Siren Song

Some strange things are so common that it's easy to forget how strange they are.  Like, for instance, standing by the side of the road.  Each time you hear a car moving toward you, you know it will sound different after whizzing by.

It happens all the time, but if you stop to think about it, it's pretty weird.  You would definitely notice if the car changed color or something on the way by,1 but the fact that it changes pitch seems normal in our everyday experience, so maybe you've never tried to work out why it happens.  Let's do that.

Sound travels in waves, so we can think in terms of peaks and troughs.  One important aspect of a sound wave is its frequency, how many peaks or troughs an observer experiences per unit of time.  Frequency is typically measured in hertz (Hz), where the unit of time is one second.  So someone listening to a 400 Hz tone would be experiencing 400 wave peaks every second.  The frequency also determines the wave's period, or how long it takes to complete a single peak-to-peak cycle.  In this case, that would be 1/400th of a second.  In other words, period is the reciprocal of frequency.

You can also describe a wave in terms of its wavelength, which is the distance between adjacent peaks or troughs.  Clearly frequency and wavelength must be related: the shorter the distance between peaks, the more frequently an observer would experience one, and vice versa.

The speed of sound is about 340 m/s.  We can use that information to write down the precise relationship between frequency and wavelength.  Check that you agree:

f = \frac{340}{\lambda} \text{  or  } \lambda=\frac{340}{f}

Let's say we have a beacon emitting sound, and two observers who are each 340 m away.  Change the frequency below to see how the associated wavelength changes.  You can also see how a change in frequency affects the sound an observer will hear.

As long as everything is stationary, everyone agrees on the frequency the beacon is emitting, but something interesting happens once things start moving.  Here the beacon moves at half the speed of sound.2

Notice that, even though the beacon's frequency hasn't changed, the two observers will no longer agree on that fact.  The forward observer is getting a shorter wavelength/higher frequency, and the rear observer is getting a longer wavelength/lower frequency.  That means the sound seems higher to the person in front, and lower to the person in back!  In fact, the person in front will experience a change in frequency (from high to low) as the beacon moves past.  This phenomenon is known as the Doppler Effect, and it explains why the car's horn in the video seemingly dropped in pitch as it passed the camera.

That's certainly interesting, but we can go further.  We can calculate the apparent frequency ahead of or behind the beacon based on its actual frequency and how fast it's moving.

Notice that, for a stationary beacon, the distance between peaks is just the wavelength.  We already knew that.   But with a moving beacon, the distance between peaks changes by whatever distance the beacon covers in a single period.  To the front observer, the distance would decrease by that amount; to the rear observer, it would increase by that amount.  Here the beacon is still moving at half the speed of sound:

So we can work out the observed wavelength each person would report for a beacon moving at speed s:

\text{Front Observer:   } \lambda_F = \frac{340 - s}{f}

\text{Rear Observer:   } \lambda_R= \frac{340 + s}{f}

So we know what wavelength each observer would report, and we already have a relationship between wavelength and frequency, so with some substitution and a little rejiggering we can write down a general rule for the observed frequencies our two recorders would report, based on the observed wavelengths and the actual frequency and speed of the beacon:

\text{Front Observer:   } f_F = \frac{340}{340-s}f

\text{Rear Observer:   } f_R= \frac{340}{340+s}f

Play around and see what the observers would hear for a 500 Hz tone from a beacon moving at a few interesting speeds.  What would it sound like if the beacon were moving at the speed of sound?  And (extra cool) what about twice the speed of sound?3

But if a car is coming at you that fast, do us all a favor and step out of the way.  Live to listen another day.

Teachers, want to have this conversation in your classroom?  Check out our lesson, Siren Song.

1.  Okay, actually the car does change color on the way by.  As we'll see, the Doppler Effect depends on the speed of an object relative to the speed of the waves it emits.  Cars can move at an appreciable fraction of the speed of sound, so we can sense the Doppler Effect with our ears.  Even the fastest cars move at a negligible speed relative to light, so the color change is too small to detect.

2.  Obviously we've slowed things down quite a bit for the visualization so you can see what's going on, but notice the beacon is moving at half the speed of the waves it emits.

3.  SPOILER ALERT: The sound would appear to have the correct frequency, but the waves arrive in reverse order.  If a jet flying at Mach 2 were blaring some music, you would hear it in the correct pitch, but backwards!  Briefly.

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