A. La Rosa Lecture Notes

PH-213 GENERAL PHYSICS ________________________________________________________________________

CHAPTER 16: WAVES-1 16.2. Interference

standing waves and resonance Interference of travelling waves

Case 1 Waves travelling in the same direction Case 1A Interference between waves of the same

frequency Case 1B Interference between waves of slightly different

frequency (This case will be done in Chapter 17) Case 1C Coherent waves

Case 2 Waves travelling in opposite direction Standing waves Resonance of waves in a string

Case 2A String attached at one end Intuitive solutions, analytical solutions Case 2B String attached at both ends Intuitive solutions, analytical solutions

CASE-1 Waves travelling in the same directionCASE-1A Interference between two harmonic waves of the same frequency w (therefore, the same k) but different phase

Δφ

Δx Δφ = k Δx

Interference of travelling waves

Interference (synonymous of addition of waves)

Interference

factor

Same phase

Peaks and valleys coincide

Example: Full constructive interference case ( φ = 0 )

different

phase

Particular case: Δx =λ/2

Example: Destructive interference case ( φ = π )

Δx

(2)

(1)

Δx

P

Q

B

P and Q are in phase

A and B are out of phase by (k) (2 Δx)

A

Example: Controlling the degree of interference ( 0 < φ < 2π )

Mirror

Mirror

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Δx

Δx

Before the reflection

After the reflection

draw

Lens

FIGURE-1

FIGURE-2

A

B

B

A

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CASE-1B Interference between waves of slightly different frequency (This case will be done in Chapter 17)

CASE-1C Coherent waves

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CASE-2 Waves travelling in the opposite directions

Standing Waves

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and this is what happens:

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f1

f2

f3

f4

f5

Fig. Solutions obtained by intuition

Resonance of waves in a string

Case 2A String fixed at one end

Intuitive solutions

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For the case when one end of the string is fixed

2

2 2 2

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Analytical solution

General solution.But we have to impose the particular boundary conditions of the problem (i.e. that the string is fixed at one end)

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satisfy

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L

Case 2B String fixed at both ends

Intuitive solutions

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Analytical solution

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This is a generalization of the example (to be worked out in the next page) for the case of having jut two particles

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