INTERFERENCE

Waves in 2 D

Constructive and Destructive Interference in 2D

Recall from waves in 1D

Crest + Crest = Anti node (constructive interference)

Trough + Trough = Anti node (constructive Interference)

Trough + Crest = node (destructive interference)

https://phet.colorado.edu/en/simulation/wave-interference

http://video.mit.edu/watch/ripple-tank-interference-of-two-point-sources-4273/

Consider wave interference in two dimensions when the waves have the same frequency (wavelength) and amplitude, and when they are vibrating at the same time (in phase).

As the successive crests and troughs travel out from each source, they interfere with each other, sometimes crest on crest, sometimes trough on trough, and sometimes crest on trough.

Waves in two dimensions show constructive and destructive interference just as waves in one dimension.

Various factors determine the kind of pattern produced:

1)the frequency at which the sources vibrate, 2)the distance between the sources, and3) the phase of the sources.

It is possible to predict what the interference pattern will look like.

A mathematical equation can be used to determine the wavelength of waves producing the interference pattern.

Examples of waves that interfere in 2 (or 3) dimensions include sound and light (Electromagnetic) waves.

Source

Trough

Crest

We will look at waves that have the same frequency (wavelength) and are in phase.

The Principle of Superposition still holds true.

The Pattern produced by wave front is more complex.

Note: This diagram in the slides do not match the one you used in your notes.

Consult your notes when you want to count the number of wavelengths.

Centralmaximum

2 nd nodal line (n = 2)

1st nodal line (n = 1)

3 rd nodal line (n = 3)

S1

4th nodal line (n = 4)

λλλλλ

S 1P 1

S2 P

1

P1

S2

λ

Let S1P1 = the distance from the 1stsource to a point on the first nodal line (n = 1)Let S2P1 = the distance from the 2nd source to a point on the first nodal line (n = 1)

Use the diagram and count the number of wavelengths between S1 and P1 and between S2 and P1.

Finding the Mathematical Relationship between Points on a nodal line and the distance from the source.

Let P1 be a point on the first nodal line

Find the difference between these two numbers S1P1 - S2P1 = 6λ -51/2 λ = 1/2 λ

Centralmaximum

2 nd nodal line (n = 2)

S1

λλλλλ

S 1P 2 S2 P

2

S2

λ

P2

Now consider a point on the 2nd nodal line

Use your diagram and count the number of wavelengths between S1 and P2 and between S2 and P2.

Centralmaximum

3 rd nodal line (n = 3)

S1

λλλλλ

S1P3

S2 P

3

S2

λP3

Now consider a point on the 3rd nodal line

Use the diagram and count the number of wavelengths between S1 and P3and between S2 and P3.

S1P1 - S2P1 = 6λ - 5 1/2 λ = 1/2 λ

S1P2 - S2P2 = 6λ - 4 1/2 λ= 1 1/2 λ

S1P2 - S2P2 = 6λ -3 1/2 λ = 2 1/2 λ

n = 1 (n - 1/2 ) = 1/2

n = 2 (n - 1/2 ) = 1 1/2

n = 3 ( n - 1/2) = 2 1/2

So what about the general form for any nodal line

S1Pn - S2Pn = (n - 1/2)λ

Nodal Line Number

Where n = the number of the nodal line.

Example 1Two sources are vibrating in phase. A point P on the 2 nd nodal line is 18 cm from source 1 and 22 cm from Source 2. It takes 2 seconds for the 1st wave to travel a distance of 15 cm. a) Find the speed of the wave.

b) Find the wavelength

c) Find the frequency

http://www.mta.ca/faculty/science/physics/suren/Interference/Ripint.html

Homework

Read Pages 20-22

Complete questions 12-13 pages 30-31

Test on Monday