Calculating Energy of a Wave

So my physics examinations are coming up and I was going through my notes on waves, but I realized that there were some discrepancies.

In my notes, the energy of a wave is directly proportional to the square of the amplitude, ie. $E \propto A^2$

However, I recalled that, in one of my physics lessons, our physics teacher told us that the energy of a wave can be calculated using $E=hf$, where $h$ is the Planck constant and $f$ the frequency.

Hence I was rather confused and tried searching google for answers but couldn't find any suitable ones. To the best extent of that research, what I found out was (apparently) (for visible light), the frequency of the wave could be used to calculate the energy of the wave, while the amplitude was used to determine the intensity of the wave.

So I was wondering, firstly, whether the above statement was correct, and secondly, in the event it is correct, whether it would be applicable to all kinds of waves, (ie. Sound waves, water waves, other EM waves, etc.), and thirdly, back to the original question, how do we calculate the energy of a wave?

Thanks. :)

Answer

Both the equations you cite are correct.

The energy carried by a wave is indeed proportional to the amplitude squared. for what it's worth, you don't even need a propagating wave, any harmonic oscillator (e.g. a pendulum) will follow that rule. The validity of this rule remains unaffected even in quantum mechanics (actually, since in QM everything can be described by a wave function, it is even more fundamental there).

The second formula expresses the energy of a single photon. A photon is the smallest quantity of radiation that can exist at that frequency. This is completely unrelated to the total energy of the wave! For instance even a small light bulb will emit something like $10^{20}$ photons each second. Each carries an energy of $hf$. Together they sum up to the total power of the beam.