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Ch17. The Principle of Linear Superposition and Interference Phenomena

The Principle of Linear Superposition

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THE PRINCIPLE OF LINEAR SUPERPOSITION

When two or more waves are present simultaneously at the same place, the resultant disturbance is the sum of the disturbances from the individual waves.

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Constructive and Destructive Interference of Sound Waves Reading

content

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

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

In either case, this means that the wave patterns do not shift relative to one another as time passes. Sources that produce waves in this fashion are called coherent sources.

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Destructive interference is the basis of a useful technique for reducing the loudness of undesirable sounds.

For two wave sources vibrating in phase, a difference in path lengths that is 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 ( , , , … ) of wavelengths leads to destructive interference.

 ,   ,   

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Interference effects can also be detected if the two speakers are fixed in position and the listener moves about the room.

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The bending of a wave around an obstacle or the edges of an opening is called diffraction. All kinds of waves exhibit diffraction.

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Example 3.   Designing a Loudspeaker for Wide Dispersion

A 1500-Hz sound and a 8500-Hz sound each emerges from a loudspeaker through a circular opening whose diameter is 0.30 m . Assuming that the speed of sound in air is 343 m/s, find the diffraction angle q  for each sound.

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Beats

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The number of times per second that the loudness rises and falls is the beat frequency and is the difference between the two sound frequencies.

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A 10-Hz sound wave and a 12-Hz sound wave, when added together, produce a wave with a beat frequency of 2 Hz. The drawings show the pressure patterns (in blue) of the individual waves and the pressure pattern (in red) that results when the two overlap. The time interval shown is one second.

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Check Your Understanding 2 A tuning fork of unknown frequency and a tuning fork of frequency of 384 Hz produce 6 beats in 2 seconds. When a small piece of putty is attached to the tuning fork of unknown frequency, the beat frequency decreases. What is the frequency of that tuning fork?

387 HZ

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Transverse Standing Waves

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The antinodes are places where maximum vibration occurs.

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The nodes are places that do not vibrate at all.

The frequencies in this series (f1, 2f1, 3f1, etc.) are called harmonics.

Frequencies above the fundamental are overtones.

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Standing waves arise because identical waves travel on the string in opposite directions and combine in accord with the principle of linear superposition. A standing wave is said to be standing because it does not travel in one direction or the other, as do the individual waves that produce it.

f1 = v/(2L)

orfnn =

v = v

fn(2L/n) = v

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Check Your Understanding 3 A standing wave that corresponds to the fourth harmonic is set up on a string that is fixed at both ends. (a) How many loops are in this standing wave? (b) How many nodes (excluding the nodes at the ends of the string) does this standing wave have? (c) Is there a node or an antinode at the midpoint of the string? (d) If the frequency of this standing wave is 440 Hz, what is the frequency of the lowest-frequency standing wave that could be set up on this string?

(a) 4, (b) 3, (c) node, (d) 110 HZ

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Longitudinal Standing Waves

Standing wave patterns can also be formed from longitudinal waves.

Possible Waves for Open Pipe

L 2

1

L Fundamental, n = 1

2

2

L 1st Overtone, n = 2

2

3

L 2nd Overtone, n = 3

2

4

L 3rd Overtone, n = 4

All harmonics are possible for open pipes:

2 1, 2, 3, 4 . . .n

Ln

n

Characteristic Frequencies for an Open Pipe.

L 1

2

vf

LFundamental, n = 1

2

2

vf

L1st Overtone, n = 2

3

2

vf

L2nd Overtone, n = 3

4

2

vf

L3rd Overtone, n = 4

All harmonics are possible for open pipes:

1, 2, 3, 4 . . .2n

nvf n

L

Possible Waves for Closed Pipe.

1

4

1

L Fundamental, n = 1

1

4

3

L 1st Overtone, n = 3

1

4

5

L 2nd Overtone, n = 5

1

4

7

L 3rd Overtone, n = 7

Only the odd harmonics are allowed:

4 1, 3, 5, 7 . . .n

Ln

n

L

Possible Waves for Closed Pipe.

1

1

4

vf

LFundamental, n = 1

3

3

4

vf

L1st Overtone, n = 3

5

5

4

vf

L2nd Overtone, n = 5

7

7

4

vf

L3rd Overtone, n = 7

Only the odd harmonics are allowed:

L

1, 3, 5, 7 . . .4n

nvf n

L

Example 4. What length of closed pipe is needed to resonate with a fundamental frequency of 256 Hz? What is the second overtone? Assume that

the velocity of sound is 340 M/s.

Closed pipe

AN

L = ?

1, 3, 5, 7 . . .4n

nvf n

L

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(1) 340 m/s;

4 4 4(256 Hz)

v vf L

L f L = 33.2 cm

The second overtone occurs when n = 5:

f5 = 5f1 = 5(256 Hz) 2nd Ovt. = 1280 Hz2nd Ovt. = 1280 Hz

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Problem 50

0 2 4 6

t = 1 s

t = 2 s

t = 3 s

t = 4 s

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