1. Sine Waves & Phase

2. Sine Waves A sine wave is the simplest periodic wave there is
Sine waves produce a pure tone at a single frequency

3. Simple Harmonic Motion
Any motion at a single constant frequency can be represented as a sine wave
Such motion is known as simple harmonic motion
Here the amplitude may vary but the frequency does not

4. S.H.M. A pendulum swings in SHM:
once it is started off it will take the same time to swing back and forth no matter how high it gets to at the top of its swing
in other words: its frequency will stay the same no matter how high its amplitude

5. Electronic Oscillators
Today electronic oscillators are the principle source of pure tones
It is easy to specify and vary the frequency of an electronic oscillator precisely

6. Describing a Sine Wave Consider a wheel of radius 1 metre
There is a line drawn on the wheel from the centre to the edge
The height of the point where the line touches the edge is plotted as the wheel spins (at say ¼ of a turn per second)

7. Describing a Sine Wave
radius = 1
height

8. Describing a Sine Wave
To create a sine wave the height of the point where the line touches the edge is plotted as the wheel spins clockwise at constant speed

9. 0
45
90
180
270
360
0 seconds
4 seconds
2 seconds
1 second
½ a second
3 seconds

10. Phase Difference
The phase of periodic wave describes where the wave is in its cycle
Phase difference is used to describe the phase position of one wave relative to another

11. Phase Difference 180
½ 
pressure
time 

12. Phase Difference 90
pressure
time ¼

13. Phase Difference 45
pressure
time
1/8 
Wave A
Wave B

14. Phase Difference Is Wave A in front of Wave B or behind it?
It can be seen either way:
Wave A leads Wave B by 45; or
Wave B leads Wave A by 315

15. The Sine Function Sine is a mathematical function y = sin(x)
sin(0) = 0 sin(45) = 0.707
sin(90) = 1 sin(180) = 0
sin(270) = -1 sin(360) = 0

16. 0
45
90
180
270
360
x = 0, y = sin(x) = 0
x = 45, y = sin(x) = 0.707
x = 90, y = sin(x) = 1
x = 180, y = sin(x) = 0
x = 270, y = sin(x) = -1
x = 360, y = sin(x) = 0

17. Radians
One radian is the angle subtended at the centre of a circle by an arc that has a circumference that is equal to the length of the radius of a circle

18. Radians arc length radius (r) = arc length (s) 1 radian radius
angles can be measured in radians:
θ = s / r

19. Calculating Angles in Radians
angle in radians = arc length / radius θ = s / r

20. How Many Radians in a Circle?
Circumference of a circle = 2  r
For one complete revolution the arc length is the entire circumference:
θ = s / r = 2  r / r = 2 

21. Radians 2 
/2
3/2 1
- 1
phase
Graph showing a sine wave with the y axis giving phase in radians.

22. 2 radians = 360, so /2 radians = 90
Radians & Degrees
2 radians = 360, so /2 radians = 90
1 radian = 90 /  * 2  57.5

23. Common Angles Cycles 1/12 1/8 1/6 1/4 1/2 3/4 1 Degrees 0 30 45 60
1/12
1/8
1/6
1/4
1/2
3/4 1
Degrees 0
30
45
60
90
180
270
360
Radians
 / 6
 / 4
 / 3
 / 2 
3 / 2 2

24. Time Difference Calculations
Calculating the time difference between waves of identical period:
time difference =  *
phase difference in cycles

25. For Example:
If two waves of period 0.05 secs have a phase difference of 45 what is the time difference between them?
0.05 * (1/8) = secs = 6.25ms
45 in terms of cycles

26. Question 1
If two waves of period 20ms are phase shifted 90 what is the time difference between them?
0.02 * 1/4 = secs = 5ms

27. Question 2
If wave A is leading wave B by 270 degrees and both have a frequency of 200Hz, what is the time difference between the waves?

28. Question 2 - Solution Recall: frequency = 1 / period f = 1 / 
So:  = 1 / f = 1 / 200 = 0.005
0.005 * (3/4) = / 4 = s (3.75ms)

29. Question 2 - Discussion Wave A leads Wave B by 270 (3.75ms); or
90
270
Wave A leads Wave B by 270 (3.75ms); or
Wave B leads Wave A by 90 (1.25ms)

30. Phase Difference Calculations
Calculating the phase difference between waves of identical period:
phase difference = (2 / ) * time difference

31. For Example:
If two waves of period 0.05 are produced seconds apart what is their phase difference?
(2 / 0.05) * = radians

32. Question 1
If two waves of frequency 100 Hz are produced seconds apart what is their phase difference?

33. Question 1 - Solution frequency = 1 / period f = 1 / 
So:  = 1 / f = 1 / 100 = 0.01
phase difference = (2 / ) * time difference
(2 / 0.01) * =  radians
which is 180 degrees

34. Question 2
If two waves of period secs are produced seconds apart what is their phase difference?
phase difference = (2 / ) * time difference
(2 / 0.009) * = radians
20 degrees (radians * 57.5)

35. Question 3
If two waves of period 0.03s are produced seconds apart what is their phase difference?
phase difference = (2 / ) * time difference
(2 / 0.03) * = radians
30 degrees (radians * 57.5)

36. which is roughly 75 degrees
Question 4
If two waves of period s are produced seconds apart what is their phase difference?
phase difference = (2 / ) * time difference
(2 / 0.024) * = radians
which is roughly 75 degrees