Chapter 15 Mechanical Waves PowerPoint Lectures for University Physics, Thirteenth Edition Hugh D. Young and Roger A. Freedman Lectures by Wayne Anderson
Goals for Chapter 15 To study the properties and varieties of mechanical waves To relate the speed, frequency, and wavelength of periodic waves To interpret periodic waves mathematically To calculate the speed of a wave on a string To calculate the energy of mechanical waves To understand the interference of mechanical waves To analyze standing waves on a string To investigate the sound produced by stringed instruments
Introduction Earthquake waves carry enormous power as they travel through the earth. Other types of mechanical waves, such as sound waves or the vibration of the strings of a piano, carry far less energy. Overlapping waves interfere, which helps us understand musical instruments.
Types of mechanical waves A mechanical wave is a disturbance traveling through a medium. Figure 15.1 below illustrates transverse waves and longitudinal waves.
Periodic waves For a periodic wave, each particle of the medium undergoes periodic motion. The wavelength of a periodic wave is the length of one complete wave pattern. The speed of any periodic wave of frequency f is v = f.
Q15.1 If you double the wavelength of a wave on a string, what happens to the wave speed v and the wave frequency f? A. v is doubled and f is doubled. B. v is doubled and f is unchanged. C. v is unchanged and f is halved. D. v is unchanged and f is doubled. E. v is halved and f is unchanged.
A15.1 If you double the wavelength of a wave on a string, what happens to the wave speed v and the wave frequency f? A. v is doubled and f is doubled. B. v is doubled and f is unchanged. C. v is unchanged and f is halved. D. v is unchanged and f is doubled. E. v is halved and f is unchanged.
Periodic transverse waves For the transverse waves shown here in Figures 15.3 and 15.4, the particles move up and down, but the wave moves to the right.
Periodic longitudinal waves For the longitudinal waves shown here in Figures 15.6 and 15.7, the particles oscillate back and forth along the same direction that the wave moves. Follow Example 15.1.
Mathematical description of a wave The wave function, y(x,t), gives a mathematical description of a wave. In this function, y is the displacement of a particle at time t and position x. The wave function for a sinusoidal wave moving in the +x-direction is y(x,t) = Acos(kx t), where k = 2π/ is called the wave number. Figure 15.8 at the right illustrates a sinusoidal wave.
Q15.2 Which of the following wave functions describe a wave that moves in the x-direction? A. y(x,t) = A sin ( kx t) B. y(x,t) = A sin (kx + t) C. y(x,t) = A cos (kx + t) D. both B. and C. E. all of A., B., and C.
A15.2 Which of the following wave functions describe a wave that moves in the x-direction? A. y(x,t) = A sin ( kx t) B. y(x,t) = A sin (kx + t) C. y(x,t) = A cos (kx + t) D. both B. and C. E. all of A., B., and C.
Graphing the wave function The graphs in Figure 15.9 to the right look similar, but they are not identical. Graph (a) shows the shape of the string at t = 0, but graph (b) shows the displacement y as a function of time at t = 0. Refer to Problem-Solving Strategy 15.1. Follow Example 15.2.
Particle velocity and acceleration in a sinusoidal wave The graphs in Figure 15.10 below show the velocity and acceleration of particles of a string carrying a transverse wave.
Q15.3 A wave on a string is moving to the right. This graph of y(x, t) versus coordinate x for a specific time t shows the shape of part of the string at that time. 0 y a x At this time, what is the velocity of a particle of the string at x = a? A. The velocity is upward. B. The velocity is downward. C. The velocity is zero. D. not enough information given to decide
A15.3 A wave on a string is moving to the right. This graph of y(x, t) versus coordinate x for a specific time t shows the shape of part of the string at that time. 0 y a x At this time, what is the velocity of a particle of the string at x = a? A. The velocity is upward. B. The velocity is downward. C. The velocity is zero. D. not enough information given to decide
Q15.4 A wave on a string is moving to the right. This graph of y(x, t) versus coordinate x for a specific time t shows the shape of part of the string at that time. 0 y a x At this time, what is the acceleration of a particle of the string at x = a? A. The acceleration is upward. B. The acceleration is downward. C. The acceleration is zero. D. not enough information given to decide
A15.4 A wave on a string is moving to the right. This graph of y(x, t) versus coordinate x for a specific time t shows the shape of part of the string at that time. 0 y a x At this time, what is the acceleration of a particle of the string at x = a? A. The acceleration is upward. B. The acceleration is downward. C. The acceleration is zero. D. not enough information given to decide
Q15.5 A wave on a string is moving to the right. This graph of y(x, t) versus coordinate x for a specific time t shows the shape of part of the string at that time. 0 y b x At this time, what is the velocity of a particle of the string at x = b? A. The velocity is upward. B. The velocity is downward. C. The velocity is zero. D. not enough information given to decide
A15.5 A wave on a string is moving to the right. This graph of y(x, t) versus coordinate x for a specific time t shows the shape of part of the string at that time. 0 y b x At this time, what is the velocity of a particle of the string at x = b? A. The velocity is upward. B. The velocity is downward. C. The velocity is zero. D. not enough information given to decide
Q15.6 A wave on a string is moving to the right. This graph of y(x, t) versus coordinate x for a specific time t shows the shape of part of the string at that time. At this time, the velocity of a particle on the string is upward at A. only one of points 1, 2, 3, 4, 5, and 6. B. point 1 and point 4 only. C. point 2 and point 6 only. D. point 3 and point 5 only. E. three or more of points 1, 2, 3, 4, 5, and 6.
A15.6 A wave on a string is moving to the right. This graph of y(x, t) versus coordinate x for a specific time t shows the shape of part of the string at that time. At this time, the velocity of a particle on the string is upward at A. only one of points 1, 2, 3, 4, 5, and 6. B. point 1 and point 4 only. C. point 2 and point 6 only. D. point 3 and point 5 only. E. three or more of points 1, 2, 3, 4, 5, and 6.
The speed of a wave on a string Follow the first method using Figure 15.11 above. Follow the second method using Figure 15.13 at the right. F The result is v.
Q15.7 Two identical strings are each under the same tension. Each string has a sinusoidal wave with the same average power P av. If the wave on string #2 has twice the amplitude of the wave on string #1, the wavelength of the wave on string #2 must be A. 4 times the wavelength of the wave on string #1. B. twice the wavelength of the wave on string #1. C. the same as the wavelength of the wave on string #1. D. 1/2 of the wavelength of the wave on string #1. E. 1/4 of the wavelength of the wave on string #1.
A15.7 Two identical strings are each under the same tension. Each string has a sinusoidal wave with the same average power P av. If the wave on string #2 has twice the amplitude of the wave on string #1, the wavelength of the wave on string #2 must be A. 4 times the wavelength of the wave on string #1. B. twice the wavelength of the wave on string #1. C. the same as the wavelength of the wave on string #1. D. 1/2 of the wavelength of the wave on string #1. E. 1/4 of the wavelength of the wave on string #1.
Calculating wave speed Follow Example 15.3 and refer to Figure 15.14 below.
Q15.8 The four strings of a musical instrument are all made of the same material and are under the same tension, but have different thicknesses. Waves travel A. fastest on the thickest string. B. fastest on the thinnest string. C. at the same speed on all strings. D. not enough information given to decide
A15.8 The four strings of a musical instrument are all made of the same material and are under the same tension, but have different thicknesses. Waves travel A. fastest on the thickest string. B. fastest on the thinnest string. C. at the same speed on all strings. D. not enough information given to decide
Power in a wave A wave transfers power along a string because it transfers energy. The average power is proportional to the square of the amplitude and to the square of the frequency. This result is true for all waves. Follow Example 15.4.
Wave intensity The intensity of a wave is the average power it carries per unit area. If the waves spread out uniformly in all directions and no energy is absorbed, the intensity I at any distance r from a wave source is inversely proportional to r 2 : I 1/r 2. (See Figure 15.17 at the right.) Follow Example 15.5.
Boundary conditions When a wave reflects from a fixed end, the pulse inverts as it reflects. See Figure 15.19(a) at the right. When a wave reflects from a free end, the pulse reflects without inverting. See Figure 15.19(b) at the right.
Wave interference and superposition Interference is the result of overlapping waves. Principle of superposition: When two or more waves overlap, the total displacement is the sum of the displacements of the individual waves. Study Figures 15.20 and 15.21 at the right.
Standing waves on a string Waves traveling in opposite directions on a taut string interfere with each other. The result is a standing wave pattern that does not move on the string. Destructive interference occurs where the wave displacements cancel, and constructive interference occurs where the displacements add. At the nodes no motion occurs, and at the antinodes the amplitude of the motion is greatest. Figure 15.23 on the next slide shows photographs of several standing wave patterns.
Photos of standing waves on a string Some time exposures of standing waves on a stretched string.
The formation of a standing wave In Figure 15.24, a wave to the left combines with a wave to the right to form a standing wave. Refer to Problem- Solving Strategy 15.2 and follow Example 15.6.
Normal modes of a string For a taut string fixed at both ends, the possible wavelengths are n = 2L/n and the possible frequencies are f n = n v/2l = nf 1, where n = 1, 2, 3, f 1 is the fundamental frequency, f 2 is the second harmonic (first overtone), f 3 is the third harmonic (second overtone), etc. Figure 15.26 illustrates the first four harmonics.
Standing waves and musical instruments A stringed instrument is tuned to the correct frequency (pitch) by varying the tension. Longer strings produce bass notes and shorter strings produce treble notes. (See Figure 15.29 below.) Follow Examples 15.7 and 15.8.
Q15.9 While a guitar string is vibrating, you gently touch the midpoint of the string to ensure that the string does not vibrate at that point. The lowest-frequency standing wave that could be present on the string A. vibrates at the fundamental frequency. B. vibrates at twice the fundamental frequency. C. vibrates at 3 times the fundamental frequency. D. vibrates at 4 times the fundamental frequency. E. not enough information given to decide
A15.9 While a guitar string is vibrating, you gently touch the midpoint of the string to ensure that the string does not vibrate at that point. The lowest-frequency standing wave that could be present on the string A. vibrates at the fundamental frequency. B. vibrates at twice the fundamental frequency. C. vibrates at 3 times the fundamental frequency. D. vibrates at 4 times the fundamental frequency. E. not enough information given to decide