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Physics 2Class 16

Wave superposition, interference, and reflection

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Adding waves: superposition

When two waves are incident on the same place at the same time, their amplitudes can usually just be added.

In the next few slides we will look at some special cases: (both with two waves at the same frequency and with the same amplitude)– same frequency and amplitude

» superposition at a point in space» same direction, but different phase» opposite directions

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Adding wavesIn most simple systems, the amplitude of two waves that cross or overlap can just be added together.

1 2( , ) ( , ) ( , )y x t y x t y x t

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Adding two equal amplitude waves at a point

1 2sin( ) sin( )

2 cos sin2 2

.

Let and

then*:

amplitude oscillating part

The overall amplitude depends only on

sum

y A t y A t

y A t

-sin sin 2cos( )sin( )2 2

*We used the relation: A B A BA B

(law of cosines)

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iClicker check 16.1

For which phase difference is the superposition amplitude a maximum?

a) =0b) =/2c) =

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iClicker check 16.2

For which phase difference is the superposition amplitude a minimum?

a) =0b) =/2c) =

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Adding two waves that have the same frequency and direction, but different phase

1 2

1 2

( , ) ( , ) ( , )sin( ) sin( )

2 cos sin( )2 2

Let and

then*:

sum

sum

y x t y x t y x ty A kx t y A kx t

y A kx t

• What happens when the phase difference is 0?• What happens when the phase difference is /2?• What happens when the phase difference is ?

Three cases:=0=

2=3

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Constructive interference when =0.Destructive interference when =.

Link to animation of two sine waves adding in and out of phase (Kettering)

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What happens when a wave is incident on an immovable object?It reflects, travelling in the opposite direction

and upside down.

(Now watch patiently while your instructor plays with the demonstration.)

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Same frequency, opposite directions

1

2

sin( ); ( )sin( ); ( )

2 sin cos ; ( ! ( ))standingsum

y A kx t right travellingy A kx t left travelling

y A kx t not travelling

• What happens when kx is 0?• What happens when kx is /2?• What happens when kx is ?

Link to animation of two sine waves traveling in opposite directions

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Standing waves

Interfering waves traveling in opposite directions can produce fixed points called nodes.

y1 = ym sin(kx – t) y2 = ym sin(kx + t)

v=/k yT = y1 + y2 = 2ym cos(t) sin(kx)yT=0 when kx = 0, , 2...yT(t)=maximum when kx = /2, 3/2, 5/2 ...

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Standing waves - ends fixed

Amplitude will resonate when an integer number of half-wavelengths fit in the opening.

Example: violin

Fundamental mode (1st harmonic, n = 1)

2nd harmonic, n = 25th harmonic, n = 5

𝑛 𝜆2 =𝐿

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Both Ends Fixed

2,2

2

2

n

n

LL nn

v vfLnvf nL

𝑛=1,2,3…

𝑛=1,2,3 ,…

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=4L/n, n odd

Fundamental mode (1st harmonic), n = 1

3rd harmonic, n = 39th harmonic, n = 9

Standing waves - one end free

Free end will be an anti-node at resonance.

Demo: spring (slinky) with one end free.

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One End Open (e.g. Organ Pipe)

4,4

1,3,5,7,...

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

where

n

n

LL nn

nv vf

Ln

vf n nL

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Standing waves - both ends free

2L/n5th harmonic, n = 5

2nd harmonic, n = 2

Fundamental mode (1st harmonic), n = 1

Example: wind instrument

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Both Ends Open (e.g. Organ Pipe)

L n Ln

n

f v vLn

f n vL

n

n

n

22

1 23 4

2

21 23

,

, , , , .. .

, , , .. .

wh e re

2,2

1,2,3,4,...

2

1,2,3,...2

where

n

n

LL nn

nv vf

Ln

vf n nL