doPhysics · Waves
Interference and Reflection of Sound
Two sources in phase make fixed loud and quiet places, decided by the path difference. The quiet spots of two speakers, noise-cancelling headphones, Quincke's tube, sound reflected at a rigid wall and at an open end, and echoes.
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Why this chapter
Why this chapter exists
Sound rarely reaches us by one path alone. Two speakers send two waves, a hall sends back reflections from every wall, and a cliff returns a shout as an echo. This chapter follows sound when it meets other sound and when it meets a boundary. Two coherent sources build a map of loud and quiet places, set by the path difference. A wall and an open end each reflect sound by their own rule. These ideas explain noise-cancelling headphones, the acoustics of a concert hall and the depth finder of a ship. They also prepare the standing waves in pipes, in the next chapter.
You will learn
By the end of this chapter you will
- State the principle of superposition for sound, and relate the path difference \(\Delta L = |L_2 - L_1|\) to the phase difference at a point.
- Use the conditions \(\Delta L = n\lambda\) (loud) and \(\Delta L = (n + \tfrac12)\lambda\) (quiet), and find the resultant amplitude and intensity.
- Locate the quiet spots of two speakers exactly, and estimate them with \(y \approx (n + \tfrac12)\lambda L/d\).
- Explain noise-cancelling headphones, dead spots in a hall and the aiming of an ultrasound beam by interference.
- Use Quincke's tube to measure a wavelength, from minima \(\lambda/2\) apart in slide.
- Derive how sound reflects at a rigid wall and at an open end, for both the displacement and the pressure.
- Use \(t = 2d/v\) for echoes, the 17 m condition, echo sounding and SONAR, and explain reverberation.
Before you begin
Before you begin
From the chapter Superposition and Reflection, you should know the principle of superposition and the phase difference \(\Delta\varphi = (2\pi/\lambda)\Delta x\). You should also know the fixed-end and free-end rules for a string. From the chapter Propagation of Sound Waves, you should know the displacement wave \(s(x, t)\), the pressure wave \(\Delta P = -B\,\partial s/\partial x\), and \(v = f\lambda\). You should be able to use Pythagoras' theorem and the chain rule.
Lessons
1
Two Sources, Loud and Quiet Places
Two sound waves pass through each other, and their pressure variations add. Two speakers in phase make a fixed map of loud and quiet places, decided by the path difference \(\Delta L = |L_2 - L_1|\). Loud where \(\Delta L = n\lambda\), quiet where \(\Delta L = (n + \tfrac12)\lambda\).
Sound waves add their pressuresPath difference sets the phaseConditions for loud and quiet spots
2
Finding the Quiet Spots
The first quiet spot of two speakers found exactly, by a short numerical search, and estimated by the small-angle formula \(y \approx L\lambda/(2d)\). Interference at work in noise-cancelling headphones, concert halls and ultrasonic imaging.
Finding the first quiet spotThe small angle estimate of the quiet spotsInterference at work
3
Quincke's Tube
One sound split into two paths and joined again. A sliding U-tube changes the path difference by twice its slide, so successive minima are \(\lambda\) apart in path and \(\lambda/2\) apart in slide, which measures the wavelength and the speed of sound.
The sliding tube sets the path differenceSuccessive minima give the wavelength
4
Sound Reflected at a Wall and at an Open End
A rigid wall holds the air still, so it is a displacement node and a pressure antinode. The displacement wave returns inverted and the pressure wave upright. An open end holds the pressure fixed, so it does the opposite. The string rules of reflection become the rules for the ends of a pipe.
A rigid wall is a displacement nodeAn open end is a pressure nodeThe two ends of an air column
5
Echoes
An echo is a reflected sound heard apart from the original. The ear keeps a sound for about 0.1 s, so the reflector must be at least 17 m away at 340 m/s. Echo sounding and SONAR measure depths from echo times, and reverberation decides how a hall sounds.
An echo needs a tenth of a secondEcho sounding and sonarReverberation in a hall
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- ambient pressure
- The steady pressure \(P_0\) of the medium when no sound is present, about \(1.013 \times 10^5\,\text{Pa}\) for air at sea level.
- coherent sources
- Sources that emit waves of the same frequency with a phase difference that stays constant in time. Only they give a stable interference pattern.
- constructive interference
- Interference in which the waves arrive in phase, crest on crest, so that their amplitudes add. For sources in phase it needs \(\Delta L = n\lambda\).
- destructive interference
- Interference in which the waves arrive out of phase, crest on trough, so that their amplitudes subtract, down to silence. For sources in phase it needs \(\Delta L = (n + \tfrac12)\lambda\).
- displacement antinode
- A place where the displacement of the air swings the most. An open end of a pipe is a displacement antinode.
- displacement node
- A place where the displacement of the air is always zero, such as a rigid wall or the closed end of a pipe.
- displacement wave
- The description of sound by the shift \(s(x, t)\) of each particle from its mean position, along the direction of travel.
- echo
- A reflected sound heard as a separate sound after the original, because it returns at least about \(0.1\,\text{s}\) later.
- echo sounding
- Finding the depth of water by timing the echo of a sound pulse from the bottom, with \(d = vt/2\).
- interference
- The combining of two or more waves according to the principle of superposition. Two sources of one pure tone give a stable pattern of loud and quiet places.
- path difference
- The difference \(\Delta L = |L_2 - L_1|\) between the distances that two waves travel to reach a point.
- persistence of hearing
- The lasting of the sensation of a sound for about \(0.1\,\text{s}\) after the sound stops. Two sounds closer together than this are heard as one.
- phase difference
- The difference between the phases of two waves at a point. A path difference \(\Delta L\) gives \(\Delta\varphi = (2\pi/\lambda)\Delta L\) for sources in phase.
- pressure antinode
- A place where the pressure variation swings the most. A closed end of a pipe is a pressure antinode, because the air piles up against it.
- pressure node
- A place where the pressure variation is always zero, such as the open end of a pipe, where the outside air holds the pressure fixed.
- pressure variation
- The small change \(\Delta P\) of pressure above or below the ambient pressure, caused by a sound wave.
- principle of superposition
- When waves overlap, the disturbance at each point is the algebraic sum of the disturbances the waves would give alone. For sound, \(\Delta P = \Delta P_1 + \Delta P_2 + \dots\)
- Quincke's tube
- An apparatus that divides the sound of one source between two tubes and recombines it at a detector. One tube slides like a trombone, so the path difference changes by twice the slide.
- reverberation
- The lingering of a sound in a closed space after its source stops, caused by many reflections that the ear cannot separate.
- reverberation time
- The time for the sound level in a hall to fall by \(60\,\text{dB}\), to a millionth of its intensity, after the source stops.
- SONAR
- Sound navigation and ranging: finding and locating objects under water by sending pulses of sound, usually ultrasound, and timing their echoes.
- ultrasound
- Sound with a frequency above about \(20\,\text{kHz}\), too high for the human ear to hear.