Waves are all around us. Sound waves carry voices through the air, light waves bring images to our eyes, water waves roll across the ocean, and seismic waves travel through the Earth during an earthquake. A wave is a disturbance that travels through a medium (or through space) transporting energy and momentum from one place to another without transporting the matter of the medium itself.
In this chapter we study mechanical waves - waves that require a material medium, such as waves on a string and sound waves in air. We distinguish transverse waves, in which the particles oscillate perpendicular to the direction of propagation, from longitudinal waves, in which they oscillate parallel to it. We then study the mathematical description of waves, their speed in various media, and their superposition.
The concepts of interference, beats, standing waves, harmonics, and the Doppler effect complete the chapter. These phenomena explain how musical instruments produce their characteristic sounds, how two sources of sound can combine, and how the pitch of an approaching or receding siren changes - the Doppler effect.
In a transverse wave, the particles of the medium oscillate perpendicular to the direction of propagation of the wave. Waves on a stretched string and the ripples on water surfaces are examples of transverse waves. Transverse waves can travel through solids and on surfaces of liquids, but not through the bulk of fluids, which cannot sustain shear stress.
In a longitudinal wave, the particles oscillate parallel to the direction of propagation. Sound waves in air are longitudinal waves, consisting of alternate compressions (regions of higher pressure) and rarefactions (regions of lower pressure) that travel through the medium. Longitudinal waves can travel through solids, liquids, and gases.
Both types of wave transport energy without transporting matter. As a wave passes, the particles of the medium oscillate about their mean positions but do not travel along with the wave. A floating cork on water merely bobs up and down as a wave passes, while the wave itself moves across the surface.
The displacement of the medium particles in a wave varies periodically in both space and time. The wavelength, lambda, is the distance between two successive points that are in the same phase, such as two successive crests or compressions. The amplitude A is the maximum displacement of a particle from its mean position. The time period T is the time taken for one complete oscillation of a particle, and the frequency f = 1/T is the number of oscillations per unit time.
The speed of the wave, v, is the distance travelled by the wave in unit time. Over one period, the wave travels a distance of one wavelength, so:
v = f * lambda
This relation holds for all waves, mechanical and electromagnetic. The frequency of a wave is fixed by its source, so when a wave enters a new medium and its speed changes, its wavelength changes correspondingly to keep v = f lambda.
A sinusoidal wave travelling along the positive x-direction can be described by the wave equation:
y = A sin(kx - omega t)
where k = 2 pi / lambda is the wave number and omega = 2 pi f is the angular frequency. The wave speed is related to these by:
v = omega / k = f lambda
If the wave travels along the negative x-direction, the equation becomes y = A sin(kx + omega t). At a fixed position, the equation describes simple harmonic motion of the particles; at a fixed time, it gives a snapshot of the wave shape. The speed of a particle of the medium is different from the speed of the wave itself - the particle oscillates about its mean position while the wave travels through the medium.
The speed of a transverse wave on a stretched string depends on the tension T in the string and the mass per unit length (linear mass density) mu:
v = sqrt(T / mu)
A string under higher tension or with lower mass per unit length carries waves faster. The speed of a longitudinal (sound) wave in a solid depends on its bulk modulus and density. The speed of sound in air, from Newton's formula and the adiabatic correction of Laplace, is:
v = sqrt(gamma P / rho)
At standard temperature and pressure, the speed of sound in air is about 332 m/s. The speed of sound increases with temperature and is greater in liquids and solids than in gases, because the particles are closer together and transfer energy more readily.
When two or more waves meet at a point, the resultant displacement is the vector sum of the individual displacements. This is the principle of superposition. It applies to waves of all kinds and is the basis of interference, beats, and standing waves.
When two waves of the same frequency and similar amplitude travel through the same region, they interfere. If they meet in phase, their amplitudes add and constructive interference occurs, producing a maximum in intensity. If they meet in opposite phase, destructive interference occurs and the amplitude is reduced; for equal amplitudes, it becomes zero.
The conditions for constructive and destructive interference are expressed in terms of the path difference. If the path difference is an integer multiple of the wavelength, the waves interfere constructively; if it is an odd multiple of half a wavelength, they interfere destructively. These conditions explain the bright and dark fringes in Young's double-slit experiment and the alternating loud and soft regions when two loudspeakers play the same tone.
When two waves of slightly different frequencies travel in the same direction, their superposition produces a periodic variation in the loudness of the resultant sound, called beats. The loudness rises and falls because the two waves alternately reinforce and cancel each other. The beat frequency is the difference of the two frequencies:
f_beat = |f1 - f2|
The phenomenon of beats is used to tune musical instruments: two sources producing no beats are in unison. The human ear can distinguish beats up to about 10 per second. Beat frequency is always the absolute difference of the two frequencies, so it is a positive quantity.
When two identical waves travelling in opposite directions superpose, they form standing (stationary) waves. In a standing wave, certain points called nodes have zero displacement at all times, while points called antinodes have maximum displacement. The nodes and antinodes are fixed in space; the wave does not appear to travel.
A standing wave on a string fixed at both ends has nodes at the ends. The lowest frequency of oscillation, called the fundamental frequency, is:
f1 = v / (2 L)
where L is the length of the string. The higher frequencies, called harmonics, are integer multiples of the fundamental: f_n = n f1, with n = 1, 2, 3, ... For a pipe open at both ends, the frequencies are also integer multiples of the fundamental, but for a pipe closed at one end, only odd harmonics are possible:
f_n = (2n - 1) f1
Standing waves explain how a guitar string and the air column in a flute produce their characteristic tones.
The Doppler effect is the change in the observed frequency of a wave when there is relative motion between the source and the observer. When the source and observer move toward each other, the observed frequency is higher; when they move apart, the observed frequency is lower. The general formula for the observed frequency f' when the source moves with speed v_s and the observer with speed v_o is:
f' = f * (v + v_o) / (v - v_s)
where v is the speed of the wave in the medium, with the signs chosen according to the direction of motion. For a source approaching a stationary observer, f' = f v / (v - v_s), which is greater than f.
The Doppler effect explains why the pitch of an approaching train's horn is higher and lower when the train recedes. It is also used in radar speed guns, in astronomy to measure the motion of stars and galaxies, and in medical ultrasound imaging.
| Wave Property | Symbol | Relation |
|---|---|---|
| Wavelength | lambda | distance between successive crests |
| Frequency | f | f = 1/T |
| Wave speed | v | v = f lambda |
| Wave number | k | k = 2 pi / lambda |
| Angular frequency | omega | omega = 2 pi f |
| Wave equation | y | y = A sin(kx - omega t) |
| Quantity | Formula |
|---|---|
| Speed on a string | v = sqrt(T/mu) |
| Speed of sound | v = sqrt(gamma P / rho) |
| Beat frequency | f_beat = |
| Fundamental string frequency | f1 = v/(2L) |
| Doppler effect | f' = f(v + v_o)/(v - v_s) |
In this chapter we studied waves and their properties. We distinguished transverse waves, in which particles oscillate perpendicular to the direction of propagation, from longitudinal waves such as sound. The fundamental relation v = f lambda and the wave equation y = A sin(kx - omega t) describe sinusoidal waves completely. We learned the speed of waves on a string, v = sqrt(T/mu), and of sound in air, and studied the superposition of waves through interference, beats with frequency |f1 - f2|, and standing waves with their nodes, antinodes, and harmonics. Finally, the Doppler effect explained the change in observed frequency due to relative motion. Waves tie together all the oscillatory phenomena of the chapter on oscillations, showing how energy is transferred through space, and they prepare the way for the study of sound, light, and modern physics in higher classes.