Why is the frequency of a wave not always directly proportional to energy? In physics and chemistry, I learned that energy is directly proportional to the frequency of a wave $E=hf$ for light. However, in biology, the opposite is true -  energy is high when frequency is low. (For example, in sound waves). Why does this discrepancy exist and why isn't there only one relationship between frequency and energy?
 A: In quantum mechanics waves of frequency $\omega$ can be considered to be composed of harmonic oscillators with that frequency. And the energy can only change by multiples of $\hbar \omega$. The amplitudes of the waves like in photons do not determine the energy change. In classical waves, like sound waves or electromagnetic waves, the wave intensity at a given frequency depends on the square of the amplitude of the wave, which, in principle, can take any value.   
A: Without digging down a bit further, the equation $E = h\nu$ can at first be confusing. It does not relate the energy in a wave to its frequency, $\nu$ but rather the energy of one quantum (one photon) to the photon's frequency. That's easily seen by just looking at the units of Planck's constant, $h$ which is Joule-sec/photon. This applies only to electromagnetic waves that, in quantum mechanics, can also be interpreted in terms of particles (quanta).
For other types of waves the energy is figured differently depending on the way energy propagates through a medium. The energy density in ocean waves, for example, does not follow Plank's equation but rather
$$E = \frac{1}{16}\rho g H^2$$
where $\rho$ is the density of the sea water, $g$ the gravitational acceleration, and $H$ the mean wave height. The energy is not carried by photons but rather by gravitational potentials that move upwards and downwards (wave crests and troughs) as the wave propagates along the sea surface. This is why ocean waves are classified as gravity (not gravitational) waves. Sound waves have yet another physical model that describes their energy content.
