Energy Considerations in Waves

## Question:

From questions asked in our homework, I have some other questions:

Q1: In the SHM case, we have constant energy, p.e. + k.e. In the wave case, is it the total energy will also be constant? Even though it may not be a spring? Like if we use a string instead, with no elasticity?

How can we prove that if we are asked to in the exams? Do we have to prove different cases mathematically?

Q2: In the SHM case, we just have one medium only. But in the wave case, we have two, or even more mediums. What extra factors do we have to note in calculating the energy? Like from denser medium to less dense medium, the k.e. of a wave will change e.t.c? If the total energy is really constant, then will a wave traveling slower when entering a new medium gains some p.e.? Why does the extra energy come from?

Q3: I think waves do not necessarily be waves in shm pattern of movement. Then how come we can use the energy formular which is in the same format as a shm case?

Thanks!

It is a very general principle that oscillatory systems, that is systems that change state in a periodic way, have two mechanisms for storing energy and exchange the energy of the system back and forth between these two mechanisms. In the case of the simple harmonic oscillator the two mechanisms are the kinetic and potential energy of the system. If there is no loss of energy due to friction, the total remains constant.

In the case of a wave in a vibrating string, the string when it is stretched has potential energy and when it is moving has kinetic energy. The wave is just a manifestation of the energy exchange between these two. Again the total energy is constant if friction is neglected.

In the case of a sound wave, the potential energy is the result of pressure differences between the compressions and rarefactions of the wave. The kinetic energy shows up in the movement of the air molecules as the wave passes. Again the total energy is conserved if friction is neglected. Of course if the sound wave is allowed to spread out as it moves, the energy is conserved only if you add it up over the whole surface over which the wave has spread. At any point the energy will appear to decrease as the surface becomes larger.

In the case of electro-magnetic waves, the energy of the wave is shared between the energy stored in the electric field and that stored in the magnetic field. As the wave develops, the energy is passed back and forth between these two mechanisms. If you think back to the SHM, the potential energy was stored in the spring and the kinetic energy was attributed to the mass of the moving parts. In the electro-magnetic wave the electric field is sort of spring like and the magnetic field sort mass like. The total energy is conserved if there is no loss due to heating, which is equivalent to friction in mechanical systems.

The fact that in the absence of some loss mechanism like friction or heating, energy is conserved is a direct consequence of the principle of conservation of momentum. More fundamentally it follows from the idea that the amount of anything you have is equal to the amount you started with minus that which was taken away. If none is taken away, then the property is conserved.