Beats and the Doppler Effect
Two tones of nearly equal frequency heard together as beats, \(f_{\text{beat}} = |f_1 - f_2|\), and tuning by beats. The change of frequency heard when a source or an observer moves, \(f' = f(c \pm v_O)/(c \mp v_S)\), and the wavefronts of a moving source.
Why this chapter
Two sounds of nearly the same pitch do not simply add. Heard together, they throb, and the number of throbs in a second is the difference of their frequencies. Musicians tune by listening for these beats. A sound also changes its pitch when its source or its listener moves: the horn of a passing train drops as the engine goes by. This is the Doppler effect. The chapter derives both from the superposition of waves and from the time between the arrivals of wave maxima. It ends with the wavefronts of a moving source. They show why a moving source changes the wavelength in the air, while a moving listener does not.
You will learn
- Derive the resultant of two waves of nearly equal frequency, and explain why the beat frequency is \(\Delta f\) and the pitch heard is the average.
- Find an unknown frequency from a beat rate, and use waxing or filing of a fork to decide between the two possible answers.
- Derive the frequency heard by a moving observer from the relative speed \(c \pm v_O\).
- Derive the frequency from a moving source from the arrival times of two maxima, and combine both into \(f' = f(c \pm v_O)/(c \mp v_S)\) with its sign rule.
- Use only the component of velocity along the line joining source and observer.
- Explain with wavefronts why a moving source changes the wavelength, \(\lambda' = (v \mp v_S)T\), while a moving observer does not.
Before you begin
From the chapter Superposition, Damped and Forced Oscillations you should know how two SHMs add. From the chapters on sound you should know the frequency, period and wavelength of a sound wave, and \(v = \lambda f\). Sound travels at a fixed speed in air. From the chapter Relative Velocity in One Dimension you should know how two speeds along one line add or subtract. You should know the identity \(\cos a + \cos b = 2\cos\frac{a-b}{2}\cos\frac{a+b}{2}\).
Lessons
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%Glossary
The key words of this chapter, with the meaning this chapter gives them.
- amplitude envelope
- The slowly varying amplitude \(2A\cos(2\pi \tfrac{\Delta f}{2} t)\) of the resultant of two close frequencies. It outlines the fast oscillation and makes the loudness pulse.
- apparent frequency
- The frequency \(f'\) that an observer actually receives, the number of maxima arriving per second. It differs from the source frequency when either one moves.
- beats
- The periodic rise and fall in the loudness of sound when two waves of slightly different frequencies are superposed. The number of beats per second is \(|f_1 - f_2|\).
- compression
- A region of a sound wave where the air is squeezed together, with a pressure above average. A compression maximum is its centre.
- Doppler effect
- The change in the frequency received by an observer when the source, the observer, or both move. The motion that counts is along the line joining them.
- in phase
- Two oscillations are in phase at an instant when they reach their crests together. Their displacements then add, and the sound is loud.
- intensity
- The power carried by a sound wave per unit area. It is proportional to the square of the amplitude, and it sets the loudness.
- medium
- The material through which a mechanical wave travels, such as air. The speed of sound \(c\), and every speed in the Doppler formula, is measured relative to it.
- observer
- The listener, or the instrument, that receives the waves and counts their maxima.
- out of phase
- Two oscillations are out of phase at an instant when the crest of one meets the trough of the other. Their displacements then cancel, and the sound is quiet.
- period
- The time \(T = 1/f\) between the emissions of two consecutive maxima by the source.
- pitch
- How high or low a sound seems to the ear. It is set by the frequency; during beats it is the average of the two frequencies.
- relative speed
- The rate at which the distance between two moving things changes. Along one line it is the sum of their speeds when they move towards each other.
- source
- The body that emits the waves, such as a horn, a siren or a whistle. Its own frequency is \(f\).
- subsonic
- Slower than the speed of sound in the medium. The Doppler formula of this chapter holds only for subsonic sources.
- superposition
- The principle that the displacement caused by two waves at a point is the sum of the displacements that each wave would cause alone.
- tuning fork
- A U-shaped steel bar on a stem whose prongs vibrate at one fixed frequency when struck. Wax on the prongs lowers the frequency; filing the ends of the prongs raises it.
- unison
- Two sounds of exactly the same frequency. In unison the beat frequency is zero and the combined tone is steady.
- wavefront
- A surface of constant phase, such as a compression maximum, spreading outward from the source. From a point source in air it is a sphere, drawn as a circle.
- wavelength
- The distance between two consecutive crests of a wave in the medium at one instant. The source and the medium set it; a moving observer does not change it.