Physics 102 SuperpositionPhysics 102 Superposition

Moza M. Al-RabbanProfessor of Physics

mmr@qu.edu.qa

Lecture 6Lecture 6

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The Goal of this chapter is to understand and use the idea of superposition

The Goal of this chapter is to understand and use the idea of superposition

• Apply the principle of superposition.• Understand how standing waves are generated.• Calculate the allowed wavelengths and

frequencies of standing waves.• Understand how waves cause constructive and

destructive interference.• Calculate the beat frequency between two nearly

equal frequencies.

•

3

Waves vs. ParticlesWaves vs. Particles

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Colliding WavesColliding WavesThe Principle of Superposition

The Principle of Superposition

When two or more waves are simultaneously present at a single point in space, the displacement of the medium at that point is the sum of the displacements due to each individual wave.

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The Principle of Superposition

The Principle of Superposition

When two or more waves are simultaneously present at a single point in space, the displacement of the medium at that point is the sum of the displacements due to each individual wave.

i

inet DDDD 21

6

Stop To ThinkStop To Think

Two pulses on a string approach each other at speed of 1 m/s. What is the shape of the string at t= 6 s?

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Standing WavesStanding Waves

A standing wave is the superposition of two A standing wave is the superposition of two waves.waves.

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Standing WavesStanding Waves

We will assume that the two waves have the same frequency, the same wavelength, and the same amplitude.

They are identical waves except that one travel to the right and the other to the left.

What happens as these two waves pass through each other?

The blue dot shows that the wave in (b) is moving neither right nor left.

This is a wave, but it is not traveling wave. It is a standing wave.

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Nodes and AntinodesNodes and Antinodes

The points that never move! Spaced by /2, are called nodes.

Halfway between nodes are points where the particle of medium oscillate with maximum displacement. These points of maximum amplitude are called antinodes, and they are also spaced /2 apart.

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Two waves 1 and 2 are said to be in phase at a point where D1 is always equal to D2.

This is called a point of constructive interference.

The antinodes of standing wave are points of constructive interference between the two traveling waves.

In contrast, two waves are said to be out of phase at points where D1 is always equal to –D2.

Their superposition gives a wave with zero amplitude== no wave at all!

This is a point of destructive interference.

The nodes of a standing wave are points of destructive interference.

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2CAI The points of maximum intensity occur where the standing wave oscillates with the largest amplitude (the antinodes) and that the intensity is zero at the nodes.

If this is a sound wave, the loudness periodically varies from zero( no sound) to maximum and back to zero.

The key idea is that the intensity is maximum at points of constructive interference and zero ( if the waves have equal amplitude) at points of destructive interference.

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The Mathematics of Standing The Mathematics of Standing WavesWaves

The Mathematics of Standing The Mathematics of Standing WavesWaves

tkxatkxaDDtxD LR sinsin,

tkxaDR sin

A sinusoidal wave traveling top the right along the x-axis

An equivalent wave traveling to the left is

tkxaDL sin

sincoscossinsin

tkxa

tkxtkxatkxtkxatxD

cos)sin2(

)sincoscos(sin)sincoscos(sin,

txAtxD cos)(),( Where the amplitude function A(x) is defined as

kxaxA sin2)(

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NotesNotes

txAtxD cos)(),(

kxaxA sin2)( The amplitude reaches a maximum value Amax = 2a at points where sin kx =1.

The displacement is neither a function of (x-vt) or (x+vt) , hence it is not a traveling wave.

The cos t , describes a medium in which each point oscillates in simple harmonic motion with frequency f= /2.

The function A(x) =2a sin kx determines the amplitude of the oscillation for a particle at position x.

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txAtxD cos)(),(

The amplitude of oscillation, given by A(x), varies from point to point in the medium.

The nodes of the standing wave are the points at which the amplitude is zero. They are located at positions x for which

0sin2)( kxaxA

That is true if ,3,2,1,02

mmx

kx mm

Thus the position xm of the mth node is

,3,2,1,02

mmxm

Where m is an integer.

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Example 1: Node spacing on a string

A very long string has a linear density of 5.0 g/m and is stretched with a tension of 8.0 N. 100 Hz waves with amplitudes of 2.0 mm are generated at the ends of string.

a. What is the node spacing along the resulting standing wave?

b. What is the maximum displacement of the string?

MODEL Two counter-propagation waves of equal frequency create a standing wave.SOLVE

a. The speed of the waves on the string is smmkg

NTv s /40

/0050.0

0.8

And thus the wavelength is

cmmHzsm

fv

4040.0100

/40

Consequently, the spacing between adjacent nodes is /2 = 20 cm.

b. The maximum displacement, at the antinodes where sin kx = 1, is

mmaA 0.42max

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Reflection and TransmissionReflection and Transmission When a traveling wave moving on a string reaches a point where the mass density ( and velocity) change, the wave “splits” into a reflected wave and an ongoing wave. If the mass density goes down and the velocity increases, the amplitude of the reflected wave is positive. If the mass density goes up and the velocity decreases, the amplitude of the reflected wave is negative.

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Reflection from a BoundaryReflection from a Boundary When a traveling wave moving on a string reaches a fixed boundary, the wave is reflected.

Because there is no transmitted wave, all the wave’s energy is reflected. Hence, the amplitude of a wave reflected from a boundary is unchanged.

With respect to the incident wave, the amplitude of the reflected wave is equal in magnitude and opposite sign.

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Standing Waves on a StringStanding Waves on a String

If a string is plucked at the center, traveling waves move in both directions, are reflected at the boundaries, and a standing wave results.

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Boundary ConditionBoundary Condition• A boundary condition is a mathematical

statements of any constraint that must be obeyed at the boundary or edge of a medium.

Because the string is tied down at the ends, the displacement at x = 0 and x = L must be zero at all times.

Thus the standing-wave boundary conditions are

D(x = 0,t)=0 and D(x = L,t)=0 . Or,

we require nodes at both ends of the string.

But,

tkxatxD cos)sin2(,

It satisfies the boundary condition D(x = 0,t)=0

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The second boundary condition, at x = L, requires

This condition will be met at all times if

2a sin kL = 0 ( boundary condition at x = L)

This will be true if sin kL =0, i.e.

D(x = L,t)=0

,3,2,12

mmL

kL

kL must be multiple of m, but m=0 is excluded because L can’t be zero.

For a string of fixed length L, the only quantity in this equation that can be vary is . That is, the boundary condition can be satisfied only if the wavelength has one of the values

,3,2,12

mmL

m

A standing wave can exist on the string only if its wavelength is one of the values given by this equation.

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Standing Wave Normal ModesStanding Wave Normal Modes

,3,2,12

mmL

m

; 1, 2,3,4,2 / 2m

m

v v vf m m

L m L

The lowest allowed frequency which correspond to wavelength 1:

Lv

f21 Fundamental Fundamental

frequencyfrequency

The allowed standing wave frequencies are all integer multiples of fundamental frequency.

The higher frequency standing waves are called harmonicsharmonics, with m=2 wave called second harmonic, the m=3 wave called third harmonic, and so on.

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Standing Wave Normal ModesStanding Wave Normal ModesHere are the first four possible standing waves on a string of fixed length L.

The possible standing waves are called the normal modesnormal modes of the string.

Each mode, numbered by the integer m, has a unique wavelength and frequency.

Keep in mind that

These drawings simply show the envelope, or outer edge, of oscillations. The string is continuously oscillating at all positions between these edges.

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Notes about the normal modes of a stringNotes about the normal modes of a string

• m is the number of antinodes on the standing wave, not the number of nodes.

• The fundamental mode, with m=1, has 1 = 2L, not 1 =L.

• The frequencies of the normal modes form an arithmetic series:

The fundamental frequency can be found as the difference between the frequencies of any two adjacent modes.

,4,3,2, 1111 ffff

1f

mm ffff 11

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Example 2: a standing wave on a string

A 2.50 m long string vibrates as a 100 Hz standing wave with nodes 1.00 m and 1.50 m from one end of the string and at no points in between these two. Which harmonic is this, and what is the string’s fundamental frequency?

Model: The nodes of a standing wave are spaced /2 apart.

SOLVE:

If there are no nodes between the two at 1.0 m and 1.5 m. then the node spacing is /2 =0.50 m.

The number of 1.50 m wide segments that fit into a 2.50 m length is five, so this is m=5 mode and 100 Hz is the fifth harmonic.

The harmonic frequencies are

Hence, the fundamental frequency is

1mffm

Hzf

f 205

51

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Clicker Question 2Clicker Question 2

A standing wave on a string vibrates as shown.

If the tension is quadrupled while the frequency and length remain the same, which diagram represents the new vibration?

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Standing Electromagnetic Standing Electromagnetic WavesWaves

Standing Electromagnetic Standing Electromagnetic WavesWaves

Another example of a standing wave are the electromagnetic waves in a laser cavity that is bounded by two reflecting mirrors.

Suppose that a laser cavity has length L=30 cm containing visible light with wavelength =600 nm. Then the mode number m is:

-7

2 2(0.3 m)1,000,000

(6.0 10 m)

Lm

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Example:Standing Light Wave in a Laser

Example:Standing Light Wave in a Laser

A helium-neon laser emits red light of = 632.9924 nm. The distance between the laser mirrors is 310.372 mm.

(a) In what standing-wave mode does the laser operate?

(b) What is the next longest wavelength that could make standing waves in the laser cavity?

2 /L m

2 2(310.372 mm)

(632.9924 nm)

980,650

Lm

' 1 980,649m m

2'

1 1 1 1/(1 1/ ) ( / )

632.9924 nm 0.00064 nm

632.9930 nm

L m

m m mm m

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Example:Cold Spots in a Microwave Oven

Example:Cold Spots in a Microwave Oven

“Cold spots”, i.e. locations where objects are not adequately heated in a microwave oven are found to be 1.25 cm apart.

What is the frequency of the microwaves?

node 1.25 cm / 2 so 2.50 cmd

810(3.00 10 m/s)

1.20 10 Hz 12.0 GHz(0.0250 m)

cf

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Standing Sound Waves and Musical Standing Sound Waves and Musical AcousticsAcoustics

Standing Sound Waves and Musical Standing Sound Waves and Musical AcousticsAcoustics

A long narrow column of air such as the air in a tube or pipe can support a longitudinal standing sound wave. Such a tube may be open or closed at the ends. The closed end of a column of air must be a node. An open end of an air column is required to be an antinode.

In the example shown here, both ends are closed and the standing wave mode is m=2. There are nodes at each end and one in the center, and there are two antinodes at the quarter wave locations.

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Example 5: Singing in the ShowerA shower stall is 2.45 m tall. For what frequencies less than 500 Hz can there be vertical standing sound waves in the shower stall?

1

(343 m/s)70 Hz

2 2(2.45 m)

vf

L

70 Hz, 140 Hz, 210 Hz, 280 Hz, 350 Hz, 420 Hz, and 490 Hz,mf

1, 2, 3, 4, 5, 6, and 7m

Model: The shower stall, at least to a first approximation, is a column of air 2.45 m long. It is closed at the ends by the ceiling and floor. Assume a room temperature speed of sound.

The possible standing wave frequencies are integer multiples of the fundamental frequency.

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Pipes and ModesPipes and Modes

1

1

2/

1,2,3,4,

2

m

m

Lm

m mv

f m mfL

1

1

4/

1,3,5,7,

4

m

m

Lm

m mv

f m mfL

Open-Open or Closed-Closed Open-Closed

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Clicker Question 2Clicker Question 2 An open-open tube of air supports standing waves of frequencies of 300 Hz and 400 Hz, with no frequencies between these two.

The second harmonic (m=2) of this tube has frequency:

(a) 100 Hz; (b) 200 Hz; (c) 400 Hz; (d) 600 Hz; (e) 800 Hz.

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Musical InstrumentsMusical Instruments

The vibrating string of a stringed instrument is the equivalent of a closed-closed pipe. This means it will have both odd and even harmonics. Its fundamental frequency is:

string1

1

2 2

v Tsf

L L

Many woodwind instruments are effectively an open-closed pipe. This means they have only odd harmonics. Their fundamental frequency will be:

sound1 4

vf

L

Note that for wind instruments, L is the only adjustable parameter, while for stringed instruments, L, Ts and can, in principle, be varied. However, wind instruments can be played at relatively pure harmonic frequencies, while strings cannot.

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Example:The Notes of a Clarinet

Example:The Notes of a Clarinet

A clarinet (an open-closed instrument) is 66 cm long. The speed of sound in warm air is 350 m/s.

What are the frequencies of the lowest note on a clarinet and of the next highest harmonic?

1

(350 m/s)133 Hz

4 4(0.66 m)

vf

L

3 13 399 Hzf f

End of Lecture 6End of Lecture 6