Doppler, Interference and Resonance - Physics for Pilot Assessments

Three wave effects sit together in this lesson because all three come from the same place: what happens when the tidy picture of one wave from one stationary source is disturbed. By the end of it you should be able to explain why a siren changes pitch as it passes without the siren itself changing, work out what happens when two waves arrive at the same point together, and say what resonance is and why matching a frequency is the whole of it.

The Doppler effect

An ambulance approaches with its siren on, passes you, and drives away, and the note you hear drops noticeably at the moment it goes by. The siren has been sounding one steady note throughout. What changed is the rate at which crests reach you, and that is worth working through slowly because the mechanism is the whole answer.

Imagine the source sending out one crest, then moving forwards a little, then sending out the next. The second crest starts from a point closer to you than the first did, so it has less distance to cover and less catching up to do — the gap between crests in front of a moving source is squeezed shorter than it would otherwise be. Squeezed spacing means a shorter wavelength, and since the speed of sound in the air is fixed by the air and not by the source, a shorter wavelength arriving at a fixed speed means more crests reach your ear each second. More crests per second is a higher frequency, and a higher frequency is heard as a higher pitch. Behind the source the opposite happens: each crest is emitted further away than the last, the spacing is stretched, and the pitch you hear is lower than the note being sounded.

So an approaching source sounds higher than it really is, and a retreating one lower, and the change happens as it passes rather than gradually along the way. That is what a Doppler shift is, and how big it gets depends on how fast the source is moving compared with the speed of the wave — a walking pace produces a shift far too small to notice, while a fast vehicle produces an obvious one. The effect works just as well if the source is still and you are the one moving, since what matters is the rate at which crests are met, and it applies to every kind of wave rather than only to sound.

Now the point every examiner tests. The source's own frequency does not change at any moment during this. The siren is vibrating at exactly the same rate the whole time, and a passenger sitting in the ambulance hears no change at all, because the ambulance is not moving relative to them. What changes is the frequency received by a listener the source is moving towards or away from. Doppler is a fact about arrival, not about emission.

Example 1 — three listeners, one siren

A fire engine drives down a straight road at a steady speed, sounding a siren at one fixed frequency. Listener A stands ahead of it, listener B is the driver, and listener C stands behind it, watching it drive away. Compare the pitch each one hears with the pitch the siren is actually sounding.

Listener A is being approached, so each crest is emitted from a point closer to A than the one before: the spacing between crests is squeezed, more crests arrive each second, and A hears a pitch higher than the siren's own.
Listener B moves with the source, so the distance between B and the siren never changes and the crests arrive exactly as often as they are made: B hears the true pitch, unchanged for the whole journey.
Listener C is being left behind, so each crest is emitted from further away than the last: the spacing is stretched, fewer crests arrive each second, and C hears a pitch lower than the siren's own.
At no point did the siren itself change. One emitted frequency, three received frequencies — which is the whole content of the Doppler effect.

Two waves in the same place: superposition

Waves do not collide and they do not block one another. When two waves pass through the same point at the same moment, the displacement of the material at that point is simply the sum of what each wave would have produced on its own — that is the principle of superposition — and afterwards each wave carries on as though nothing had happened. Adding displacements means paying attention to sign: two crests arriving together add to make a bigger crest, two troughs add to make a deeper trough, and a crest arriving with a trough of the same size adds to zero, since one is pushing the material up by exactly as much as the other is pulling it down.

Two identical waves in step add to one of twice the amplitude: crest on crest adds. The same two waves half a wavelength out of step add to nothing at all, because crest on trough cancels — the material is being pushed and pulled equally at every point.

Two waves overlapping in this way are said to interfere, which in physics means nothing more disruptive than adding together. Where they arrive in step, meaning crest with crest, the result is larger than either alone and the interference is called constructive. Where they arrive out of step by half a wavelength — crest with trough, a condition described as being in antiphase — the result is smaller than either alone, and the interference is called destructive. If the two waves have equal amplitudes, destructive interference gives exactly zero: no displacement, no energy arriving, silence in the case of sound. That is not energy being destroyed. It is energy that never arrived at this point, because it went to the points where the interference was constructive instead.

What decides whether two waves from the same source arrive in step is how far each has travelled to get there. If one path is longer than the other by exactly one whole wavelength, the extra wavelength puts the wave right back where it started in its cycle, and the two arrive in step; the same is true for two whole wavelengths, or any whole number of them, including none at all.

Condition for constructive interference

\text{path difference} = n \lambda

$\text{path difference}$: how much further one wave has travelled than the other, in metres (m)
$n$: any whole number: 0, 1, 2, 3 and so on — no units
$\lambda$: the wavelength, in metres (m)

If instead one path is longer by half a wavelength, the extra half-cycle arrives with a trough where the other wave has a crest, and the two cancel; and adding any number of whole wavelengths on top of that half keeps them cancelling.

Condition for destructive interference

\text{path difference} = \left(n + \tfrac{1}{2}\right) \lambda

$\text{path difference}$: how much further one wave has travelled than the other, in metres (m)
$n$: any whole number: 0, 1, 2, 3 and so on — no units
$\lambda$: the wavelength, in metres (m)

The two conditions are worth reading as one idea: whole wavelengths reinforce, half wavelengths cancel, and everything between gives a partial result. Walk across a room with two loudspeakers playing the same steady note and you pass through alternating loud and quiet positions for exactly this reason — your two distances to the speakers change as you move, and with them the path difference.

Example 2 — adding two waves

Wave A has an amplitude of 5 units and wave B an amplitude of 3 units, and they have the same wavelength. What is the amplitude of the result when they arrive in step, and when they arrive in antiphase? And what if both had an amplitude of 3?

In step, the displacements add with the same sign: $5 + 3 = 8\ \text{units}$
In antiphase, one is displacing the material one way while the other displaces it the other way, so they subtract: $5 - 3 = 2\ \text{units}$
So the result lies between 2 and 8 units depending only on how the two line up. Nothing about either wave has changed.
If both amplitudes were 3, then in step the result is 3 + 3 = 6 units, and in antiphase 3 − 3 = 0. Exactly equal waves in antiphase cancel completely, which unequal ones never quite do.

Example 3 — a quiet spot between two speakers

Two loudspeakers play the same steady 170-hertz note in air, where sound travels at 340 metres per second. At a certain point one speaker is 1 metre further away than the other. Is that point loud or quiet? What about a point where the difference is 2 metres?

First find the wavelength, using the wave equation from the earlier lesson: $\lambda = \frac{v}{f} = \frac{340}{170} = 2\ \text{m}$
At the first point the path difference is 1 metre, which is half of the 2-metre wavelength — a half-wavelength difference.
A half-wavelength path difference means the waves arrive in antiphase, so they interfere destructively and the point is quiet. With equal loudspeakers it would be silent.
At the second point the path difference is 2 metres, which is exactly one whole wavelength, so the waves arrive in step and interfere constructively: that point is louder than either speaker alone.
Note that both answers came from comparing the path difference with the wavelength, so the wavelength had to be found first. Comparing the path difference with the frequency would be comparing a distance with a rate, which is meaningless.

Resonance and natural frequency

Anything that can oscillate has a natural frequency: the rate it oscillates at when you disturb it once and leave it alone. A pendulum released from one side swings at its own rate, a ruler held over the edge of a desk and twanged vibrates at its own rate, and a wine glass rung with a fingernail sounds its own note. That frequency is a property of the object — its size, its mass, its stiffness — and not of how hard you hit it.

Now drive that object repeatedly instead of hitting it once. If your pushes come at the object's natural frequency, each one arrives in step with the motion already there — you push while it is already moving away from you — so each push adds energy to what is stored, and the amplitude builds up cycle after cycle. That build-up is resonance. Push at the wrong frequency and some of your pushes oppose the motion instead, energy goes in and comes back out, and the amplitude stays small. It is exactly what you do to a child on a swing: the pushes are small, but timed to the swing's own rate, and the amplitude grows to something no single push could have produced.

Resonance explains a long list of otherwise unrelated things: a washing machine that shakes violently at one particular spin speed and is quiet above and below it, a loose trim panel that buzzes only at one engine setting, a wine glass shattered by a sustained note at its own frequency, and the swaying of bridges and tall buildings under a rhythmic load. The engineering response is always one of two things — shift the natural frequency away from the frequencies the object will meet, or add damping so that energy is removed as fast as it is fed in. Damping is what stops resonance running away, and it is the reason most real objects merely hum rather than destroying themselves.

Example 4 — the shaking washing machine

A washing machine shakes violently at one particular spin speed, yet runs smoothly both below and above it. Explain what is happening, and what a designer could do about it.

The spinning drum gives the case one small push per revolution, so the pushes arrive at a frequency set by the spin speed — a driving frequency the machine sweeps through as it speeds up.
The case on its feet is a thing that can oscillate, so it has a natural frequency of its own. At one particular spin speed the driving frequency matches it: every push then arrives in step with the motion already there, each one adds energy, and the amplitude builds. That is resonance.
Below and above that speed the pushes fall out of step with the motion, so some of them oppose it: energy goes in and comes back out, and the amplitude stays small. The pushes are no weaker at the resonant speed — they are merely better timed.
The designer's two options are the standard ones: move the natural frequency away from the spin speeds the machine uses, by stiffening the case or adding mass, or add damping so energy is removed as fast as the drum feeds it in.

Standing waves, briefly

One special case of superposition deserves a mention because it is where resonance and interference meet. Send a wave along a string fixed at both ends and it reflects off the far end and travels back through the wave still coming the other way. The two superpose, and at certain frequencies the result is a pattern that does not appear to travel at all: some points never move, and others oscillate with a large amplitude, always in the same places. That is a standing wave; the stationary points are called nodes and the maximally moving points antinodes.

It only happens at particular frequencies, because the ends of the string are pinned and so must be nodes, and that only works if the string's length holds a whole number of half-wavelengths.

Standing waves on a string fixed at both ends

L = n \frac{\lambda}{2}

$L$: the length of the string between its fixed ends, in metres (m)
$n$: a whole number of half-wavelengths: 1, 2, 3 and so on — no units
$\lambda$: the wavelength of the wave on the string, in metres (m)

The lowest frequency that fits — one half-wavelength in the whole length — is called the fundamental, and the higher ones that also fit are the harmonics. This is why a guitar string sounds a definite note rather than a noise, why shortening it with a finger raises the note, and why tightening it does too: a shorter length or a faster wave on a tighter string both push the fitting frequencies up.

Example 5 — the note a string sounds

A string 0.6 metres long is fixed at both ends, and waves travel along it at 240 metres per second. What is the wavelength of its fundamental, and what frequency does the string sound?

The fundamental fits one half-wavelength into the string's length, so half a wavelength is 0.6 metres and the whole wavelength is 2 × 0.6 = 1.2 metres.
Now use the wave equation from earlier in the module, rearranged for frequency: $f = \frac{v}{\lambda} = \frac{240}{1.2}$
Clear the decimal by multiplying top and bottom by 10: $\frac{2400}{12} = 200$ , so the string sounds 200 hertz.
Check it forwards: 200 × 1.2 = 240 metres per second, the wave speed on the string. And notice the wavelength here is a property of the wave on the string — the sound that then travels through the air at 340 metres per second has the same 200-hertz frequency but its own, different wavelength.

Why it matters in aviation

Airborne weather radar uses the Doppler shift to tell movement from mere presence. Returns from raindrops being carried towards the aircraft come back at a slightly higher frequency and those being carried away at a slightly lower one, so comparing the shift across a cloud reveals how the air inside it is moving rather than only how much water is in it. That is what lets a display separate heavy but smooth rain from a region of turbulence and strong vertical movement worth going round.

Resonance carries a warning instead of a benefit. Any structure has frequencies at which small repeated inputs build a large oscillation, and an aircraft in flight sits in a stream of air well able to supply repeated inputs — buffet, shed vortices, the movement of a control surface. If a wing or a control responded strongly at a frequency the airflow can excite, the amplitude could grow instead of dying away, with the structure absorbing energy faster than it can shed it. Designers therefore place natural frequencies deliberately clear of that range, stiffen structures to move them, and mass-balance control surfaces so that the oscillation is damped rather than fed.

What the test asks

For Doppler, the distractor claims the source changes its frequency. It does not: an option saying that the siren sounds a higher note as it approaches, or that the driver hears the pitch change, is wrong, and the correct statement is always about the frequency received by a listener the source is approaching or leaving. Watch also for the option that has the shifts the wrong way round, with the pitch rising as the source departs. For interference, the trap is treating destructive interference as waves cancelling each other permanently or destroying energy; they pass through and carry on, and the energy has gone to the constructive points instead. Expect a path-difference item where the trap is comparing the path difference with the frequency rather than with the wavelength, so the wavelength has to be calculated first. For resonance, the distractor defines it as any loud sound, or as a sound getting louder because the source got louder, when the definition is a driving frequency matching the object's own natural frequency.