Waves Interference, Schemes and Mind Maps of Particle Physics

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Waves

Interference

Lana Sheridan

De Anza College

May 22, 2020

Last time

• (^) transverse speed of an element of the medium
• (^) energy transfer by a sine wave

Rate of Energy Transfer in Sine Wave

dK =

μ dx A^2 ω^2 cos^2 (kx − ωt)

dU =

μA^2 ω^2 cos^2 (kx − ωt) dx

Adding dU + dK gives

dE = μω^2 A^2 cos^2 (kx − ωt) dx

Integrating over one wavelength gives the energy per wavelength:

Rate of Energy Transfer in Sine Wave

dK =

μ dx A^2 ω^2 cos^2 (kx − ωt)

dU =

μA^2 ω^2 cos^2 (kx − ωt) dx

Adding dU + dK gives

dE = μω^2 A^2 cos^2 (kx − ωt) dx

Integrating over one wavelength gives the energy per wavelength:

Eλ = μω^2 A^2

∫ (^) λ

0

cos^2 (kx − ωt) dx

= μω^2 A^2

λ 2

Question

Quick Quiz 16.5^1 Which of the following, taken by itself, would be most effective in increasing the rate at which energy is transferred by a wave traveling along a string?

(A) reducing the linear mass density of the string by one half (B) doubling the wavelength of the wave (C) doubling the tension in the string (D) doubling the amplitude of the wave

(^1) Serway & Jewett, page 496.

Question

Quick Quiz 16.5^1 Which of the following, taken by itself, would be most effective in increasing the rate at which energy is transferred by a wave traveling along a string?

(A) reducing the linear mass density of the string by one half (B) doubling the wavelength of the wave (C) doubling the tension in the string

(D) doubling the amplitude of the wave ←

(^1) Serway & Jewett, page 496.

Interference of Waves (Reminder from Lab)

Waves that exist at the same time in the same position in space add together.

superposition principle If two or more traveling waves are moving through a medium, the resultant value of the wave function at any point is the algebraic sum of the values of the wave functions of the individual waves.

This works because the wave equation we are studying is linear.

This means solutions to the wave equations can be added:

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

y is the resultant wave function.

Interference of Waves: Constructive Interference

b

a

y 1 y 2

y 1 y 2

!

When the pulses overlap, the

wave function is the sum of

the individual wave functions.

When the crests of the two

pulses align, the amplitude is

the sum of the individual

W

wa

th

W

pu

b

a

b

c

y 1 y 2

y 1 y 2

!

When the pulses overlap, the

wave function is the sum of

the individual wave functions.

When the crests of the two

pulses align, the amplitude is

the sum of the individual

amplitudes.

W

wa

th

W

pu

th

ind

b

c

b

c

y 1 y 2

y 1! y 2

!

When the crests of the two

pulses align, the amplitude is

the sum of the individual

amplitudes.

When the pulses no longer

overlap, they have not been

permanently affected by the

interference.

W

pu

th

in

W

ov

pe

in

b

c

c

d

y 2 y 1

y 1! y 2

pulses align, the amplitude is

the sum of the individual

amplitudes.

When the pulses no longer

overlap, they have not been

permanently affected by the

interference.

W

pu

th

in

W

ov

pe

in

c

d

Superposition of Sine Waves

Consider two sine waves with the same wavelength and amplitude, but different phases, that interfere.

y 1 (x, t) = A sin(kx − ωt) y 2 (x, t) = A sin(kx − ωt + φ)

Add them together to find the resultant wave function, using the identity:

sin θ + sin ψ = 2 cos

θ − ψ 2

sin

θ + ψ 2

Then y (x, t) =

[

2 A cos

φ 2

)]

sin(kx − ωt +

φ 2

New amplitude Sine oscillation

Dependence on Phase Difference

The amplitude of the resultant wave is A′^ = 2 A cos

φ 2

, where φ is the phase difference.

For what value of φ is A′^ maximized?

Dependence on Phase Difference

If φ = π, −π, 3π, − 3 π, etc. A′^ = 0. Destructive interference.

position and Standing Waves

y

x

x

x

y

y 1 y (^2) y

y y^ y (^1) y 2

! 60°

y

f

f! 180°

f! 0°

The individual waves are in phase and therefore indistinguishable.

Constructive interference: the amplitudes add.

The individual waves are 180° out of phase.

Destructive interference: the waves cancel.

This intermediate result is neither constructive nor destructive.

b

c

a

Interference of Two Sine Waves (equal wavelength)

y (x, t) =

[

2 A cos

φ 2

)]

sin(kx − ωt +

φ 2

x

x

y y

y y^

y 1

y 2

f! 60°

f! 180°

b

c

Summary

• energy transfer by a sine wave
• interference

Homework

• (^) WebAssign due Tuesday night