Why an Ambulance Siren Drops in Pitch as It Speeds Past

Ripples spread across a pond, sound crosses a classroom and a sitar string hums, yet in each case the medium only vibrates in place while energy travels onwards. The same ideas explain musical notes, the throbbing of two nearly matched tones, and the changing pitch of a passing siren.

This lesson covers transverse and longitudinal waves and wave speed, standing waves in strings and pipes, beats, and the Doppler effect.

**In a transverse wave the particles vibrate at right angles to the direction of travel, in a longitudinal wave they vibrate along it, and every wave obeys $v=f\lambda$.

Two kinds of wave:

- Transverse — crests and troughs, as on a stretched string
- Longitudinal — compressions and rarefactions, as in sound, which travels through solids, liquids and gases

Deriving $v=f\lambda$.** In one time period T the wave advances by one wavelength, so

$$v=\frac{\lambda }{T}=f\lambda$$

Speed in a medium. On a string with tension F and mass per unit length $\mu$, $v=\sqrt{F/\mu }$; for sound in a gas, $v=\sqrt{\gamma p/\rho }$.

Worked example 1. A 512 Hz sound travels through air at 340 m s${}^{-1}$:

$$\lambda =\frac{v}{f}=\frac{340}{512}=0.664\text{ m}$$

Worked example 2. A string with $\mu =0.010$ kg m${}^{-1}$ under a tension of 100 N carries waves at $v=\sqrt{\frac{100}{0.010}}=100$ m s${}^{-1}$.

An everyday example. A Mexican wave in a cricket stadium travels round the stands while each spectator only stands up and sits down.

The substance. Frequency is set by the source, while speed is set by the medium — when sound passes from air into water, its frequency stays the same but its speed and wavelength increase.

When two identical waves travelling in opposite directions superpose, they form a standing wave with fixed nodes and antinodes, and only certain wavelengths fit a given string or pipe, giving its harmonics.

Superposition. The resultant displacement at a point is the algebraic sum of the individual displacements. In a standing wave, nodes have zero displacement and lie $\frac{\lambda }{2}$ apart, with antinodes midway between them.

Allowed frequencies:

- String fixed at both ends and open pipe: $f_n=\frac{nv}{2L}$, with $n=1,2,3,\dots$ — all harmonics
- Closed pipe, with a node at the closed end and an antinode at the open end: $f_n=\frac{nv}{4L}$, with $n=1,3,5,\dots$ — odd harmonics only

Worked example. With $v=340$ m s${}^{-1}$ and $L=0.50$ m:

$$f_{\text{open}}=\frac{340}{2\times 0.50}=340\text{ Hz}{f}_{\text{closed}}=\frac{340}{4\times 0.50}=170\text{ Hz}$$

The open pipe's next harmonics are 680 Hz and 1020 Hz; the closed pipe's are 510 Hz and 850 Hz.

An everyday example. A flute player covering and uncovering holes changes the effective length of the vibrating air column, and so the note it produces.

The substance. A closed pipe sounds an octave lower than an open pipe of the same length — here 170 Hz against 340 Hz.

**Beats are the regular rise and fall in loudness heard when two sounds of slightly different frequencies play together, and the beat frequency is the difference of the two frequencies, $f_b=|f_1-f_2|$.

How beats form. The two waves drift in and out of step: in step they add to a loud sound, and out of step they cancel to a soft one. The ear hears beats clearly only when the difference is a few per second.

Worked example 1.** Tuning forks of 256 Hz and 260 Hz sounded together give

$$f_b=260-256=4\text{ beats per second}$$

Worked example 2 — an unknown fork. A fork gives 5 beats per second with a 512 Hz fork, so it is 507 or 517 Hz. Adding wax lowers its frequency, and the beats fall to 3 per second — it moved closer to 512 Hz, so it was 517 Hz.

An everyday example. A tabla player tuning the drum against a harmonium listens for the beats to slow down and vanish, which means the two frequencies now match.

The substance. Beats are not a new pitch — the sound is heard near the average of the two frequencies, while its loudness swells and fades at their difference.

**The Doppler effect is the change in observed frequency when a source or observer moves; for motion towards each other, $f^{\prime }=f\frac{v+v_o}{v-v_s}$, where v is the speed of sound.

Special cases:**

- Source approaching a stationary observer: $f^{\prime }=f\frac{v}{v-v_s}$, a higher pitch
- Source moving away: $f^{\prime }=f\frac{v}{v+v_s}$, a lower pitch
- Observer approaching a stationary source: $f^{\prime }=f\frac{v+v_o}{v}$

Worked example 1 — a siren. A 700 Hz siren moves at 20 m s${}^{-1}$, with sound at 340 m s${}^{-1}$:

$$f_{\text{approach}}=700\times \frac{340}{320}=744\text{ Hz}{f}_{\text{recede}}=700\times \frac{340}{360}=661\text{ Hz}$$

Worked example 2 — a moving observer. A cyclist riding at 10 m s${}^{-1}$ towards a stationary 500 Hz horn hears $500\times \frac{350}{340}=515$ Hz.

An everyday example. Standing beside a highway as a honking truck passes, you hear the horn's pitch drop suddenly at the moment it goes by.

The substance. A moving source and a moving observer at the same speed give different shifts — the source's motion squeezes the wavelength, while the observer's motion changes how many wavefronts arrive each second.