The Doppler effect.

Step 1 of 5

A moving source

The ripple tank again, and this time the bead is fixed to a cart that can cross the tank. Source speed is measured against the wave speed: $0.5$⁠ means half the wave speed, which here is 25 cm/s. The Frequency is fixed at 6 Hz, and time is slowed down 5 times.

While the bead stands still, the wavelength in every direction is $\lambda = \dfrac{25}{6} \approx 4.2$$\lambda =$$\dfrac{25}{6} \approx$$4.2$⁠ centimetres. Predict what will change when the bead moves, then look.

Step 2 of 5

What you hear ahead of the source

In each cycle, $T = \dfrac{1}{f}$⁠, the bead sends out one crest. In that time the crest before it has moved on $vT = \lambda$⁠, but the bead, moving at speed $u$⁠, has moved on $uT$⁠ the same way. So in front of it the gap between crests is smaller:

$\lambda' = \lambda - uT = \lambda\left(1 - \frac{u}{v}\right)$$\lambda' =$$\lambda - uT =$$\lambda\left(1 - \frac{u}{v}\right)$⁠

And the wave still travels at $v$⁠, so a listener standing in front of the source receives the crests at a frequency

$f' = \frac{v}{\lambda'} = \frac{f}{1 - u/v}$$f' =$$\frac{v}{\lambda'} =$$\frac{f}{1 - u/v}$⁠

Behind the source everything is reversed: $\lambda(1 + \tfrac{u}{v})$⁠, and the frequency $\dfrac{f}{1 + u/v}$⁠. This is the Doppler effect. Source speed and Frequency are fixed now.

Step 3 of 5

Ten hertz ahead

The bead’s Frequency is fixed at 6 Hz. Find the source speed at which a listener in front of it receives 10 Hz.

Try it: after checking, slide Source speed up to 1 × wave speed and then to 1.2. At the wave speed all the crests pile up into one front just ahead of the bead, and beyond it they trail behind it in a V: the wave of a sonic boom. No wave reaches the water in front of such a source, and the formula $\dfrac{f}{1 - u/v}$⁠ no longer applies.

Back to the experiment ↑

Step 4 of 5

A passing siren

An ambulance with a siren does exactly what the bead does. The driver moves with the siren, so they hear the frequency it plays. Someone standing on the pavement hears a different frequency, which depends on whether the ambulance is coming towards them or going away.

Step 5 of 5

A real siren

The formula works for sound too. The speed of sound in air is 343 m/s, and a fast ambulance travels at about 30 m/s, so $\dfrac{u}{v}$⁠ is much smaller than in the tank. The change in pitch is smaller, but you still hear it clearly.