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

0 45 90 180 270 360 0 seconds 4 seconds 2 seconds 1 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

Phase Difference 180 ½  pressure time 

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(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 (r) = arc length (s) 1 radian 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 2  /2 3/2 1 - 1 phase Graph showing a sine wave with the y axis giving phase in radians.

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

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

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 Recall: frequency = 1 / period f = 1 /  So:  = 1 / f = 1 / 200 = 0.005 0.005 * (3/4) = 0.015 / 4 = 0.00375s (3.75ms)

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)

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)

which is roughly 75 degrees 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