Sound waves, beats and the Doppler effect

Year 9 introduced sound, echoes and the speed of sound. Here we model sound as a pressure wave, connect pitch and loudness to wave properties with numbers, and explain beats and the Doppler effect.

Sound as a longitudinal pressure wave

A vibrating object (a speaker cone, a guitar string, your vocal folds) pushes the air next to it back and forth. Each push squeezes air molecules together into a compression; each pull-back leaves a rarefaction where they are spread out. The molecules oscillate parallel to the direction the sound travels, so sound in air is a longitudinal wave.

Bunched-up molecules mean higher pressure, so a sound wave can also be described as a pressure wave: the pressure rises and falls a tiny amount above and below normal atmospheric pressure. A microphone, and your eardrum, respond to these pressure changes.

One subtle point that HSC questions test: the pressure variation and the displacement of the molecules are a quarter of a wavelength out of step. At the centre of a compression the molecules on each side have moved in towards it, but the molecules at the centre itself are at their rest position (zero displacement) while the pressure is greatest. Where displacement is largest, the pressure is normal.

The speed of sound in dry air is about 331 m/s at 0 °C and increases by about 0.6 m/s for every degree, giving 343 m/s at 20 °C. Many textbooks and exams round this to 340 m/s; use whatever value the question gives.

Pitch and loudness

• Pitch is how high or low a sound seems. It depends on the frequency: the higher the frequency, the higher the pitch. Young people can hear roughly 20 Hz to 20 000 Hz; the upper limit falls with age. Above 20 kHz is ultrasound; below 20 Hz is infrasound.
• Loudness is how strong a sound seems. It depends mainly on the amplitude of the pressure variation, which determines the energy the wave carries. Doubling the amplitude carries four times the energy.

On a microphone and oscilloscope trace (a displacement–time graph of the pressure), a higher pitch shows more waves across the screen and a louder sound shows taller waves.

Intensity and the inverse square law

The intensity of a wave is the power it carries per unit area, measured at right angles to the direction of travel:

$I = \dfrac{P}{A}$   (W m$^{-2}$)

A small source sending sound out equally in all directions spreads its power over the surface of an ever-growing sphere of area $4\pi r^2$, so

$I = \dfrac{P}{4\pi r^2} \quad \Rightarrow \quad I \propto \dfrac{1}{r^2} \quad \Rightarrow \quad I_1 r_1^2 = I_2 r_2^2$

Double the distance and the intensity drops to a quarter; triple it and it drops to a ninth. This assumes a point source, no absorption and no reflections. Indoors, echoes off walls mean the intensity falls more slowly than the law predicts.

Because the ear handles intensities from about $10^{-12}$ W m$^{-2}$ (the threshold of hearing) up to about 1 W m$^{-2}$ (painful), sound levels are given on the logarithmic decibel scale. Each increase of 10 dB is a tenfold increase in intensity; a fall of 6 dB is close to a quarter, which is what you get by doubling your distance from a source.

Beats

When two sounds of slightly different frequency $f_1$ and $f_2$ play together, their waves drift in and out of step. When compressions coincide they interfere constructively and the sound is loud; a little later a compression of one meets a rarefaction of the other and the sound is soft. The loudness rises and falls regularly. Each rise and fall is a beat, and the number of beats per second is

$f_{\text{beat}} = |f_2 - f_1|$

A correction to a common description. Some books say that beats produce "a third frequency that is heard". That is misleading. You do not hear a separate note at $f_2 - f_1$ (4 Hz would be far too low to hear as a pitch anyway). You hear one tone at the average frequency, $\frac{f_1 + f_2}{2}$, whose loudness pulses $|f_2 - f_1|$ times per second. The beat frequency is the frequency of the loudness variation, not of a new sound.

Musicians use beats to tune: play a string together with a reference note and adjust the tension until the beats slow down and disappear. Beats are only heard clearly when the difference is less than about 10 to 15 Hz; above that the ear hears a rough sound or two separate notes.

The Doppler effect

The Doppler effect is the change in observed frequency when the source of a wave and the observer move relative to each other. If they are getting closer, the observed frequency is higher; if they are moving apart, it is lower.

Moving source. The source emits each compression from a little further forward than the last, so ahead of it the compressions are closer together (shorter wavelength) and behind it they are further apart. The speed of sound in the air is unchanged, so the observer ahead receives more compressions per second: a higher frequency.

Moving observer. The wavelength in the air is unchanged, but an observer moving towards the source meets the compressions faster than they would standing still, so receives more per second.

Both cases combine into one formula, measured with the speeds as positive when the source and observer move towards each other:

$f' = f\,\dfrac{(v_{\text{wave}} + v_{\text{observer}})}{(v_{\text{wave}} - v_{\text{source}})}$

If the observer moves away from the source, $v_{\text{observer}}$ is negative; if the source moves away from the observer, $v_{\text{source}}$ is negative. Check every answer with common sense: approaching means $f' > f$.

The Doppler effect applies to all waves. Doppler ultrasound measures the speed of blood in arteries from the frequency shift of sound reflected off moving blood cells. For light, the shift of spectral lines from distant galaxies towards the red end (redshift) showed that the universe is expanding, which you will meet in Year 12.