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Copyright © 2012 Pearson Education Inc.

PowerPoint® Lectures for

University Physics, Thirteenth Edition

– Hugh D. Young and Roger A. Freedman

Lectures by Wayne Anderson

Chapter 15

Mechanical Waves


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.


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.

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


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.


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.

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


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.



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.


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, 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

Q15.3

x

y

0 a












A15.3

x

y

0 a






At this time, what is the acceleration of a


A. The acceleration is upward.

B. The acceleration is downward.

C. The acceleration is zero.


Q15.4

x

y

0 a






At this time, what is the acceleration of a


A. The acceleration is upward.

B. The acceleration is downward.

C. The acceleration is zero.


A15.4

x

y

0 a







particle of the string at x = b?





Q15.5

x

y

0 b







particle of the string at x = b?





A15.5

x

y

0 b






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.

Q15.6

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

At this time, the velocity of a particle on the string is upward at


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.

• The result is .Fv


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.

Q15.7

Two identical strings are each under the same tension. Each string

has a sinusoidal wave with the same average power Pav.

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


Two identical strings are each under the same tension. Each string

has a sinusoidal wave with the same average power Pav.

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


Calculating wave speed

• Follow Example 15.3 and refer to Figure 15.14 below.


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.


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.


A15.8


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.



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 r2: I 1/r2.

(See Figure 15.17 at the right.)



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 super-

position: When two

or more waves

overlap, the total

displacement is the

sum of the displace-

ments 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 fn = n v/2L = nf1, where

n = 1, 2, 3, …

• f1 is the fundamental

frequency, f2 is the second

harmonic (first overtone), f3

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.


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

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

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