Physics · Oscillations and waves · Grade 11

Waves and the Doppler effect

Why does an ambulance siren sound higher as it approaches?

The situation

An ambulance went past in the street with its siren sounding: as it approached, its sound was high-pitched, and as it moved away, low — even though the siren sounds at the same frequency. Noise-cancelling headphones, in turn, emit a wave in antiphase to the noise and “switch it off”. And behind a plane flying faster than sound you hear a blow like an explosion. We can see all of these in waves on the surface of water.

Topic

Interference and the Doppler effect. Waves on the surface of water, a siren and a supersonic aeroplane

Key ideas

Interference
Doppler effect
Mach cone

Formulas

λ = \frac{c}{f}

Wavelength: divide the propagation speed $c$ by the frequency $f$ .

Δ r = r_{1} - r_{2} = m λ

Interference maximum (sources in phase), $m = 0, \pm 1, \pm 2, \dots$

Δ r = (m + \frac{1}{2}) λ

Interference minimum — nodal lines (hyperbolas).

f^{'} = f \frac{c}{c \mp u}

Doppler effect: if the source approaches the observer (−) the frequency increases, if it recedes (+) it decreases.

\sin α = \frac{c}{u} = \frac{1}{M}, \cos θ = \frac{c / n}{v} = \frac{1}{n β}

Half-angle of the Mach cone; $M = u / c$ is the Mach number. For light the same cone is Vavilov–Cherenkov radiation: $θ = 90^{\circ} - α$ is the angle between the direction of the radiation and the particle's velocity, $n$ is the refractive index, $β = v / c$ .

Theory

Superposition

When two waves reach one point, the displacements simply add. If crest meets crest — reinforcement (maximum); if crest meets trough — cancellation (minimum). Which of these occurs is decided by the difference in the distances to the sources, $Δ r$ .

Why are the nodal lines hyperbolas?

The set of points for which the difference of distances is constant is a hyperbola with its foci at the sources. That is why the still lines on the water spread out like “rays”. If the distance between the sources $d$ increases or $λ$ decreases, the lines become more numerous: their number is about $2 d / λ$ .

Doppler effect

A moving source emits each successive crest from a position shifted forwards. Ahead the crests are bunched together ( $λ$ shortens, $f$ rises), behind they are spread out. The source's own frequency does not change — what changes is the frequency that reaches the observer. Doctors' ultrasound Doppler devices measure the speed of blood flow from this shift; ultrasound, infrasound and piezoelectric sources — in the “Sound: speed, pitch, ultrasound” experiment.

The sound barrier

If a source moves faster than the wave ( $u > c$ ), the waves cannot get ahead and pile up into a cone-shaped shock wave. The V-shaped wake behind a ship and the “boom” of a supersonic aircraft ( $\sin α = c / u$ ) are exactly this.

Cherenkov (Vavilov–Cherenkov) radiation

Nothing travels faster than light in a vacuum, but in a medium light is slower: in water $c / n \approx 0.75 c$ ( $n = 1.33$ ). If a charged particle travels in the medium at a speed $v > c / n$ , the electromagnetic waves emitted by atoms along its path pile up into a single cone front, just like the Mach cone in water — this is Vavilov–Cherenkov radiation. The radiation goes off at an angle $\cos θ = 1 / (n β)$ to the direction of the particle: this is exactly the $θ = 90^{\circ} - α$ of this scene, only with $c / n$ in place of $c$ and $v$ in place of $u$ . The threshold is $β > 1 / n$ : for an electron in water the kinetic energy $T > 0.26$ MeV; as $β \to 1$ , $θ \to 41^{\circ}$ . The spectrum is stronger at short wavelengths (the Frank–Tamm formula), which is why a reactor pool glows sky-blue. The phenomenon was discovered in 1934 by P. A. Cherenkov under the supervision of S. I. Vavilov, and its theory was created in 1937 by I. E. Tamm and I. M. Frank. Detectors measure a particle's speed from the cone angle; the passage of particles through matter and bremsstrahlung — in the “Detecting particles” experiment.

Did you know?

In air at 20 °C the speed of sound is about 343 m/s. So every 3 seconds of delay after a flash of lightning means a distance of about 1 km.
Christian Doppler predicted this effect theoretically in 1842. In 1845 Buys Ballot tested it by ear in the Netherlands, carrying trumpeters on a train.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

Swing — Grade 10
Spring pendulum — Grade 10
Waves on a rope and a spring — Grade 10
Sound: speed, pitch, ultrasound — Grade 10
The dutar string and sound — Grade 11
Bridge resonance — Grade 11
Coupled oscillations — Grade 11
Nonlinear oscillations — Year 2
Fourier: the spectrum of sound — Year 3