Sine Waves & Phase
Sine Waves
• A sine wave is the simplest periodic wave
there is
• Sine waves produce a pure tone at a single
frequency
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
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
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
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)
Describing a Sine Wave
radius = 1
height
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
½ a second
0 seconds
0
45
1 second
90
2 seconds
180
3 seconds
270
4 seconds
360
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
Phase Difference 180

pressure
time
½
Phase Difference 90
pressure
time
¼
Phase Difference 45
pressure
time
1/8 
Wave A
Wave B
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
The Sine Function
• Sine is a mathematical function
• y = sin(x)
sin(0) = 0
sin(90) = 1
sin(270) = -1
sin(45) = 0.707
sin(180) = 0
sin(360) = 0
x = 45, y = sin(x) = 0.707
x = 0, y = sin(x) = 0
0
45
x = 90, y = sin(x) = 1
90
x = 180, y = sin(x) = 0
180
x = 270, y = sin(x) = -1
270
x = 360, y = sin(x) = 0
360
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
Radians
arc length
1 radian
radius (r) = arc length (s)
radius
angles can be measured in
radians:
θ =s/r
Calculating Angles in Radians
angle in radians = arc length / radius
θ
=
s
/
r
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
Radians
1
0
/2

3/2
2 phase
-1
Graph showing a sine wave with the y axis giving phase
in radians.
Radians & Degrees
2 radians = 360, so /2 radians = 90
1 radian = 90 /  * 2  57.5
Common Angles
Cycles
0
1/12
1/8
1/6
1/4
1/2
3/4
1
Degrees
0
30
45
60
90
180
270
360
Radians
0
/6
/4
/3
/2

3 / 2
2
Time Difference Calculations
Calculating the time difference between
waves of identical period:
phase
difference
time difference =  *
in cycles
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) = 0.00625 secs = 6.25ms
45 in terms of cycles
Question 1
If two waves of period 20ms are phase
shifted 90 what is the time difference
between them?
0.02 * 1/4 = 0.005 secs = 5ms
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?
Question 2 - Solution
Recall:
So:
frequency = 1 / period
f
= 1 / 
 = 1 / f = 1 / 200 = 0.005
0.005 * (3/4) = 0.015 / 4 = 0.00375s (3.75ms)
Question 2 - Discussion
Wave A
Wave B
90
270
Wave A leads Wave B by 270 (3.75ms); or
Wave B leads Wave A by 90 (1.25ms)
Phase Difference Calculations
Calculating the phase difference between
waves of identical period:
phase difference = (2 / ) * time difference
For Example:
If two waves of period 0.05 are produced
0.00625 seconds apart what is their phase
difference?
(2 / 0.05) * 0.00625 = 0.7853 radians
Question 1
If two waves of frequency 100 Hz are
produced 0.005 seconds apart what is their
phase difference?
Question 1 - Solution
frequency = 1 / period
f
= 1 / 
So:
 = 1 / f = 1 / 100 = 0.01
phase difference = (2 / ) * time difference
(2 / 0.01) * 0.005 =  radians
which is 180 degrees
Question 2
If two waves of period 0.009 secs are
produced 0.0005 seconds apart what is their
phase difference?
phase difference = (2 / ) * time difference
(2 / 0.009) * 0.0005 = 0.34906585 radians
20 degrees (radians * 57.5)
Question 3
If two waves of period 0.03s are produced
0.0025 seconds apart what is their phase
difference?
phase difference = (2 / ) * time difference
(2 / 0.03) * 0.0025 = 0.523598775 radians
30 degrees (radians * 57.5)
Question 4
If two waves of period 0.024 s are produced
0.005 seconds apart what is their phase
difference?
phase difference = (2 / ) * time difference
(2 / 0.024) * 0.005 = 1.308996939 radians
which is roughly 75 degrees