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)

radius = 1

height

Describing a Sine Wave

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

0 45

90 180

270 360

0 seconds

4 seconds

2 seconds1 second

½ a second

3 seconds

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

½

pressure

time

Phase Difference 180

Phase Difference 90

¼

pressure

time

Phase Difference 45

1/8

pressure

time

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(45) = 0.707

sin(90) = 1 sin(180) = 0

sin(270) = -1 sin(360) = 0

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

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

radius

1 radian radius (r) = arc length (s)

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

2/2 3/2

1

- 1

0phase

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:

time difference = * phase 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

0.005 * (3/4) = 0.015 / 4 = 0.00375s (3.75ms)

So: = 1 / f = 1 / 200 = 0.005

Recall: frequency = 1 / period

f = 1 /

Wave A leads Wave B by 270 (3.75ms); or

Question 2 - Discussion

270

Wave A Wave B 90

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

(2 / 0.01) * 0.005 = radians

So: = 1 / f = 1 / 100 = 0.01

frequency = 1 / period

f = 1 /

phase difference = (2 / ) * time difference

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?

(2 / 0.009) * 0.0005 = 0.34906585 radians

phase difference = (2 / ) * time difference

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?

(2 / 0.03) * 0.0025 = 0.523598775 radians

phase difference = (2 / ) * time difference

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?

(2 / 0.024) * 0.005 = 1.308996939 radians

phase difference = (2 / ) * time difference

which is roughly 75 degrees