# Phase Difference after Resonance

If you take the keys out of your pocket and swing them very slowly back and forth from your key chain, emulating a driven pendulum, you’ll notice that the keys swing back and forth in phase with your hand. Now, if you slowly start to speed up the swinging, you’ll notice that eventually you’ll hit a resonance frequency, where the keys will swing back and forth with a much greater amplitude.

If you keep slowly increasing the frequency of your swing beyond the resonance frequency, you’ll see that the keys don’t swing up as high. Also, you will notice that the keys now seem to be swaying out of phase with your hand (i.e. your hand is going in one direction while the keys are moving in the opposite direction!). This change of phase by 180 degrees between the driving force and the position of the oscillator is a ubiquitous feature of damped harmonic motion at frequencies higher than the resonance frequency. Why does this happen?

To understand this phenomenon, it helps to write down the equation for damped, driven harmonic motion. This could be describing a mass on a spring, a pendulum, a resistor-inductor-capacitor circuit, or something more exotic. Anyway, the relevant equation looks like this:

$\underbrace{\ddot{x}}_{inertial} + \overbrace{b\dot{x}}^{damping} + \underbrace{\omega_0^2 x}_{restoring} = \overbrace{F(t)}^{driving}$

Let’s describe in words what each of the terms means. The first term describes the resistance to change or inertia of the system. The second term represents the damping of the system, which is usually quite small. The third term gives us the pullback or restoring force, while the last term on the right-hand side represents the external driving force.

With this nomenclature in place, let’s move on to what actually causes the phase change. First, we have to turn this differential equation into an algebraic equation by doing a Fourier transform (or similarly assuming a sinusoidal dependence of everything). This leaves us with the following equation:

$(-\omega^2 + i\omega b + \omega_0^2 )x_0e^{i\omega t} = F_0e^{i\omega t}$

Now we can more easily visualize what is going on if we concentrate on the left-hand side of the equation. Note that this equation can also suggestively be written as:

$(e^{i\pi}\omega^2 + e^{i\pi/2}\omega b + \omega_0^2 )x_0e^{i\omega t} = F_0e^{i\omega t}$

For small driving frequencies, $b << \omega << \omega_0$, we see that the restoring term is the largest. The phase difference can then be represented graphically on an Argand diagram, where we can draw the following picture:

Restoring term dominates for low frequency oscillations

Therefore, the restoring force dominates the other two terms and the phase difference between the external force and the position of the oscillator is small (approximately zero).

At resonance, however, the driving frequency is the same as the natural frequency. This causes the restoring and inertial terms to cancel each other out perfectly, resulting in an Argand diagram like this:

Equal contribution from the restoring and inertial terms

After adding the vectors, this results in the arrow pointing upward, which is equivalent to saying that there is a 90 degree phase difference between the driving force and position of the oscillator.

You can probably see where this is going now, but let’s just keep going for the sake of completeness. For the case where the driving frequency exceeds the natural frequency (or resonant frequency), $b << \omega_0 << \omega$, we see that the inertial term starts to dominate, resulting in a phase shift of 180 degrees. This is again can be represented with an Argand diagram, as seen below:

Inertial term dominates for high frequency oscillations

This expresses the fact that the inertia can no longer “keep up” with the driving force and it therefore begins to lag behind. If the mass in a mass-spring system were to be reduced, the oscillator would be able to keep up with the driver up to a higher frequency. In summary, the phase difference can be plotted against the driving frequency to yield:

This phase change can be observed in so many contexts that it would be near impossible to list them all. In condensed matter physics, for instance, when sweeping the incident frequency of light in a reflectivity experiment of a semiconductor, a phase difference arises between the photon and the phonon above the phonon frequency. The problem that actually brought me to this analysis was the ported speaker, where above the resonant frequency of the speaker cone, the air from the port and the pressure wave generated from the speaker go 180 degrees out of phase.

### 3 responses to “Phase Difference after Resonance”

1. Conner

Great explanation. I even tried it with my keys and lanyard. I think it finally clicked. Thank you.

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