1. Types of waves 2. The wave equation of motion: y(x,t) = ymsin(kx ...

WAVES 

1. Types of waves 

2. The wave equation of motion: y(x,t) = y m sin(kx – ωt + φ 0 ) 

3. Traveling waves 

- waves on a string 

- sound waves 

4. wave power and intensity 

- the decibel system 

5. Interference in 1-D 

- beats 

- reflection and transmission 

- standing waves and resonance1 

6. Interference in 2-D 

7. The Doppler effect 

PHYS 1101, Winter 2009, Prof. Clarke 1

Clicker question 1 

A simple harmonic wave is given by: y(x,t) = y m sin(kx – ωt + φ 0 ) 

Consider the wave given by: y(x,t) = 2sin(πx – 2πt). Which of 

the following functions correspond to this wave? 

a) 

2 

y 

x 

1 

2 

–2 

2 

–2 

4 

c) 

2 

y 

x 

b) 

1 

y 

x 

d) 

1 

y 

x 

–1 

2 

4 

–1 

π 

2π 

T = 2π/ω 

λ = 2π/k 

2 

–2 

4 

PHYS 1101, Winter 2009, Prof. Clarke 2 

e) 

2 

y 

x

A simple harmonic wave is given by: y(x,t) = y m sin(kx – ωt + φ 0 ) 

Consider the wave given by: y(x,t) = 2sin(πx – 2πt). Which of 

the following functions correspond to this wave? 

a) 

2 

y 


x 

1 

–2 

2 

2 

–2 

4 

c) 

2 

y 

x 

b) 

1 

y 

x 

d) 

1 

y 

x 

–1 

2 

4 

–1 

π 

2π 

T = 2π/ω 

λ = 2π/k 

2 

–2 

4 

PHYS 1101, Winter 2009, Prof. Clarke 3 

e) 

2 

y 

x

What is the phase difference between a crest in a 

wave, and the adjacent trough? 

a) 0 

b) π/4 

c) π/2 

d) π 

e) 3π/2 

f) 2π 




What is the phase difference between a crest in a 

wave, and the adjacent trough? 

a) 0 

y 

b) π/4 

c) π/2 

d) π 

0 

π/2 

π 

3π/2 

5π/2 

2π 

7π/2 

3π 

9π/2 φ = kx 

4π 

e) 3π/2 

f) 2π 



What is the frequency of this travelling wave? 

a) 0.2 Hz 

b) 2 Hz 

c) 5 Hz 

d) 10 Hz 

e) 500 Hz 



What is the frequency of this travelling wave? 

a) 0.2 Hz 

b) 2 Hz 

c) 5 Hz 

d) 10 Hz 

e) 500 Hz 

v = fλ; λ = 10 m; f = v/λ = 5 Hz 


Wavefronts 

Plane waves (1-D) 

Spherical waves (3-D) 

A 

r 

source 

Plane waves don’t spread, 

and the intensity, I = P/A, 

remains constant. 

Concentric spherical waves spread 

out over ever-larger area, A = 4πr 2 . 

With the same power, P, spread out 

over each sphere, the intensity, I, is: 

P P 

I = 

A 

= 

4πr 2 ⇒ I ∝ r –2 


“Yardsticks” for the dB system 

β = 10 log(I/I 0 ) decibels (dB) 

threshold of human hearing 

rustling leaves 

conversation 

jack hammer 

rock concert 

threshold of pain 

jet engine 

10 log(I 0 /I 0 ) = 10 log(1) = 0 dB 

10 dB 

60 dB 

90 dB 

110 dB 

120 dB 

130 dB (can cause the ear to 

bleed) 


Wave Interference 

waves approach… 

wave interference 

begins… 

constructive interference: crests 

and troughs of waves coincide 

maximum wave 

interference… 

wave interference 

ends… 

waves recede as 

though nothing 

happened. 

destructive interference: crests of 

one wave coincides with troughs 

of the other 

complete destructive interference: 

amplitudes of waves are also the 

same. 


Beats (Fig. 21-31, Knight) 

y r 

T = 2π/ω 

T = 2π/ω 

Note that there are two “beats” in each period, T’. 


Reflection and Transmission 

reflection, free end 

reflection, fixed end 

transmission, heavy → light 

transmission, light → heavy 


Standing Waves 



Two loud speakers emit waves with λ = 2.0 m. Speaker 2 is 

1.0 m to the right of speaker 1. What, if anything, must be 

done to speaker 1 to cause constructive interference between 

the two waves? 

a) move it 1.0 m to the right 

b) move it 0.5 m to the right 

c) move it 0.5 m to the left 

d) move it 1.0 m to the left 

e) nothing; there is already 

perfect constructive interference 

between the two waves 



Two loud speakers emit waves with λ = 2.0 m. Speaker 2 is 

1.0 m to the right of speaker 1. What, if anything, must be 

done to speaker 1 to cause constructive interference between 

the two waves? 

a) move it 1.0 m to the right 

b) move it 0.5 m to the right 

c) move it 0.5 m to the left 

d) move it 1.0 m to the left 

Remember, you must consider 

both the shift and the phase 

e) nothing; there is already 

perfect constructive interference 

between the two waves 



Two sources of sound waves beat at 3 Hz. It is known that 

the frequency of one source is 610 Hz. Therefore, the 

frequency of the other source must be: 

a) 604 Hz b) 607 Hz c) 613 Hz d) 616 Hz 

e) either b) or c) f) either a) or d) 



Two sources of sound waves beat at 3 Hz. It is known that 

the frequency of one source is 610 Hz. Therefore, the 

frequency of the other source must be: 

a) 604 Hz b) 607 Hz c) 613 Hz d) 616 Hz 

e) either b) or c) f) either a) or d) 

f beat = f 2 – f 1 , the difference in the two frequencies. 



A standing wave vibrates as shown to 

the right. If the tension is quadrupled 

while keeping the frequency and length 

of string constant, which standing wave 

pattern is produced? 

L 

a) b) c) 

no standing wave 

for these conditions 

v = √T/µ = fλ 

d) e) 



A standing wave vibrates as shown to 

the right. If the tension is quadrupled 

while keeping the frequency and length 

of string constant, which standing wave 

pattern is produced? 

L 

a) b) c) 

v = √T/µ = fλ 

⇒ λ ∝ √T 

no standing wave 

for these conditions 

d) e) 

⇒ if T quadruples, λ 

doubles, and only one 

loop can fit inside L 


Resonance (in pipes) 

Figure 21.15, page 657, Knight 


Wave Interference in 2-D 

Example problem 

λ 

S 1 

S 1 

r 1 

r 2 

S 2 

v 

d 

S 2 

r 1 

y 

“beach” 

“breakwater” 

r 2 

L 

lines of antinodes joins points of constructive 

interference (maximum amplitude) 

lines of nodes joins points of destructive 

interference (minimum amplitude) 




λ 

S 1 

v 

d 

S 2 

r 1 

y 

“beach” 


r 2 

L 








λ 

S 1 

v 

d 

S 2 

r 1 

y 

“beach” 


r 2 

L 






Doppler Shift 

W 1 

W 6 

v S = 0 

S 

λ 

v D = 0 

D 

Nλ = c s ∆t 


Doppler Shift 

W 1 

W 6 

S 1 

S 6 

v S 

S 

λ 

v D = 0 

D 

v S ∆t 

Nλ = (c s – v S )∆t 

Nλ = c s ∆t 


Doppler Shift 

And did you hear the one about the rumoured (and fallacious) transcript of a 

conversation between US and Canadian authorities off the coast of 

Newfoundland… 

Canadians: Please divert your course 15 degrees south to avoid a collision. 

Americans: Recommend you divert your course 15 degrees north to avoid a collision. 

C: Negative. You will have to divert your course 15 degrees to the south to avoid a collision. 

A: This is the captain of a US Navy ship. I say again, divert your course. 

C: No sir, I say again, you will have to divert your course to avoid a collision. 

A: This is the aircraft carrier USS Lincoln, the second largest ship in the 

United States’ Atlantic fleet. We are accompanied by three destroyers, three 

cruisers, and numerous support vessels. I demand that you change your 

course by 15 degrees north. I say again, that’s one-five degrees north, or 

counter-measures will be undertaken to ensure the safety of this ship. 

C: This is a lighthouse. Your call. 



Two speakers are in phase and emit equal-amplitude sound 

waves with the same wavelength of 1.0 m. The interference 

of the two sound waves at point P can best be described as: 

P 

a) perfect constructive 

interference 

b) partial constructive 

interference 

c) partial destructive 

interference 

d) perfect destructive 

interference 



Two speakers are in phase and emit equal-amplitude sound 

waves with the same wavelength of 1.0 m. The interference 

of the two sound waves at point P can best be described as: 

P 

a) perfect constructive 

interference 

b) partial constructive 

interference 

c) partial destructive 

interference 

d) perfect destructive 

interference 



A source of sound, S, moves to the right as shown. In its 

own frame of reference, it emits a sound at frequency f 0 . 

Compare the frequencies that Amy and Zack hear. 

S 

a) f Amy < f 0 < f Zack b) f Amy > f 0 < f Zack 

c) f Amy > f 0 > f Zack d) f Amy = f 0 < f Zack 

e) f Amy = f 0 > f Zack f) f Amy = f 0 = f Zack 



A source of sound, S, moves to the right as shown. In its 

own frame of reference, it emits a sound at frequency f 0 . 

Compare the frequencies that Amy and Zack hear. 

S 

a) f Amy < f 0 < f Zack b) f Amy > f 0 < f Zack 

c) f Amy > f 0 > f Zack d) f Amy = f 0 < f Zack 

e) f Amy = f 0 > f Zack f) f Amy = f 0 = f Zack

1. Types of waves 2. The wave equation of motion: y(x,t) = ymsin(kx ...

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