General Physics I

Spring 2011

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Superposition and Standing Waves

Superposition• Two solid objects cannot occupy

the same space at the same time. (The outer electrons repel each other, so you cannot have interpenetration of the objects.) However, two (or more) wavescan occupy the same space at the same time. When two waves are at the same place

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are at the same place simultaneously, the total displacement of the medium at a given point in space is the sum of the displacements that each wave would cause by itself at that that point. This phenomenon is called the principle ofsuperposition.

Waves pass through each other.

Constructive and Destructive Interference• Superposition is a general

principle that applies to many

phenomena, not just waves. In

the case of waves, however,

superposition is given a special

name: interference. Thus,

interference is simply the

superposition of waves.

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• If the superposition of two waves

leads to a displacement of the

medium at each point in space

that is greater in magnitude than

the individual displacement due

to either wave by itself, one has

constructive interference.

Where these waves overlap,

there is constructive

interference because the

total displacement is

greater than the individual

displacement due to either wave

by itself.

Constructive and Destructive Interference• If the superposition of two

waves leads to a displacement

of the medium at each point in

space that is smaller in

magnitude than the individual

displacement due to either

wave by itself, one has

destructive interference.

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destructive interference.

Where these waves overlap, there is

destructive interference because the

total displacement is smaller than the

individual displacement due to either

wave by itself.

Workbook: Chapter 16, Questions 1, 2

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Superposition of Two Waves Traveling in

Opposite Directions• Consider two traveling waves

having the same amplitude and wavelength that are moving in opposite directions in a medium. In the region of space where the two waves interfere, the overall disturbance does not travel in one direction or the other. The No oscillation at these

Largest-amplitude

oscillation at these points

1

2

3

5

4

1

1

1

2

2

2

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one direction or the other. The particles of the medium simply oscillate continuously. There are points at which the amplitude of oscillation is large. There are also points at which there is no oscillation at all. This type of wave disturbance is called a standing wave.

Standing wave(The sketch shows the

displacement of the medium

at five different times, indicated

by corresponding numbers.)

No oscillation at these

points

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Standing Waves• The behavior of the standing wave at a given point in space

as time progresses can be understood by superposing

(adding) the displacement of each wave at that point at

different times.

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Standing Waves• The points in a standing wave

that do not oscillate are called

nodes.

• The points in a standing wave

that oscillate with maximum

amplitude are antinodes.

• The distance between two

successive nodes (or two

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successive nodes (or two

successive antinodes) is equal

to one-half of the wavelength of

the standing wave. The

wavelength of the standing

wave is equal to that of the

individual traveling waves

whose interference creates the

standing wave.Standing Water Waves

Workbook: Chapter 16, Question 3

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Standing Waves on a String

• The musical notes that emanate

from stringed instruments are

initially produced by standing

waves on the strings. The vibrating

strings produce sound waves that

are amplified by a sounding board

or electronic amplifier. These

waves are what we hear.

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waves are what we hear.

• We will focus on the standing

waves generated on a string.

These standing waves are also

produced by two waves traveling

in opposite directions. The

oppositely directed waves are due

to reflections at the ends of the

string.

A wave pulse being reflected.

Standing Waves on a String

• A wave traveling in a medium is

generally reflected when it

encounters a boundary with a

second medium. If the wave

traveling on a string encounters a

boundary with a heavier string, the

wave is inverted upon reflection.

The same is true when a wave is

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The same is true when a wave is

reflected at a boundary where the

end of the string is fixed.

• The reason is that when the wave

reaches the boundary and exerts

an upward force on it, the fixed

boundary exerts a downward force

on the string (action/reaction!),

which inverts the reflected wave.

Standing Waves on a String

• If the wave traveling on a string

encounters a boundary with a

lighter string, the wave is not

inverted upon reflection. The

same is true when a wave is

reflected at a boundary where

the end of the string is free.

• The reason is that when the

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• The reason is that when the

wave reaches the boundary and

exerts an upward force on it, the

string at the boundary is free to

move and therefore moves in

the direction of the force. It

follows that the reflected wave

must have the same orientation

as the incoming wave.

Standing Waves on a String Fixed at Both

Ends

• If a string fixed at both ends is plucked in the middle, waves move outward from the middle and are eventually reflected at each end. The forward-going and reflected waves eventually interfere with each other and

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interfere with each other and will produce a standing wave only for certain wavelengths. The allowed wavelengths are those that give standing waves that fit an integer number of half-wavelengths on the string, with the ends always being nodes.

Standing Waves on a String Fixed at Both

Ends• The allowed standing waves are

called the resonant modes of the

string. The longest wavelength

mode corresponds to two nodes

at the ends and an antinode in the

middle. Since the distance

between two successive nodes is

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between two successive nodes is

a half wavelength, it follows that

the length of the string is equal to

half of a wavelength, i.e.,

The second-longest wavelength

mode has a node in the middle

and two antinodes on either side

(as well as the nodes at the ends).

For this mode,

12 .Lλ =

2.Lλ =

Standing Waves on a String Fixed at Both

Ends• Continuing in this way, we find that the wavelengths of the

allowed standing wave modes for a string fixed at both ends is given by

• The frequency of a given mode is calculated from f = v/λ:

2 . 1,2,3...mL mmλ = =

. 1,2,3,...v vf m m

= = =

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• Note that

where f1 is the lowest allowed frequency, called the fundamental frequency. All allowed frequencies are integer multiples of the fundamental. The fundamental frequency is also known as the first harmonic (m = 1). The m = 2 mode is called the second harmonic and so forth. Collectively, modes with m > 1 are called higher harmonics.

. 1,2,3,...2m

m

v vf m mLλ

= = =

,1 1,2,3...mf mf m= =

Stringed Musical Instruments

• The pitch of a note from a stringed instrument corresponds to

the fundamental frequency. The quality or timber of the sound

depends on the amplitudes of the higher harmonics.

• To tune a string, you change the fundamental frequency to the

desired value. This is done by changing the tension in the

string. Recall that the wave speed in a string depends on the tension Ts as well as the mass per unit length µ: .sv T µ=

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Thus, the fundamental frequency for a string is given bys

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1 .2

sTvfL µλ

= =

Workbook: Chapter 16, Questions 5, 6

Textbook: Chapter 16, Problems 13, 14, 44,

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