Physics 2A

Chapters 15: Traveling Waves and Sound and

16: Superposition and Standing Waves

“We are what we believe we are.” – Benjamin Cardozo

“We would accomplish many more things if we did not think of them as impossible”
C. Malesherbez

“The only limit to our realization of tomorrow will be our doubts of today. Let us move forward
with strong and active faith.” – Franklin Delano Roosevelt

Reading: pages 476 – 490; 496 – 500

Outline:

⇒ introduction to waves (PowerPoint)

Mechanical waves
EM waves and Matter Waves
transverse and longitudinal waves
⇒ graphical description of waves
amplitude, wavelength, period, frequency, and wave speed
⇒ traveling waves
waves on a string
sound waves
⇒ sound and light waves (PowerPoint)
⇒ the Doppler effect
moving source, moving observer, and general case
shock waves

Reading: pages 507 – 516; 523 - 530

Outline:

⇒ the principle of superposition

constructive and destructive interference
interference of waves from two sources
⇒ standing waves
⇒ standing waves on a string
⇒ beats (PowerPoint)
 Problem Solving
Many of the problems involving waves on a string deal with the relationships v = λ f = λ /T,
where v is the wave speed, λ is the wavelength, f is the frequency, and T is the period. Typical
problems might give you the wavelength and frequency, then ask for the wave speed, or might
give you the wave speed and period, then ask for the wavelength.

Sometimes the quantities are given by describing the motion. For example, a problem might tell
you that the string at one point takes a certain time to go from its equilibrium position to
maximum displacement. This, of course, is one-fourth the period. In other problems, you may be
asked how long it takes a particle on a string to move through a total distance. You must then
recognize that a particle on the string moves through a distance a 4A (where A is the amplitude)
during a time equal to the period.

Some problems deal with the wave speed. For waves on a string, the fundamental equation is
v = Ts / µ , where Ts is the tension in the string and µ is the linear mass density. The tension
may not be given directly but, if the problem asks for the wave speed, sufficient information will
be given to calculate it.
 vo 
1± v 
Nearly all Doppler shift problems can be solved using f = 
vs  0
f
1 
 v 
where v is speed of sound, vS is the speed of the source, vo is the speed of the observer, f0 is the
frequency of the source, and f is the frequency detected by the observer. The upper sign in the
numerator refers to a situation in which the observer is moving toward the source; the lower sign
refers to a situation in which the observer is moving away from the source. The upper sign in the
denominator refers to a situation in which the source is moving toward the observer; the lower
sign refers to a situation in which the source is moving away from the observer.

Some problems deal with the production of beats by two sound waves with nearly the same
frequency. You may be given the frequency f1 of one of the waves and the beat frequency fbeat,
= f 1 − f 2 , it is given by
then asked for the frequency f2 of the other wave. Since fbeat
f=2 f 1 ± fbeat . You require more information to determine which sign to use in this equation.
One way to give this information is to tell you what happens to the beat frequency if f1 is
increased (or decreased). If the beat frequency increases when f1 increases, then f1 must be
greater than f2 and f2 = f1 - fbeat. If the beat frequency decreases, then f1 must be less than f2 and
f2 = f1 + fbeat.

Some problems deal with standing waves on a string. If you are told the distance between
successive nodes or successive antinodes, double the distance to find the wavelength. If you are
told the distance between a node and a neighboring antinode, multiply it by 4 to find the
wavelength.

If a standing wave is generated in a string with both ends fixed, the wave pattern must have a
node at each end of the string. This means the length L of the string and the wavelength λ of the
traveling waves must be related by L = nλ/2, where n is an integer.
KNIG5491_02_ch15_pp477-506.qxd 6/19/09 3:07 PM Page 500

500 CHAPTER 15 Traveling Waves and Sound

SUMMARY
The goal of Chapter 15 has been to learn the basic properties of traveling waves.

GENERAL PRINCIPLES
The Wave Model Mechanical waves require a material medium. The speed of the
wave is a property of the medium, not the wave. The speed does
This model is based on the idea of a traveling wave, which is an not depend on the size or shape of the wave.
organized disturbance traveling at a well-defined wave speed v.
• For a wave on a m
__ v
• In transverse waves the string, the string Ts m 5 L Ts
particles of the medium move vstring =
perpendicular to the direction
v is the medium. Am
in which the wave travels. • A sound wave is a
wave of compressions In a gas:
• In longitudinal waves the vsound
v and rarefactions of a gRT
particles of the medium move medium such as air. vsound =
parallel to the direction in A M
which the wave travels.
Electromagnetic waves are waves of the electromagnetic field.
A wave transfers energy, but there is no material or substance They do not require a medium. All electromagnetic waves travel at
transferred. the same speed in a vacuum, c = 3.00 * 108 m/s.

IMPORTANT CONCEPTS
Graphical representation of waves Mathematical representation of waves The intensity of a wave is the
A snapshot graph is a picture of a wave at one Sinusoidal waves are produced by a ratio of the power to the area:
instant in time. For a periodic wave, the source moving with simple harmonic P
wavelength l is the distance between crests. motion. The equation for a sinusoidal I=
A
wave is a function of position and time:
y l vr For a spherical wave the power
A
decreases with the surface area
y1x, t2 = A cos a2pa ⫾ b b
Fixed t: x
x t
0 of the spherical wave fronts:
2A l T

A history graph is a graph of the displacement of +: wave travels to left Psource Wave
fronts
one point in a medium versus time. For a periodic -: wave travels to right l l
wave, the period T is the time between crests. r
For sinusoidal and other periodic waves:
y T
A
Fixed x: t
1 Psource
0 T= v = fl
2A f I=
4pr 2

APPLICATIONS
The loudness of a sound is given by the The Doppler effect is a shift in frequency when there is relative motion of a wave
sound intensity level. This is a logarithmic source (frequency f0 , wave speed v) and an observer.
function of intensity and is in units of
Moving source, stationary observer:
decibels.
• The usual reference level is the Receding source: Approaching source:
quietest sound that can be heard: vrs
ƒ0 f0
ƒ- = 0 1 2 3 f+ =
I0 = 1.0 * 10 -12 2
W/m 1 + vs/v 1 - vs/v

• The sound intensity level in dB is Moving observer, stationary source: Reflection from a moving object:
computed relative to this value:
f + = a1 + bf
Approaching vo vo
For vo V v, ¢ƒ = ⫾2ƒ0
b = 110 dB2 log 10 a b
I the source: v 0 v
I0
Moving away When an object moves faster than the
f- = a1 - bf
vo
• A sound at the reference level from the source: v 0 wave speed in a medium, a shock wave
corresponds to 0 dB. is formed.
KNIG5491_02_ch16_pp507-541.qxd 6/19/09 3:08 PM Page 530

530 CHAPTER 16 Superposition and Standing Waves

SUMMARY
The goal of Chapter 16 has been to use the idea of superposition to understand the phenomena
of interference and standing waves.

GENERAL PRINCIPLES
Principle of Interference
Superposition In general, the superposition of two or more waves into a single wave is called interference.
The displacement of a Constructive interference occurs when Destructive interference occurs when crests
medium when more than one crests are aligned with crests and troughs with are aligned with troughs. We say the waves
wave is present is the sum of troughs. We say the waves are in phase. It are out of phase. It occurs when the path-
the displacements due to each occurs when the path-length difference ¢d is length difference ¢d is a whole number of
individual wave. a whole number of wavelengths. wavelengths plus half a wavelength.
v v

2 1
v v
Dp Dd=l Dp
Dd= 12l

0 x 0 x

IMPORTANT CONCEPTS
Standing Waves A standing wave on a string has a A standing sound wave in a tube can have
node at each end. Possible modes: different boundary conditions: open-open,
Two identical traveling waves closed-closed, or open-closed.
moving in opposite directions m⫽1
create a standing wave. Open-open Dp in tube

fm = m a b
v
Antinodes 2L m51
m⫽2
m = 1, 2, 3, Á

Closed-closed Dp in tube
m⫽3

fm = m a b
v
m51
2L
Nodes
m = 1, 2, 3, Á
Node spacing is 12 l. L
Dp in tube
The boundary conditions deter- Open-closed
fm = m a b = mf1
2L v
mine which standing-wave fre- lm =
fm = m a b
v m51
quencies and wavelengths are m 2L
4L
allowed. The allowed standing
m = 1, 2, 3, Á m = 1, 3, 5, Á L
waves are modes of the system.

APPLICATIONS
Beats (loud-soft-loud-soft modulations Standing waves are multiples of a
of intensity) are produced when two fundamental frequency, the frequency
waves of slightly different frequencies of the lowest mode. The higher modes
are superimposed. are the higher harmonics. Fundamental frequency
Relative intensity

For sound, the fundamental frequency

determines the perceived pitch; the Higher harmonics
t higher harmonics determine the tone
quality.

Our vocal cords create a range of har-

f (Hz)
Loud Soft Loud Soft Loud Soft Loud monics. The mix of higher harmonics is 262 524 786 1048 1572 2096
changed by our vocal tract to create dif-
fbeat = ƒ f1 - f2 ƒ ferent vowel sounds.
 Questions and Example Problems from Chapters 15 and 16
Question 1
Two cars, one behind the other, are traveling in the same direction at the same speed. Does either
driver hear the other’s horn at a frequency that is different from that heard when both cars are at
rest?

Question 2
Refer to the figure below. As you walk along a line that is perpendicular to the line between the
speakers and passes through the overlap point, you do not observe the loudness to change from
loud to faint to loud. However, as you walk along a line through the overlap point and parallel to
the line between the speakers, you do observe the loudness to alternate between faint and loud.
Explain why your observations are different in the two cases.
Problem 1
A person lying on an air mattress in the ocean rises and falls through one complete cycle every
five seconds. The crests of the wave causing the motion are 20.0 m apart. Determine (a) the
frequency and (b) the speed of the wave.

Problem 2
The linear density of the A string on a violin is 7.8 × 10-4 kg/m. A wave on the string has a
frequency of 440 Hz and a wavelength of 65 cm. What is the tension in the string?
Problem 3
The middle C string on a piano is under a tension of 944 N. The period and wavelength of a
wave on this string are 3.82 ms and 1.26 m, respectively. Find the linear density of the string.

Problem 4
Two submarines are underwater and approaching each other head-on. Sub A has a speed of 12
m/s and sub B has a speed of 8 m/s. Sub A sends out a 1550 Hz sonar wave that travels at a
speed of 1522 m/s. (a) What is the frequency detected by sub B? (b) Part of the sonar wave is
reflected from B and returns to A. What frequency does A detect for this reflected wave?
Problem 5
The security alarm on a parked car goes off and produces a frequency of 960 Hz. The speed of
sound is 343 m/s. As you drive toward this parked car, pass it, and drive away, you observe the
frequency to change by 95 Hz. At what speed are you driving?

Problem 6
Two loudspeakers emit sound waves along the x-axis. The sound has maximum intensity when
the speakers are 20 cm apart. The sound intensity decreases as the distance between the
speakers is increased, reaching zero at a separation of 30 cm. (a) What is the wavelength of the
sound? (b) If the distance between the speakers continues to increase, at what separation will the
sound intensity again be a maximum?
Problem 7
A pair of in-phase stereo speakers are placed next to each other, 0.60 m apart. You stand directly
in front of one of the speakers, 1.0 m from the speaker. What is the lowest frequency that will
produce constructive interference at your location?

Problem 8
Two out-of-tune flutes play the same note. One produces a tone that has a frequency of 262 Hz,
while the other produces 266 Hz. When a tuning fork is sounded together with the 262-Hz tone,
a beat frequency of 1 Hz is produced. When the same tuning fork is sounded together with the
266 Hz tone, a beat frequency of 3 Hz is produced. What is the frequency of the tuning fork?
Problem 9
A string of length 0.28 m is fixed at both ends. The string is plucked and a standing wave is set
up that is vibrating at its second harmonic. The traveling waves that make up the standing waves
have a speed of 140 m/s. What is the frequency of vibration?

Problem 10
On a cello, the string with the largest linear density (1.56 × 10-2 kg/m) is the C string. The string
produces a fundamental frequency of 65.4 Hz and has a length of 0.800 m between the two fixed
ends. Find the tension in the string.

Problem 11
The figure shows a standing wave oscillating at 100 Hz on a string. What is the wave speed?