I am just really confused about what determines energy of a wave?! I have heard that the energy of a wave is proportional to the amplitude squared but I have also heard that frequency determines a wave's energy. Which is the correct answer? Can they both be true?
Classically, it is just the amplitude. You're probably confusing this with photons, which have frequency proportional to their energy.
Amplitude squared for almost all physical waves- water, pressure, electromagnetic etc. For photons however, which are massless particles, the energy is proportional to the frequency, with the proportionality constant h (planck's constant) E=h freq You have to be Einstein to discover things like that- and in fact he did. He deduced the above relation from the 'photo-electric effect'. It's completely counterintuitive that frequency should have anything to do with the energy of the wave, but that's what quantum-mechanics gives. Edit: And even more weird: The amplitude squared for photon quantum waves is the probability that the photon is found in that place when a measurement is made. Nobody understands that one- but it seems to be how the universe works.
Isn't that the same thing? I mean, aren't EM waves composed of photons (or rather they are one and the same thing) and therefore whatever determines the energy of a photon would also determine the energy of the EM waves (i.e the frequency)?
The energy of *any* classical wave is determined by the amplitude. If you want to describe an EM wave in terms of photons, given a particular amplitude (and hence total energy), the number of photons composing the wave is greater if the frequency is low. You should think of frequency (or perhaps more suggestively, wavelength) as determining the density of photons in an EM wave.
I see. That makes a lot of sense and it is a pretty nice way to look at the relationship between photons and EM waves. Thanks!
For sound waves (including seismic waves) power transported is proportional to the squared amplitude times the squared frequency. For water waves the dependence is more complex, but is some cases it is also proportional to the squared amplitude and squared frequency.