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Two Point Source Interference PatternPowerPoint Presentation

Two Point Source Interference Pattern

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### Two Point Source Interference Pattern

A Mathematical Analysis

Nodal Lines

- Nodes are areas of destructive interference and antinodes are the opposite (constructive)
- In standing waves, nodes are the particles that appear to stand still
- Nodal lines occur in areas of destructive interference (crest+trough or trough+crest)
- Nodal lines in light interference would be dark
- An antinodal line appears in the centre of an interference pattern when the two frequencies of the sources match)
- The “count” of the nodal and antinodal lines increases as one moves away from the centre antinodal line

Point A, on antinodal line 1

PD = | S1A - S2A | = | 5 - 6 | = 1

Point B, on antinodal line 1

PD = | S1B - S2B | = | 3 - 4 | = 1

Point C, on antinodal line 2

PD = | S1C - S2C | = | 4 - 6 | = 2

Changing Notation Somewhat

- Look only at nodal lines
- Remember that nodal line numbers get larger as you move out from centre
- The first nodal line is just to the right of the right bisector of the line joining the sources
- Instead of A,B,C, etc., we will call the general point, P, and use a subscript to denote the nodal line on which it sits

Point D, on nodal line 1

PD = | P1S1 – P1S2 | = | 5–4.5 | = 0.5

Point E, on the nodal line 2

PD = | P2S1– P2S2| = | 3.5–5 | = 1.5

Red: Antinodal Lines

General Relationship

- Note that this is only for nodal lines (destructive interference)
- Antinodal lines would have (n-1) instead

PD = | PnS1 – PnS2 | =(n-0.5)l

Nodal Line Analysis

- Where q is the angle for the nth nodal line from the main nodal line (right bisector)
- l is the wavelength
- d is the source spacing

Boundary Conditions

- qn cannot be larger than 1, the RHS cannot be larger than 1
- The largest n that satifies this condition will be seen in the interference pattern – count them!
- By measuring d and counting nodal lines, we can approximate l

What angle does q measure?

Make note of what each variable means

using the diagram to the right.

Example 1

- Two point sources generate identical waves that interfere in a ripple tank. The sources are located 5.0 cm apart, and the frequency of the waves is 8.0 Hz. A point on the first nodal line is located 10 cm from one source and 11 cm from the other.
- What is the wavelength of the waves?
- [2.0 cm/s]
- What is the speed of the waves?
- [16 cm/s]

Example 2

- A ripple tank experiment has given the following data from 2 point sources operating in phase: n=3, x3=35cm, L=77, d=6.0cm, q3=25°, 5 crests=4.2cm. Using 3 methods, determine the wavelength of the waves.

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