# Does the energy of a sound wave depend on frequency?

The energy of electromagnetic waves is said to be dependent on frequency. Is the energy of a sound wave also dependent on frequency?

• Hi! Look here: en.wikipedia.org/wiki/Sound_energy Oct 1, 2020 at 7:17
• When you say "the energy of electromagnetic waves is said to be dependent on frequency" are you sure you aren't confusing the energy in a photon with the energy in the wave (composed of many photons)? Oct 1, 2020 at 16:54

## 1 Answer

First let's think what we mean by the energy of a wave. It can be defined as the amount of energy that an emitter has to give up, or a receiver take in, in order to emit or absorb the wave. If the wave is being emitted or absorbed continuously then we could talk about the energy crossing any given plane per unit time, or else the energy per unit length of the wave.

In the case of electromagnetic waves, the primary equation for this quantity is $${\bf S} = {\bf E} \times {\bf H}$$ This is called the Poynting vector and it gives the flux, which is amount of energy per unit area per unit time flowing past a plane at right angles to the vector. Notice that there is no mention of frequency in this formula. This means that two waves having fields $$\bf E$$ and $$\bf H$$ of the same amplitude will have the same energy flux, no matter what their frequencies may be.

Now in the question it said that energy was related to frequency. I think this is a reference to the formula $$E_p = h f$$ where $$E_p$$ is energy, $$f$$ is frequency and $$h$$ is Planck's constant. This is not a formula for the energy "in the wave"; it is a formula for how much of the energy in the wave is assigned to each photon. I put a subscript $$p$$ to act as a reminder of this (and to make sure we don't muddle it with electric field). It follows that the number of photons crossing a plane, per unit area and per unit time, is $$N = \frac{\bf S}{E_p} = \frac{ {\bf E} \times {\bf H} }{h f}.$$

Coming now to sound waves, the energy is to do with the kinetic energy and potential energy of the matter which is transmitting the wave. As the matter particles move to and fro, they have kinetic energy, and the restoring forces on them (pressure or tension) give rise to potential energy. The result is that the energy flux is proportional to the square of the amplitude. The energy per unit volume is $$\frac{p^2}{2 \rho_0 v_s^2} + \frac{1}{2} \rho v^2$$ where $$p$$ is pressure, $$\rho$$ is density, $$v_s$$ is the speed of sound and $$v$$ is the speed of the movement in the medium. Multiply this formula by the speed of sound in order to get energy flux.

Now the question was, does this depend on frequency? The answer is that it may do, but this depends also on other things. If the amplitude of the wave displacements is fixed and the frequency is increased, then the kinetic energy of the particle motion will be increased, and therefore so will the energy flux. On the other hand, if the amplitude of the speed oscillations of the vibrating thing producing the wave stays fixed as the frequency changes, then the kinetic energy will not change and neither will the energy flux. In that case the position amplitude goes down as the frequency goes up and the two effects cancel.

Finally, in quantum physics we can associate particles with sound waves. The particles are called phonons. Each phonon has an energy related to its frequency by $$E_p = h f$$. That formula is pretty much universal for all types of wave motion.