# 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?

• What waves are you thinking of in biology? The sound waves of lion roars? Or the pecking sounds of woodpeckers? Or the chants of whales under the ocean? – freecharly Oct 14 '16 at 18:45
• Where did you see that in sound waves energy is high when frequency is low? – freecharly Oct 14 '16 at 18:47
• I am studying the physiology of the ear and how the vibrations of the basilar membrane reach farther for lower frequency sound waves than higher frequency sound waves. My professor explained that lower frequency waves travel farther because they are higher in energy. He used an example of a subwoofer, saying that you can 'feel' the beat of a subwoofer, but we don't really 'feel' higher frequency sound waves because they are lower in energy. This seems to contradict what I learned in physics. – notorious Oct 14 '16 at 18:50
• Amplitude and intensity are just as important as frequency when it comes to sound. – Bill Alsept Oct 14 '16 at 18:54
• penetration of materials is not equivalent to energy. Waves with similar amplitudes always carry more energy at higher frequencies. Subwoofers often put out higher amplitudes AND low sound frequencies penetrate large solid objects better (like walls, floors, chest cavities, etc) – Jim Oct 14 '16 at 19:06

## 2 Answers

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.

• Quantization of energy according to $E = h\nu$ is not restricted to electromagnetic waves it equally applies to sound waves, e.g. in crystalline solids, whose quanta are called phonons. – freecharly Oct 14 '16 at 21:42

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.