Energy in a wave (Physics)

I’m a high school physics teacher and everyone describes waves as something like “a disturbance of particles that carries energy from one place to another,” but we rarely talk about the energy of the wave, except for the energy of a photon (E=hf).

Let’s take a transverse wave in a spring. As far as I can remember, the energy of a wave is proportional to the square of the amplitude. And it makes sense that a higher amplitude would mean more energy if I think of it in terms of the kinetic energy of the particles of the medium (the spring) when it is passing through the equilibrium (flat) position. (With a fixed frequency) a bigger amplitude means the particles are traveling a bigger distance in the same amount of time, so they have more (average) speed, and thus there is more energy in the wave. … Or instead of the kinetic energy, it’s easy to see that the spring gets more stretched for higher amplitudes, so when the spring is momentarily stopped at the instant of greatest amplitude there is more potential energy in the spring than with a lower amplitude wave.

HERE IS MY QUESTION: Is the frequency of the wave (including a wave in a rope, spring, water, etc., not just for photons) a factor in the energy of the wave, or in the energy the wave delievers in a second (i.e. the wave’s power and therefore its intensity)?

It makes sense to me that a higher frequency (with the same amplitude) would give more energy because the particles of the medium would be moving more quickly than with with a lower frequency, because the particles cover the same distance from peak to trough in less time. So I’m guessing that a higher frequency would mean more energy. But I’ve searched the internet and our book and I can’t find any mention of an equation for the energy of a wave, or how the frequency would or would not affect the wave’s energy.

Just using “linear logic.” Two identical springs, one bouncing at twice the rate of the other. Means twice the speed (at max) thus twice the acceleration, thus twice the force… Gotter mean more energy.

i.e., just take one single portion of one cycle. It takes more energy to slow a fast-moving object to a halt than to slow a slow-moving, otherwise identical, object to a halt.

Naive physics, but, well, inevitable, innit?

You’re familiar with the formula for energy of a photon, E = hf (that’s Planck’s constant times the frequency)? Well, that actually applies for a single quantum of any wave. And the amplitude of the wave determines the number of quanta you have. So, yes, for a given amplitude, the energy in a wave is proportional to its frequency.

Sorry for the hijack, but how do you determine the number of quanta from the amplitude?


What “E=hf” means is that there is a minimum amplitude, and that there is a second lowest amplitude, and a third lowest, and so on. So if you measure a wave with wavelength f and energy 5hf, it has the fifth lowest possible amplitude, corresponding to 5 quanta of frequency f. The relationship between the exact physical amplitude (say, measured in cm) and the number of quanta is dependent on the medium through which the wave is propagating. But generally speaking the physical amplitude is not a meaningful/measurable quantity. It is more meaningful to talk in terms of more useful things like quanta, energy, and frequency.

Hmm… it makses sense that E=hf says that there is a smallest amount of matter (a quantum), so that A=n*Ao where Ao is the amplitude of one quantum.

It sounds like we’re saying E=nhf… How does the amplitude fit in? It’s not just n=A/Ao, so E=A/Ao*hf, is it? That doesn’t seem to make sense. I swear there was something about the square of the amplitude (from math class) but perhaps I’m getting it mixed with quantum mechanics.

Speaking of which, perhaps in here I’m mixing quantum mechanics and a continuous wave, which problematic (for me, at least). I suppose the energy of a continuous wave would be infinite, since there is supposedly and infinite number of particles vibrating. So that’s probably why we never talk about the energy of a wave, but we might talk about the power it delivers, which leads us to intensity and the like. Ah… I think it was the (root mean) square of the amplitude is the power delivered? Is that it? And then the power leads to the intensity of the wave…

It says there is a smallest “amplitude” of oscillation for a given frequency.

No. First of all, it depends on what theory we are talking about. Are we talking about waves in water, waves on a drum, electromagnetic waves…? In general it would be correct to say that A=Ao*f(n), where f(n) is some function of n. For a classical electromagnetic wave, the energy density is proportional to the square of the amplitude of the wave’s oscillating electric field, so f(n)=sqrt(n). But it quantum mechanics it’s not so clear anymore what the precise relationship is between amplitude and the number of quanta (although of course in the limit of large number of quanta it reduces to the classical relationship), because individual photons don’t behave anymore as though they are classical waves with B and E field amplitudes. B and E fields are made out of photons, not the other way around, so it is ultimately a confused question you are asking!