Waves - Frequency and energy/Amplitude and energy

Question

Been wondering about this for a while and I hope someone can explain it. Thanks for taking the time for reading this and perhaps answering the question :-)

In my book it says: If you double the frequency of a wave, the energy gets quadrupled. Why does is it not doubled? My reason is that the amount of waves in a given time is doubled - therefore the energy is also doubled - what am I missing?

It also states: If you double the amplitude of a wave, the energy gets quadrupled. Again I don't understand why. My logic says it would be doubled and not quadrupled. If I have a wave of amplitude 1 and double the height, the area under it doubles - it does not quadruple. What am I missing?

Hope someone can help me, I have asked a lot of clever people and googled all I can.

Pierre

Answer 1

In a wave, the particles of the medium are all oscillating. The particles pass on the oscillations from one to another, hence the oscillations lag more and more behind, the further the particles are from the source – but that doesn't matter; all that matters is that the particles are oscillating.

The simplest type of oscillation is sinusoidal, or 'simple harmonic'. That means that the particle's displacement at time t is$$x=A \sin(2\pi ft+\phi).$$ Here, A is a constant, the amplitude, f is the frequency, and $\phi$ is the phase constant (which is increasingly negative the further the particle is from the source). For our purposes we lose nothing by putting $\phi=0.$

The particle's velocity is given by$$v=\frac{dx}{dt}=A2\pi f \cos 2\pi ft$$ so its kinetic energy is$$E_k=\tfrac{1}{2}mv^2=2 \pi^2 mA^2f^2 \cos^2 2\pi ft.$$

The mean kinetic energy of the particle over a complete cycle is $\pi^2 mA^2f^2$. We can show that the particle has an equal mean amount of potential energy, so the particle energy is proportional to $A^2$ and $f^2$ ! And that applies to each particle in the path of the wave.