The total energy density in a harmonic wave on a stretched string is given by
$$\frac{1}{2}p A^2 \omega^2 sin^2(kx-\omega t).$$
We can see that this energy oscillates between a maximum and a minimum. So the energy is maximum at 0 displacement when the string is stretched and at its maximum speed (both KE and PE density are maximum at the same time) and minimum when the displacement is maximum as it is unstretched and doesnt have any velocity.
This makes sense but I am having trouble merging this with SHM oscillations. In SHM the KE and PE are not in phase. And if we consider each particle of the wave acting as a shm oscillator then would the PE not be maximum at the maximum displacement?
Best Answer
PE and KE that we are talking here are of a small part of string. PE and KE are maximum when the element passes through its mean position as velocity is maximum and string part is most stretched. At crest (or trough) velocity is zero and string part is not stretched so both PE and KE are zero of that string part.
Now to relate it with SHM, let us take SHM of spring-block system. In the string, the string part is like a block and rest of the string is like the spring. PE that we talk in SHM is of the spring and not stored in the block, whereas in the wave on string we are talking of PE stored in the string part i.e. block. That is why the two PE we are talking are different. The PE stored in the rest of the string is behaving as PE stored in the spring. Hope it helps!