# Waves - Frequency and energy/Amplitude and energy

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

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.