1© Manhattan Press (H.K.) Ltd. Constructive and destructive interference Mathematical approach...

© Manhattan Press (H.K.) Ltd. 1

• Constructive and destructive Constructive and destructive interference interference

• • Mathematical approachMathematical approach

9.8 Interference of 9.8 Interference of water waveswater waves


9.8 Interference of water waves (SB p. 65)

Interference of water wavesInterference of water waves

Interference is the effect produced by the superposition of waves from two coherent sources passing through the same region.




Two wave sources are said to be coherent if:

- the phase difference between the sources is constant, and

- two waves should have same frequency, and

- two waves should have comparable amplitudes for interference to occur.

The interference pattern produced in a ripple tank using two sources of circular waves which are in phase with each other.



Constructive & destructive interferenceConstructive & destructive interference

The two sources S1 and S2 are in phase and coherent. Therefore, the wavelengths of waves from S1 and S2 are the same, say λ.




At A1, produce double crest, constructive interference

occurs, antinode

At A2, produce double trough, constructive

interference occurs, antinode

For all the points along the line

joining A1 and A2, the distances from

S1 and S2 are equal. It is

antinodal line which are joining all the antinodes.



Constructive interferenceConstructive interference

For constructive interference to occur,

n = 0, 1, 2, 3, …Path difference = n



At X1, superposition of a crest and a trough produces zero amplitude,

destructive interference occurs,

node

At X2, superposition of a crest and a trough produce zero amplitude, destructive interference occurs, node

The line joining all the nodes is called

a nodal line.

Destructive interferenceDestructive interference




For destructive interference to occur,

n = 0, 1, 2, 3, …

For the point X1, n = 1,

For the point Y1, n = 2,

Path difference = (n )21

21

21 - 1 difference Path

211

21 - 2 difference Path

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Example 9Example 9



Mathematical approachMathematical approach

By the principle of superposition, the resultant displacement y due to the two waves is:

This can be simplified using the relationship:

where y0 is the amplitude of the resultant displacement.

)sin( sin 2 1 tatayy

2cos

22sin sin sin BABABA

2)(cos

2sin2 ttttay

2cos

2cos

2cos

2sin2 ta

2 sin

2cos 2 ta

2 sin 0 ty



Constructive interferenceConstructive interference

Constructive interference occurs when the waves have no phase difference, i.e., = 0. Thus, the resultant amplitude is given by

Maximum amplitude is obtained for = 2, 4, …etc

a

ay

2 20cos2 0




Destructive interference occurs when the waves are out of phase, i.e., = . The resultant amplitude is

Minimum amplitude is obtained for = 3, 5, …etc

0 2

cos2 0

ay



Phasor diagramPhasor diagram

A phasor diagram is used to obtain the amplitude at the other points in the medium where the phase difference is not an integral multiple of .

A phasor is a rotating vector used to represent a sinusoidally varying quantity, e.g. the displacement y at a point in a wave.



Suppose the displacement y1 at a point P due to a wave motion can be represented by the equation

tyy sin 01

As time t increases, the tip of the phasor rotates counter-clockwise around the circle with constant angular speed ω.

Its projection on the y-axis, y1 which is equalto y0 sinωt represents the displacement at the point P at any time t.




Suppose the displacement at the point P due to another wave is represented by

)sin( 02 tyy

Then by the Principle of Superposition of Waves, the resultant of displacement y at the point P due to the two waves is given by

21 yyy




The resultant displacement y can be obtained using a phasor diagram by finding the vector sum of y1 and y2.

The amplitude of y is given by the length Y of the rotating vector OA.




To find the length Y of the resultant displacement, it is easier to draw the phasor diagram for y1 and y2 at time t = 0 as shown.


OB represents the phasor for y1.

AB represents the phasor for y2.

OA of length Y represents the phasor for y = y1 + y2.The value of Y can be calculated using the cosine rule since OB = BA = y0 and angle OBA = 180o .

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Example 10Example 10


End



QQ: : Two wave generators S1 and S2 placed 4 m apart in

a water tank, produce water waves of wavelength

1 m. P is a point 3 m from S1 as shown in the

figure.

Solution

Assuming each generator produces a wave at P which has an amplitude A.When the generators are operating together and in phase, what is the resultantamplitude at P?


Solution:Solution:

Distance of P from S2 = = 5 m

Path difference = PS2 PS1

= (5 3) m = 2 m = 2

Since the path difference is 2λ, constructive interference occurs. Thus, the resultant amplitude at P is A + A = 2A.

22 4 3

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QQ: : (a) Write down a progressive wave equation.

With the aid of suitable diagrams, explain the meanings of the quantities appearing in your equation.

(b) Sketch two similar sinusoidal waves each of amplitude A and with phase differences such that, when superposed, the waves would produce(i) constructive interference,(ii) destructive interference.



QQ: : (c) If two waves have exactly a phase

difference of 60° and are superposed, find(i) by means of a phasor diagram, the amplitude of the resultant wave in terms of A,(ii) the ratio of the power carried by the resultant wave to the total power carried by the two component waves considered separately.Does the result obtained contradict the principle of conservation of energy?

Solution



Solution:Solution:(a) The progressive wave equation is: y = a sin( ) where a = amplitude, y = displacement of a particle at a distance x from O at time t, λ= wavelength, = angular frequency of wave.

xt 2 -



Solution (cont’d):Solution (cont’d):(b) (i) Constructive interference (ii) Destructive interference Phase difference = radians



Solution (cont’d):Solution (cont’d):(c) (i) If phase difference = 60°, the amplitude of the resultant wave is given by the length of PR in the phasor diagram below.

In the triangle PQR, using the cosine rule, PR2 = PQ2 + QR2 – 2(PQ)(QR) cos120° = A2 + A2 – 2A2 cos120° = 3A2

PR = A

Therefore, the amplitude of the resultant wave is A.

3

3



Solution (cont’d):Solution (cont’d):

(ii) Power carried by wave Intensity (Amplitude)2

The result does not contradict the Principle of Conservation of Energy because where constructive interference occurs, the power carried by the resultant wave is greater than the sum of the power carried by the two waves. The extra power comes from areas where destructive interference occurs.

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23

3

2

2

2

AA

A

waves twoby the carriedpower Totalwaveresultant by carriedPower


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