Concepts: beats, the Doppler effect and exercise practice

6.1 Beats

When two sound waves of slightly different frequencies reach the ear together, the resultant intensity rises and falls periodically. The loudness waxes and wanes, and this phenomenon is called beats. Superposition at a fixed point gives a resultant whose amplitude changes slowly with time, because the two waves drift in and out of phase.

The beat frequency is the difference between the two frequencies. Two tuning forks of 256 and 260 hertz produce four beats per second. The average of the two frequencies is the frequency of the combined tone that is heard, while the envelope of the sound varies at the beat frequency.

Beats are used in practice to tune instruments. A musician compares the sound of an instrument with a standard fork and adjusts the string or pipe until the beats slow down and disappear, at which point the frequencies are equal. Beats can also be used to find an unknown frequency: if a fork of known frequency produces a known number of beats with the unknown source, the unknown frequency is either the sum or the difference, and a second comparison resolves the ambiguity.

6.2 The Doppler effect

The Doppler effect is the change in the observed frequency of a wave when the source and the observer move relative to each other. When a source approaches, the wavefronts ahead of it are crowded together, so the wavelength in that direction is shortened and the observed frequency is higher. When the source recedes, the wavefronts are stretched and the observed frequency is lower.

The general relation for sound uses the speed of sound in the medium. The observed frequency is the source frequency multiplied by the medium speed plus or minus the observer speed, divided by the medium speed minus or plus the source speed.

The upper sign in the numerator applies when the observer moves towards the source, and the upper sign in the denominator applies when the source moves towards the observer. The observer's motion changes the number of wavefronts met per second, while the source's motion changes the wavelength, so the two effects enter the formula differently. For light, relativity modifies the result, but the qualitative rule that approach raises the frequency and recession lowers it remains.

The Doppler effect explains the change in pitch of an ambulance siren as it passes: the pitch is high while the vehicle approaches and drops suddenly as it goes away. It is also the basis of radar speed measurement and of the redshift used in astronomy.

6.3 The exercise set

The chapter exercise set runs from 14.1 to 14.21 and covers all the topics. The first few questions apply the string speed formula: 14.1 finds the time for a jerk to travel the length of a string, and 14.3 finds the tension needed for a transverse wave to travel at the speed of sound in air. Question 14.2 combines free fall with the time for sound to travel back up a tower.

Questions 14.5 to 14.11 test the progressive wave equation. Exercise 14.8 asks for the speed, direction, amplitude, frequency, initial phase and crest separation from the equation of a transverse harmonic wave. Exercise 14.10 finds phase differences for several path differences, and 14.11 interprets a standing-wave equation as a superposition of two travelling waves.

Questions 14.12 to 14.21 cover organ pipes, resonance, beats and the Doppler effect. The pipe questions use the closed or open formula and often the end correction. The beat questions use the difference of frequencies. The Doppler questions use the sign rule for approach and recession, and a few combine relative motion with a stationary medium.

6.4 Worked practice

Exercise 14.1: a string of mass 2.50 kilograms and length 20.0 metres is under a tension of 200 newtons. The linear density is 2.50 divided by 20.0, which is 0.125 kilogram per metre. The wave speed is the square root of 200 divided by 0.125, which is the square root of 1600, so 40 metres per second. The time for the disturbance to travel 20.0 metres is 20.0 divided by 40, which is 0.50 second.

Exercise 14.2: a stone dropped from a 300 metre tower falls in the time given by 300 equals half of 9.8 times t squared, which gives t about 7.8 seconds. The splash sound then takes 300 divided by 340, about 0.88 second, to travel back up. The splash is heard about 8.7 seconds after the stone is dropped.

Exercise 14.6: a bat emits 1000 kilohertz ultrasound. In air at 340 metres per second the wavelength is 340 divided by 10 to the sixth, which is 0.34 millimetre. The reflected sound stays in air and keeps that wavelength. The transmitted sound enters water at 1486 metres per second, so its wavelength becomes 1486 divided by 10 to the sixth, about 1.49 millimetres. The frequency does not change when sound crosses a boundary; the wavelength changes because the speed changes.