Ch 17 Linear Superposition and Interference

Chapter 17Chapter 17The Principle of Linear

Superposition and Interference Phenomena

AP Learning ObjectivesAP Learning ObjectivesWAVES AND OPTICS Wave motion (including sound)

• Standing wavesStudents should understand the physics of standing waves, so they can: Sketch possible standing wave modes for a stretched

string that is fixed at both ends, and determine the amplitude, wavelength, and frequency of such standing waves.

Describe possible standing sound waves in a pipe that has either open or closed ends, and determine the wavelength and frequency of such standing waves.

• SuperpositionStudents should understand the principle of superposition, so they can apply it to traveling waves moving in opposite directions, and describe how a standing wave may be formed by superposition

Table of ContentsTable of Contents

1. The Principle of Linear Superposition

2. Constructive and Destructive Interference of

Sound Waves

3. Diffraction

4. Beats

5. Transverse Standing Waves

6. Longitudinal Standing Waves

7. Complex Sound Waves

Chapter 17:Chapter 17: The Principle of Linear Superposition The Principle of Linear Superposition

and Interference Phenomenaand Interference Phenomena

Section 1:The Principle of Linear

Superposition

When the pulses merge, the Slinkyassumes a shape that is the sum ofthe shapes of the individual pulses.

SuperpositionSuperposition

When the pulses merge, the Slinkyassumes a shape that is the sum ofthe shapes of the individual pulses.

SuperpositionSuperposition

Principle of SuperpositionPrinciple of Superposition

When two or more waves pass a particular point in a medium simultaneously, the resulting displacement at that point in the medium is the sum of the displacements due to each individual wave.

The waves interfere with each other.

17.1.1. The graph shows two waves at time t = 0 s, one moving toward the right at 2.0 cm/s and the other moving toward the left at 2.0 cm/s. What will the amplitude be at x = 0 at time t = 0.5 s?

a) +1 cm

b) zero cm

c) 1 cm

d) 2 cm

e) 3 cm

17.1.2. Two waves are traveling along a string. The graph shows the position of the waves at time t = 0.0 s. One wave with a maximum amplitude of 0.5 cm is traveling toward the right at 0.5 cm/s. The second wave with a maximum amplitude of 2.0 cm is traveling toward the left at 2.0 cm/s. At what elapsed time will the two waves completely overlap and what will the maximum amplitude be at that time?

a) 2.0 s, 1.5 cm

b) 1.3 s, 2.5 cm

c) 1.0 s, 1.5 cm

d) 1.0 s, 2.5 cm

e) 1.3 s, 0.0 cm



Section 2:Constructive and Destructive Interference of Sound Waves

Types of InterferenceTypes of Interference

If the waves are “in phase”, that is crests and

troughs are aligned, the amplitude is increased.

This is called constructive interference.

If the waves are “out off phase”, that is crests

and troughs are completely misaligned, the

amplitude is decreased and can even be zero.

This is called destructive interference.

When two waves always meet condensation-to-condensation and rarefaction-to-rarefaction, they are said to be exactly in phase and to exhibit constructive interference.

Constructive InterferenceConstructive Interference

Constructive InterferenceConstructive Interference

Animation courtesy of Dr. Dan Russell, Kettering University

When two waves always meet condensation-to-rarefaction, they are said to be exactly out of phase and to exhibit destructive interference.

Destructive InterferenceDestructive Interference

Sample ProblemSample Problem

SolutionSolution

Noise Cancelling Noise Cancelling HeadphonesHeadphones

If the wave patters do not shift relative to one another as time passes,the sources are said to be coherent.

For two wave sources vibrating in phase, a difference in path lengths thatis zero or an integer number (1, 2, 3, . . ) of wavelengths leads to constructive interference; a difference in path lengths that is a half-integer number(½ , 1 ½, 2 ½, . .) of wavelengths leads to destructive interference.

Coherent WavesCoherent Waves

Example 1 What Does a Listener Hear?Two in-phase loudspeakers, A and B, are separated by 3.20 m. A listener is stationedat C, which is 2.40 m in front of speaker B.Both speakers are playing identical 214-Hz tones, and the speed of sound is 343 m/s.Does the listener hear a loud sound, or no sound?

Calculate the path length difference.

m 1.60m 40.2m 40.2m 20.3 22

Calculate the wavelength.

m 60.1Hz 214

sm343

f

v

Because the path length difference is equal to an integer (1) number of wavelengths, there is constructive interference, whichmeans there is a loud sound.

Conceptual Example 2 Out-Of-Phase Speakers

To make a speaker operate, two wires must be connected between the speaker and the amplifier. To ensure that the diaphragms of the two speakers vibrate in phase, it is necessary to make these connectionsin exactly the same way. If the wires for onespeaker are not connected just as they are for the other, the diaphragms will vibrateout of phase. Suppose in the connections are made so that the speaker diaphragms vibrate out of phase, everything else remaining the same. In each case, what kind of interference would result in the overlap point?

17.2.1. A radio station has a transmitting tower that transmits its signal (electromagnetic waves) uniformly in all directions on the west end of Main Street. They are considering building a second, identical transmitter at the east end of Main Street, ten miles due east of the first transmitter. The same signal is to be broadcast at the same time from both towers. As you drive ten miles east to west on Main Street, what would you hear as you listen to the radio station broadcast from these two towers?

a) The signal gets stronger as you drive the first five miles, but then the signal decreases as you travel the final five miles.

b) The signal is somewhat stronger than when there was just one tower and there is no variation in signal strength as you drive the ten miles.

c) The signal alternates between increasing strength and decreasing strength as you drive the ten miles.

d) The signal is the same as it was with just one tower. For the first five miles, you receive the signal from the east tower. For the second five miles, you receive the signal from the west tower.

e) To answer this question, one must know the amplitude of the broadcast signal.

17.2.2. A tuning fork, like the one shown in the drawing, is tapped and begins to vibrate. When you place it next to you ear as shown, you can hear a distinctive tone. The dashed lines in the picture indicate possible axes of rotation. Consider each if the five axes shown. About which of these axes can you rotate the tuning fork without producing constructive or destructive interference at the ear as it is rotated?

a) A only

b) B only

c) C only

d) D only

e) D and E only

17.2.3. Two identical speakers are emitting a constant tone that has a wavelength of 0.50 m. Speaker A is located to the left of speaker B. At which of the following locations would complete destructive interference occur?

a) 2.15 m from speaker A and 3.00 m from speaker B

b) 3.75 m from speaker A and 2.50 m from speaker B

c) 2.50 m from speaker A and 1.00 m from speaker B

d) 1.35 m from speaker A and 3.75 m from speaker B

e) 2.00 m from speaker A and 3.00 m from speaker B



Section 3:Diffraction

DiffractionDiffraction

Diffraction is defined as the bending of a wave around a barrier.

Diffraction of waves combined with interference of the diffracted waves causes “Diffraction patterns”.

Double/Multi- Slit DiffractionDouble/Multi- Slit Diffraction

Each opening creates diffraction pattern The diffraction patterns interfere with each other

Picture courtesy of Dr. John U Free, Eastern Nazarene College

Diffraction PatternsDiffraction Patterns m = d sin m: bright band number

• (m = 0 for central) : wavelength (m) d: space between slits (m) : angle defined by

central band, slit, and band n

m=0

m=1

m=2

m=1

m=2 md sin

d

Double Slit ApproximationDouble Slit Approximation

If x << L when measured

in radians

md sin

sin

L

x

mL

xd

d

Lmx

d

L

x

17.3.1. A speaker is located inside a box and emits a constant tone. There is a partition in the box that has a circular opening with the diameter shown. There is only one opening in the box to the outside. That opening is also circular with the diameter shown. A man is slowly walking alongside the box in the direction shown. What does the person hear, if anything, as he passes the outer circular opening? Notes: Not all of the waves are shown and the walls do not absorb any sound.

a) As the man walks along side the box, he hears the constant tone emitted by the speaker and its intensity increases as he is passing the circular opening.

b) As the man walks along side the box, he only hears the constant tone emitted by the speaker when he is front of the circular opening.

c) At no time does the man hear any sound from the speaker.

d) As the man walks along side the box, he hears the tone intensity alternating between itsmaximum and minimum values.

17.3.2. Consider the situation shown below. You are walking north on a street approaching a small marching band that is traveling west to east. The large, shaded rectangles in the drawing represent tall buildings. At the moment shown, which instrument(s) do you hear first?

a) flute (f)

b) snare drum (sd)

c) bass drum (bd)

d) flute (f) and snare drum (sd)



Section 4:Beats

BeatsBeats

“Beats is the word physicists use to describe the characteristic loud-soft pattern that characterizes two nearly (but not exactly) matched frequencies.

Musicians call this “being out of tune” or “bad intonation”.

Let’s hear (and see) a demo of this phenomenon.


The beat frequency is the difference between the two soundfrequencies.

Beat FrequencyBeat Frequency

17.4.1. Two waves, A and B, are superposed. For which one of the following circumstances will beats result?

a) A and B are identical waves traveling in the same direction.

b) A and B are traveling with differing speeds.

c) A and B are identical waves traveling in the opposite directions.

d) A and B are waves with slightly differing frequencies, but otherwise identical.

e) A and B are waves with slightly differing amplitudes, but otherwise identical.

17.4.2. Which of the graphs shown represent the superposition of two different waves with the smallest difference in frequency between the two waves?

a) A

b) B

c) both A and B, since the frequency difference is the same in the two cases

d) This cannot be answered since no frequency information is available.

17.4.3. Consider the following graphs, each showing the result waves from addition of two differing waves. For which graph is the frequency difference between the two original waves the smallest?

a) 1

b) 2

c) 3

d) The frequency difference is the same for graphs 1 and 3 and is the smallest.



Section 5:Transverse Standing Waves

Standing WaveStanding Wave A standing wave is a wave which is reflected back

and forth between fixed ends (of a string or pipe, for example).

Reflection may be fixed or open-ended. Superposition of the wave upon itself results in a

pattern of constructive and destructive interference and an enhanced wave.


In reflecting from the wall, a forward-traveling half-cycle becomes a backward-traveling half-cycle that is inverted.

Unless the timing is right, the newly formed and reflected cycles tend to offset one another.

Repeated reinforcement between newly created and reflected cycles causes a large amplitude standing wave to develop.

Standing wavesStanding waves

Standing Wave PatternsStanding Wave Patterns

Fixed End Standing WavesFixed End Standing Waves

Fundamental (First harmonic)

= 2L

First Overtone (Second harmonic)

= L

Second Overtone (Third harmonic)

= 2L/3

“nth” harmonic

= 2L/n

,4,3,2,1 2

nL

vnfnString fixed at both ends

HarmonicsHarmonics

Standing WavesStanding Waves

Conceptual Example 5 The Frets on a GuitarFrets allow a the player to produce a complete sequence of musical noteson a single string. Starting with the fret at the top of the neck, each successivefret shows where the player should press to get the next note in the sequence.Musicians call the sequence the chromatic scale, and every thirteenth note in it corresponds to one octave, or a doubling of the sound frequency. The spacing between the frets is greatest at the top of the neck and decreases witheach additional fret further on down. Why does the spacing decrease going down the neck?

17.5.1. A transverse standing waves is present on a plucked guitar string. What is the distance from the fixed end of a string to the nearest antinode?

a) λ/4

b) λ/2

c) 2λ/3

d) 3λ/4

e) λ

17.5.2. Which one of the following statements is true concerning the points on a string that sustain a standing wave?

a) All points undergo motion that is purely longitudinal.

b) All points vibrate with the same energy.

c) All points vibrate with different amplitudes.

d) All points undergo the same displacements.

e) All points vibrate with different frequencies.

17.5.3. A rope of length L is clamped at both ends. Which one of the following is not a possible wavelength for standing waves on this rope?

a) L/2

b) 2L/3

c) L

d) 2L

e) 4L

17.5.4. Consider a wire under tension that is driven by an oscillator. Initially, the wire is vibrating in its second harmonic mode. How does the oscillation of the wire change as the frequency is slowly increased?

a) No standing wave may be observed until the frequency matches the third harmonic mode of the wire.

b) No standing wave may be observed until the frequency matches the first harmonic mode of the wire.

c) The observed oscillation of the wire not change until the frequency matches the third harmonic mode of the wire.

d) The observed oscillation of the wire will slowly change in fractions of the harmonic between the second and third harmonic modes.

e) The observed oscillation of the wire will slowly change in fractions of the harmonic between the second and first harmonic modes.

17.5.5. Which one of the following statements explains why a piano and a guitar playing the same musical note sound different?

a) The fundamental frequency is different for each instrument.

b) The two instruments have the same fundamental frequency, but different harmonic frequencies.

c) The two instruments have the same harmonic frequencies, but different fundamental frequencies.

d) The two instruments have the same fundamental frequency and the same harmonic frequencies, but the amounts of each of the harmonics is different for the two instruments..

17.5.6. The sound emitted from a strummed guitar string is either a resonant frequency or one of its harmonics. Although the string is not being driven at its resonant frequency, no non-resonant waves are emitted. Which one of the following statements best describes why non-resonant waves are not heard?

a) Non-resonant waves are not sound waves.

b) The non-resonant waves are too quickly damped out.

c) The musician has tuned the strings so that only resonant waves will occur.

d) Any non-resonant waves will destructively interfere with each other.



Section 6:Longitudinal Standing Waves

A longitudinal standing wave pattern on a slinky.

Longitudinal Standing Longitudinal Standing WavesWaves

Fundamental (First harmonic)

= 2L

First Overtone (Second harmonic)

= L

Second Overtone (Third harmonic)

= 2L/3

“nth” harmonic

= 2L/n

Open-ended Standing Open-ended Standing Waves Waves

,4,3,2,1 2

nL

vnfnTube open at both ends

Open-ended Standing Open-ended Standing Waves Waves

Example 6 Playing a FluteWhen all the holes are closed on one type offlute, the lowest note it can sound is middleC (261.6 Hz). If the speed of sound is 343 m/s,and the flute is assumed to be a cylinder openat both ends, determine the distance L.

,4,3,2,1 2

nL

vnfn

nf

nvL

2

Hz 261.62

sm3431 m 656.0

Mixed Standing WaveMixed Standing Wave

One end open, the other end closed First harmonic

= 4L Second harmonic

= (4/3)L Third harmonic

= (4/5)L “nth” harmonic

= 4L/(n+2)

,5,3,1 4

nL

vnfnTube open at one end

Sample Problem:Sample Problem:

How long do you need to make an organ pipe (open at both ends) that produces a fundamental frequency of middle C (256 Hz)? The speed of the sound in air is 340 m/s.

A) Draw the standing wave for the first harmonic

B) Calculate the pipe length.

Sample ProblemSample Problem

How long do you need to make an organ pipe whose fundamental frequency is a middle C (256 Hz)? The pipe is closed on one end, and the speed of sound in air is 340 m/s.

A) Draw the situation:

B) Calculate the pipe length:

ResonanceResonance Resonance occurs when a vibration from one oscillator occurs

at a natural frequency for another oscillator. The first oscillator will cause the second to vibrate.

Watch:• http://www.youtube.com/watch?

v=zWKiWaiM3Pw&feature=fvw

ResonanceResonance

In some cases, the amplitude will increase until equaled by damping forces.

Watch:• http://www.youtube.com/watch?

v=JUVgouE_sg0&feature=related

17.6.1. A soft drink bottle is 15 cm tall. Joey blows across that top of the bottle just after drinking the last of his drink. What is the approximate fundamental frequency of the tone that Joey generates?

a) 230 Hz

b) 570 Hz

c) 680 Hz

d) 810 Hz

e) 1100 Hz

17.6.2. An aluminum rod of length L may be held at various points along its length, some of which are indicated in the drawing. A small hammer is then used to tap the rod. As a result, longitudinal standing waves are generated in the rod. At which position should the rod be held to generate its second harmonic?

a) A

b) B

c) C

d) D

e) E

17.6.3. Which one of the following statements concerning standing waves within a pipe open only at one end is true?

a) The standing waves have a fundamental mode have a shorter wavelength than that for the same tube with both ends open.

b) The standing waves must be transverse waves, since longitudinal waves could not exit the tube.

c) The standing waves have a greater number of harmonics than which occur for the tube when both ends are open.

d) The standing waves have fewer harmonics than which occur for the tube when both ends are open.

e) The standing waves have a fundamental mode with a smaller frequency than that which occurs when both ends of the tube are open.

17.6.4. Given that the first three resonant frequencies of an organ pipe are 200, 600, and 1000 Hz, what can you conclude about the pipe?

a) The pipe is open at both ends and has a length of 0.95 m.

b) The pipe is closed at one end and has a length of 0.95 m.

c) The pipe is closed at one end and has a length of 0.475 m.

d) The pipe is open at both ends and has a length of 0.475 m.

e) It is not possible to have a pipe with this combination of resonant frequencies.



Section 7:Complex Sound Waves

Pure SoundsPure Sounds

Sounds are longitudinal waves, but if we graph them right, we can make them look like transverse waves.

When we graph the air motion involved in a pure sound tone versus position, we get what looks like a sine or cosine function.• A tuning fork produces a

relatively pure tone. • So does a human whistle.

Complex SoundsComplex Sounds

Because of the phenomena of “superposition”

and “interference” real world waveforms may not

appear to be pure sine or cosine functions.

That is because most real world sounds are

composed of multiple frequencies.

The human voice and most musical instruments

produce complex sounds.

The OscilloscopeThe Oscilloscope

With the Oscilloscope we can view waveforms in the “time domain”. Pure tones will resemble sine or cosine functions, and complex tones will show other repeating patterns that are formed from multiple sine and cosine functions added together.

The Fourier TransformThe Fourier Transform

We can also view waveforms in the “frequency domain”.

A mathematical technique called the Fourier Transform will separate a complex waveform into its component frequencies.

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Ch 17 Linear Superposition and Interference

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