Interference and Doppler Effect

Question 1

Interference of Travelling Waves
2 harmonic waves, equal amplitude and frequency
Description of Resultant Wave

Answer 1

-resulting wave from the superposition of these two waves is also a harmonic travelling wave with the same frequency and wavelength as the two original waves

Question 2

Interference of Travelling Waves
2 harmonic waves, equal amplitude and frequency
Equation

Answer 2

resulting wave function:
y = 2A cos(Φ/2) sin(kx-ωt+Φ)

amplitude = 2A cos(Φ/2)
where A = amplitude of original waves and Φ = phase difference

Question 3

Interference of Travelling Waves
2 harmonic waves, equal amplitude and frequency
Constructive Interference

Answer 3

Φ = 0 or any even multiple of π

the two waves are in phase and interfere constructively

Question 4

Interference of Travelling Waves
2 harmonic waves, equal amplitude and frequency
Destructive Interference

Answer 4

Φ = any odd multiple of π

the two waves are in antiphase and interfere destructively

Question 5

Waves From Two Sources

Pressure Equation

Answer 5

Wave 1 is a distance r1 from ear:
P1 = P0 sin (k*r1 - ωt)
Wave 2 is a distance r2 from ear:
P2 = P0 sin k*r2 - ωt)
Pressure at Ear
P = P1 + P2 = P0 [ sin (k*r1 - ωt) + sin k*r2 - ωt)]

Question 6

Waves From Two Sources

Phase Difference

Answer 6

Φ = kr1 - kr2 = 2πΔ / λ

Question 7

Waves From Two Sources

Path Difference

Answer 7

Δ = r1 - r2

Question 8

Constructive Interference

Definition

Answer 8

maximum sound heard when path difference is 0 or an integral number of wavelengths

Question 9

Destructive Interference

Definition

Answer 9

non sound is hears, occurs when path difference is a half integral number of wavelengths

Question 10

Coherent

Definition

Answer 10

two sound sources are coherent if their phase distance is consistent, i.e. not random

Question 11

When do beats occur?

Answer 11

-occurs when 2 harmonic waves have slightly different frequency

Question 12

Beats

Equation

Answer 12

y = y1 + y2
y = 2A cos [2πt((f1-f2)/2)] cos[2πt((f1+f2)/2)]

Question 13

Beats

Equation Explained

Answer 13

y = 2A cos [2πt((f1-f2)/2)] cos[2πt((f1+f2)/2)]

Slow Amplitude Modulation, frequency of change in amplitude = f1-f2 / 2
Rapidly oscillating carrier wave, frequency of resultant wave = f1+f2 / 2

Question 14

When is a beat heard?

Answer 14

-when amplitude is maximum or minimum, twice every cycle (at frequency of amplitude modulation)

Question 15

Intensity of Two Incoherent Sources

Answer 15

-sources are incoherent so there is no interference
-intensity is isotropic
I = I1 + I2

Question 16

Doppler Effect

Observer Moving Towards Stationary Source

Answer 16

-observer detects more wavefront per unit time

f’ = f ((v+v0)/v)

f' = perceived frequency
f = actual frequency
v = wave speed
v0 = speed of observer

Question 17

Doppler Effect

Observer Moving Away From Stationary Source

Answer 17

-observer detects fewer wave fronts per unit time

f’ = f((v-v0)/v)

f' = perceived frequency
f = actual frequency
v = wave speed
v0 = speed of observer

Question 18

Doppler Effect

Source Moving Towards Stationary Observer

Answer 18

-wave length is shortened, wave fronts are closer together

f’ = f(v/(v-vs))

f' = perceived frequency
f = actual frequency
v = wave speed
vs = source velocity

Question 19

Doppler Effect

Source Moving Away From Stationary Observer

Answer 19

-wave fronts are further apart as wavelength is greater

f’ = f (v/(v+vs))

f' = perceived frequency
f = actual frequency
v = wave speed
vs = source velocity

Question 20

Doppler Effect

Source and Observer in Motion

Answer 20

f’ = f [ (v±v0) / (v∓vs)

(upper sign for towards, lower sign for away from)

f' = perceived frequency 
f = actual frequency
v = wave speed
v0 = observer speed 
vs = source speed

Question 21

Shock Waves

Answer 21

• occur when the speed of the source approaches the speed of sound
• the source travels a distance vs*t in time t
• but in that time the wavefront emitted at time t=0 has also travelled the same distance
• this creates a conical shock front

Question 22

Intensity of Two Coherent Sources In Phase

Answer 22

It = (√I1 + √I2)²

Question 23

Intensity of Two Coherent Sources Out of Phase

Answer 23

It = (√I1 - √I2)²