy density is proportional to the amplitude squared.

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

 (14.23)

The total energy density is of course the sum of these two:

where   . Using     we can write this as

 (14.24)

You can use    in place of     if you like, since they are equal. [Exercise for the student.]

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

 (14.25)

Again, you can play around with the constants; instead of     you can use     and so on.

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!

#### Footnotes

. . . have14.8
I have avoided complex exponentials here to avoid confusion when I get around to calculating the transverse speed of the string element, . The acceleration is the same as for the complex version.

Next: Water Waves Up: WAVES Previous: Classical Quantization
Jess H. Brewer - Last modified: Sun Nov 15 18:04:30 PST 2015