Unit 1.2
Phase
When we look at a sinusoidal wave we realize that the pattern repeats itself at regular intervals.
In the above diagram of a particle in circular motion, the pattern repeats itself after rotating 2π radians.
If we plotted displacement against time the pattern repeats itself after T (period) seconds.
If we plotted displacement against wavelength the pattern repeats itself after λ.
Points in Phase on a Single Sine Wave
Points that have the same displacement in a wave are in phase.
It is clear that for the first graph points in phase are 2nπ radians apart where n is an integer.
and in the case of period points in phase are nT seconds apart where n is an integer.
For λ, points in phase are nλ apart where n is an integer.
Points exactly out of Phase on a Single Sine Wave
Points that have exactly opposite displacement in a wave are exactly out of phase.
It is clear that for the first graph points exactly out of phase are ^{(n+1)}/_{2} π radians apart where n is an integer.
and in the case of period points exactly out of phase are n^{T}/_{2} seconds apart where n is an integer.
For λ, points exactly out of phase are n ^{λ}/_{2} apart where n is an integer.
Why are points that are n ^{λ}/_{2} or n ^{T}/_{2} or ^{(n+1)}/_{2 }π (where n is an integer) NOT in phase (or exactly out of phase)?
Points that are n ^{λ}/_{2} or n ^{T}/_{2} or nπ (where n is an integer) have a displacement of zero, but some points are moving up and others down. To be in phase, the directions have to be the same, thus every other point in in the same phase.
Points out of Phase
For other points out of Phase, it is convenient to express the phase difference as an angle in Radians (θ) where the closest points in phase are (λ or T or 2π radians), so the phase difference is a fraction of 2π radians.
Path difference, x = difference in distance from each source to a particular point.
Phase difference, Ø = difference in phase of the waves at a point.
In general: Ø = ^{2πx}/_{λ }radians
Coherent and Incoherent Sources
When a wave arises from the same source, no matter what the path, the waves have a relationship with each other and are considered to be coherent waves. Waves arising from separate sources don't have this relationship and are considered incoherent. Consider two sound systems playing two different tunes, these are incoherent, but the speakers from one sound sytem play the same waves and thus are coherent. Only coherent waves set up the kind of interference patterns we are investigating. Sinusoidal waves are a special case, as changes time do not affect coherency.
Sources and the waves are said to be coherent if they have the same frequencies, same wavelength, same speed, almost same amplitude and having no phase difference or a constant phase difference. When any of these are lacking, the sources are said to be incoherent. Examples of coherent sources are sound waves from two loud speakers driven by the same audio oscillator, microwaves driven by the same oscillator, and light waves coming from a laser gun. However, light from regular bulbs are incoherent because the source of the individual light waves are from different atoms in the filament or discharge tube of a fluorescent bulb.
Examples
Example 1:
Two loudspeakers are 4m apart, and are connected to the same signal. A listener is three metres away from the left speaker on a line which is perpendicular to the line joining the two speakers as in the diagram. Speed of sound in air 330ms^{1}.
(a) When both speakers generate a (i) 330Hz (ii) 420Hz signal what is the phase difference between them in Radians?
(b) Would changing the frequency to 440Hz affect the loudness at the listeners position?.
Answer:
(a) (i) by Pythagoras's Theorem:
Distance of right speaker from person = √(4^{2}+ 3^{2})
= 5m
Using v= fλ.
λ= ^{v}/_{f}.
Wavelength = 330ms^{1}/_{330Hz}
Wavelength = 1m.
Number of wavelengths in left path = ^{3m}/_{1m}
= 3
Number of wavelengths in left path = ^{5m}/_{1m}
= 5
No ofwavelengths difference (^{x}/_{λ}) = 2
Points that are nλ apart where n is an integer are in phase.
Ø = 2π (^{x}/_{λ}) radians
Ø = 2π (2_{})radians.
Ø = 4π radians.
Recall: Points in phase are 2nπ radians apart where n is an integer
Constructive Interference (loud)
(a) (ii) Using v= fλ.
λ= ^{v}/_{f}.
Wavelength = 330ms^{1}/_{420Hz}
Wavelength = ^{11}/_{14 }m.
Number of wavelengths in left path = 3m _{}÷ ^{11}/_{14 }m
= ^{42}/_{11 }
Number of wavelengths in left path = 5m _{}÷ ^{11}/_{14 }m
= ^{70}/_{11 }
No of wavelengths difference (^{x}/_{λ}) = ^{28}/_{11 }
Ø = ^{2πx}/_{λ }radians
= ^{28}/_{11 }_{ }x 2 π radians.
=_{ } ^{56}/_{11 }π radians
5^{1}/_{11 }π radians
Soft sound because it is between 5 π and 6 π. Closer to 5 π (destructive interfenrence  min) and is now increasing to a max at 6 π (constructive interference) .
(b) Using v= fλ.
λ= ^{v}/_{f}.
Wavelength = ^{330 ms1}/_{440HZ}.
Wavelength = ^{3}/_{4 }m.
Number of wavelengths in left path = 3m _{}÷ ^{3}/_{4 }m
= ^{12}/_{3}
Number of wavelengths in left path = 5m _{}÷ ^{3}/_{4 }m
(^{x}/_{λ}) = ^{20}/_{3}
No of wavelengths difference (^{x}/_{λ}) = ^{20}/_{3}  ^{12}/_{3}
(^{x}/_{λ}) = ^{8}/_{3}
Ø = ^{2πx}/_{λ }radians
= ^{8}/_{3} x 2 π radians.
= ^{16}/_{3} π radians.
=5^{1}/_{3 }π radians.
We went from loud at 330Hz to soft at 420 Hz and now the loudness is increasing again.
The waves would be out of phase by ^{1}/_{3 }π radians.
How does ^{16}/_{3} π radians and ^{1}/_{3 }π radians have the same phase difference?.
Recall that points 2nπ radians apart are in phase...
Example 2:
2. Two loudspeakers both emit sounds with a frequency of 330Hz and equal amplitude A. The speakers are 4m apart and face each other. (The speed of sound in air is 330ms^{1}).
(a) What is the wavelength of the sound?
(b) What is the amplitude of the sound on a line joining both speakers (ignoring any reduction in amplitude as the sound travels)?
(i) half way between the speakers
(ii) 1.25m from the left speaker
Answer:
(a) Using v= fλ.
λ= ^{v}/_{f}.
Wavelength = ^{330Hz}/_{330 ms}1.
Wavelength = 1m.
(b) (i) Half way between the speakers is 2m
Number of wavelengths from left speaker = 2m ÷ 1m
= 2
Number of wavelengths from right speaker = 2m ÷ 1m
= 2
No of wavelengths difference (^{x}/_{λ}) = 2  2 = 0
Ø = 2π (^{x}/_{λ}) radians (Note above (^{x}/_{λ}) = 0)
Ø = 0
Both waves meet at same phase so constructive interference, therefore amplitude is a max.
Amplitude = Amplitude of left wave + amplitude of right wave
= A + A
=2A
This diagram is not necessary but may help you to visualize it...
(b) (ii) 1.2 m from the left speaker, (4  1.25 = 2.75)m from the right speaker
Number of wavelengths from left speaker = (b) (i) 1.2 m ÷ 1m
= 1.25
Number of wavelengths from right speaker = 2.75 m ÷ 1m
= 2.75
No of wavelengths difference (^{x}/_{λ}) = 2.75  1.25 = 1.5
Ø = ^{2πx}/_{λ }radians
Ø = 1.5 x 2 π radians.
3.0 π radians.
Points that are n^{ λ}/_{2} apart where n is an integer ( or are n π radians apart where ^{(n+1)}/_{2}is an integer) are exactly out of phase.
So Amplitude = A + (A) = 0.
