Chapter 15 Wave Motion Chapter opener. Caption: Waves—such as these water waves—spread outward from a source. The source in this case is a small spot of water oscillating up and down briefly where a rock was thrown in (left photo). Other kinds of waves include waves on a cord or string, which also are produced by a vibration. Waves move away from their source, but we also study waves that seem to stand still (“standing waves”). Waves reflect, and they can interfere with each other when they pass through any point at the same time.

15-2 Types of Waves: Transverse and Longitudinal Earthquakes produce both longitudinal and transverse waves. Both types can travel through solid material, but only longitudinal waves can propagate through a fluid—in the transverse direction, a fluid has no restoring force. Surface waves are waves that travel along the boundary between two media. Figure 15-9: A water wave is an example of a surface wave, which is a combination of transverse and longitudinal wave motions. Figure 15-10: How a wave breaks. The green arrows represent the local velocity of water molecules.

15-3 Energy Transported by Waves By looking at the energy of a particle of matter in the medium of a wave, we find: Then, assuming the entire medium has the same density, we find: Figure 15-11. Calculating the energy carried by a wave moving with velocity v. Therefore, the intensity is proportional to the square of the frequency and to the square of the amplitude.

15-3 Energy Transported by Waves If a wave is able to spread out three-dimensionally from its source, and the medium is uniform, the wave is spherical. Just from geometrical considerations, as long as the power output is constant, we see: Figure 15-12. Wave traveling outward from a point source has spherical shape. Two different crests (or compressions) are shown, of radius r1 and r2.

15-4 Mathematical Representation of a Traveling Wave Suppose the shape of a wave is given by: Figure 15-13. In time t, the wave moves a distance vt.

15-4 Mathematical Representation of a Traveling Wave After a time t, the wave crest has traveled a distance vt, so we write: Or: with ,

15-6 The Principle of Superposition Superposition: The displacement at any point is the vector sum of the displacements of all waves passing through that point at that instant. Fourier’s theorem: Any complex periodic wave can be written as the sum of sinusoidal waves of different amplitudes, frequencies, and phases. Figure 15-16. The superposition principle for one-dimensional waves. Composite wave formed from three sinusoidal waves of different amplitudes and frequencies (f0, 2f0, 3f0) at a certain instant in time. The amplitude of the composite wave at each point in space, at any time, is the algebraic sum of the amplitudes of the component waves. Amplitudes are shown exaggerated; for the superposition principle to hold, they must be small compared to the wavelengths.

15-6 The Principle of Superposition Conceptual Example 15-7: Making a square wave. At t = 0, three waves are given by D1 = A cos kx, D2 = -1/3A cos 3kx, and D3 = 1/5A cos 5kx, where A = 1.0 m and k = 10 m-1. Plot the sum of the three waves from x = -0.4 m to +0.4 m. (These three waves are the first three Fourier components of a “square wave.”) Solution is shown in the figure.

15-7 Reflection and Transmission A wave reaching the end of its medium, but where the medium is still free to move, will be reflected (b), and its reflection will be upright. Figure 15-18. Reflection of a wave pulse on a cord lying on a table. (a) The end of the cord is fixed to a peg. (b) The end of the cord is free to move. A wave hitting an obstacle will be reflected (a), and its reflection will be inverted.

15-7 Reflection and Transmission A wave encountering a denser medium will be partly reflected and partly transmitted; if the wave speed is less in the denser medium, the wavelength will be shorter. Figure 15-19. When a wave pulse traveling to the right along a thin cord (a) reaches a discontinuity where the cord becomes thicker and heavier, then part is reflected and part is transmitted (b).

15-7 Reflection and Transmission Two- or three-dimensional waves can be represented by wave fronts, which are curves of surfaces where all the waves have the same phase. Lines perpendicular to the wave fronts are called rays; they point in the direction of propagation of the wave. Figure 15-20. Rays, signifying the direction of motion, are always perpendicular to the wave fronts (wave crests). (a) Circular or spherical waves near the source. (b) Far from the source, the wave fronts are nearly straight or flat, and are called plane waves.

15-7 Reflection and Transmission The law of reflection: the angle of incidence equals the angle of reflection. Figure 15-21. Law of reflection: θr = θi.

15-8 Interference The superposition principle says that when two waves pass through the same point, the displacement is the arithmetic sum of the individual displacements. In the figure below, (a) exhibits destructive interference and (b) exhibits constructive interference. Figure 15-22. Two wave pulses pass each other. Where they overlap, interference occurs: (a) destructive, and (b) constructive.

15-8 Interference These graphs show the sum of two waves. In (a) they add constructively; in (b) they add destructively; and in (c) they add partially destructively. Figure 15-24. Graphs showing two identical waves, and their sum, as a function of time at three locations. In (a) the two waves interfere constructively, in (b) destructively, and in (c) partially destructively.

15-9 Standing Waves; Resonance Standing waves occur when both ends of a string are fixed. In that case, only waves which are motionless at the ends of the string can persist. There are nodes, where the amplitude is always zero, and antinodes, where the amplitude varies from zero to the maximum value. Figure 15-25. Standing waves corresponding to three resonant frequencies.

15-9 Standing Waves; Resonance The frequencies of the standing waves on a particular string are called resonant frequencies. They are also referred to as the fundamental and harmonics. Figure 15-26. (a) A string is plucked. (b) Only standing waves corresponding to resonant frequencies persist for long.

15-9 Standing Waves; Resonance The wavelengths and frequencies of standing waves are: and

15-10 Refraction If the wave enters a medium where the wave speed is different, it will be refracted—its wave fronts and rays will change direction. We can calculate the angle of refraction, which depends on both wave speeds: Figure 15-28. Refraction of waves passing a boundary.

15-10 Refraction The law of refraction works both ways—a wave going from a slower medium to a faster one would follow the red line in the other direction. Figure 15-30b. Law of refraction for waves.

15-11 Diffraction When waves encounter an obstacle, they bend around it, leaving a “shadow region.” This is called diffraction. Figure 15-31. Wave diffraction. The waves are coming from the upper left. Note how the waves, as they pass the obstacle, bend around it, into the “shadow region” behind the obstacle.

15-11 Diffraction The amount of diffraction depends on the size of the obstacle compared to the wavelength. If the obstacle is much smaller than the wavelength, the wave is barely affected (a). If the object is comparable to, or larger than, the wavelength, diffraction is much more significant (b, c, d). Figure 15-32. Water waves passing objects of various sizes. Note that the longer the wavelength compared to the size of the object, the more diffraction there is into the “shadow region.”

Summary of Chapter 15 Vibrating objects are sources of waves, which may be either pulses or continuous. Wavelength: distance between successive crests Frequency: number of crests that pass a given point per unit time Amplitude: maximum height of crest Wave velocity:

Summary of Chapter 15 Transverse wave: oscillations perpendicular to direction of wave motion Longitudinal wave: oscillations parallel to direction of wave motion Intensity: energy per unit time crossing unit area (W/m2): Angle of reflection is equal to angle of incidence

Summary of Chapter 15 When two waves pass through the same region of space, they interfere. Interference may be either constructive or destructive. Standing waves can be produced on a string with both ends fixed. The waves that persist are at the resonant frequencies. Nodes occur where there is no motion; antinodes where the amplitude is maximum. Waves refract when entering a medium of different wave speed, and diffract around obstacles.

Chapter 16 Sound Chapter opener. “If music be the food of physics, play on.” [See Shakespeare, Twelfth Night, line 1.] Stringed instruments depend on transverse standing waves on strings to produce their harmonious sounds. The sound of wind instruments originates in longitudinal standing waves of an air column. Percussion instruments create more complicated standing waves. Besides examining sources of sound, we also study the decibel scale of sound level, sound wave interference and beats, the Doppler effect, shock waves and sonic booms, and ultrasound imaging.

Units of Chapter 16 Sections – 2, 4, 6, 7 only Mathematical Representation of Longitudinal Waves Sources of Sound: Vibrating Strings Quality of Sound, and Noise; Superposition Interference of Sound Waves; Beats Doppler Effect

16-2 Mathematical Representation of Longitudinal Waves Longitudinal waves are often called pressure waves. The displacement is 90° out of phase with the pressure. Figure 16-2. Longitudinal sound wave traveling to the right, and its graphical representation in terms of pressure. Figure 16-3. Representation of a sound wave in space at a given instant in terms of (a) displacement, and (b) pressure.

16-2 Mathematical Representation of Longitudinal Waves By considering a small cylinder within the fluid, we see that the change in pressure is given by (B is the bulk modulus): Figure 16-4. Longitudinal wave in a fluid moves to the right. A thin layer of fluid, in a thin cylinder of area S and thickness Δx, changes in volume as a result of pressure variation as the wave passes. At the moment shown, the pressure will increase as the wave moves to the right, so the thickness of our layer will decrease, by an amount ΔD.

16-2 Mathematical Representation of Longitudinal Waves If the displacement is sinusoidal, we have where and

16-5 Quality of Sound, and Noise; Superposition So why does a trumpet sound different from a flute? The answer lies in overtones—which ones are present, and how strong they are, makes a big difference. The sound wave is the superposition of the fundamental and all the harmonics. Figure 16-13. The amplitudes of the fundamental and first two overtones are added at each point to get the “sum,” or composite waveform.

16-5 Quality of Sound, and Noise; Superposition This plot shows frequency spectra for a clarinet, a piano, and a violin. The differences in overtone strength are apparent. Figure 16-14. Sound spectra for different instruments. The spectra change when the instruments play different notes. The clarinet is a bit complicated: it acts like a closed tube at lower frequencies, having only odd harmonics, but at higher frequencies all harmonics occur as for an open tube.

16-6 Interference of Sound Waves; Beats Sound waves interfere in the same way that other waves do in space. Figure 16-15. Sound waves from two loudspeakers interfere. Figure 16-16. Sound waves of a single frequency from loudspeakers A and B (see Fig. 16–15) constructively interfere at C and destructively interfere at D. [Shown here are graphical representations, not the actual longitudinal sound waves.]

16-6 Interference of Sound Waves; Beats Example 16-12: Loudspeakers’ interference. Two loudspeakers are 1.00 m apart. A person stands 4.00 m from one speaker. How far must this person be from the second speaker to detect destructive interference when the speakers emit an 1150-Hz sound? Assume the temperature is 20°C. Solution: The wavelength is 0.30 m. For destructive interference, the person must be half a wavelength closer or farther away, 3.85 m or 4.15 m.

16-6 Interference of Sound Waves; Beats Waves can also interfere in time, causing a phenomenon called beats. Beats are the slow “envelope” around two waves that are relatively close in frequency. Figure 6-17. Beats occur as a result of the superposition of two sound waves of slightly different frequency.

16-6 Interference of Sound Waves; Beats If we consider two waves of the same amplitude and phase, with different frequencies, we can find the beat frequency when we add them: This represents a wave vibrating at the average frequency, with an “envelope” at the difference of the frequencies.

16-6 Interference of Sound Waves; Beats Example 16-13: Beats. A tuning fork produces a steady 400-Hz tone. When this tuning fork is struck and held near a vibrating guitar string, twenty beats are counted in five seconds. What are the possible frequencies produced by the guitar string? Solution: The beat frequency is 4 Hz, so the string is either 396 or 404 Hz.

16-7 Doppler Effect The Doppler effect occurs when a source of sound is moving with respect to an observer. A source moving toward an observer appears to have a higher frequency and shorter wavelength; a source moving away from an observer appears to have a lower frequency and longer wavelength. Figure 16-18. (a) Both observers on the sidewalk hear the same frequency from a fire truck at rest. (b) Doppler effect: observer toward whom the fire truck moves hears a higher-frequency sound, and observer behind the fire truck hears a lower-frequency sound.

16-7 Doppler Effect If we can figure out what the change in the wavelength is, we also know the change in the frequency. Figure 16-19. Determination of the frequency shift in the Doppler effect (see text). The red dot is the source.

16-7 Doppler Effect The change in the frequency is given by: If the source is moving away from the observer:

16-7 Doppler Effect If the observer is moving with respect to the source, things are a bit different. The wavelength remains the same, but the wave speed is different for the observer. Figure 16-20. Observer moving with speed vobs toward a stationary source detects wave crests passing at speed v’ = vsnd + vobs, where vsnd is the speed of the sound waves in air.

16-7 Doppler Effect We find, for an observer moving toward a stationary source: And if the observer is moving away:

16-7 Doppler Effect Example 16-14: A moving siren. The siren of a police car at rest emits at a predominant frequency of 1600 Hz. What frequency will you hear if you are at rest and the police car moves at 25.0 m/s (a) toward you, and (b) away from you? Solution: a. 1730 Hz b. 1490 Hz

16-7 Doppler Effect Example 16-15: Two Doppler shifts. A 5000-Hz sound wave is emitted by a stationary source. This sound wave reflects from an object moving toward the source. What is the frequency of the wave reflected by the moving object as detected by a detector at rest near the source? Solution: The Doppler shift is applied twice; the frequency is 5103 Hz.

16-7 Doppler Effect All four equations for the Doppler effect can be combined into one; you just have to keep track of the signs!

Summary of Chapter 16 Sound is a longitudinal wave in a medium. The strings on stringed instruments produce a fundamental tone whose wavelength is twice the length of the string; there are also various harmonics present. Sound waves exhibit interference; if two sounds are at slightly different frequencies they produce beats. The Doppler effect is the shift in frequency of a sound due to motion of the source or the observer.