Two point source interference pattern
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Two Point Source Interference Pattern. A Mathematical Analysis. 2 Point Interference Pattern. 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

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Two Point Source Interference Pattern

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Two point source interference pattern

Two Point Source Interference Pattern

A Mathematical Analysis


2 point interference pattern

2 Point Interference Pattern


Nodal lines

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

Point A, on antinodal line 1

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


Point b on antinodal line 1

Point B, on antinodal line 1

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


Point c on antinodal line 2

Point C, on antinodal line 2

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


Changing notation somewhat

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

Point D, on nodal line 1

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


Point e on the nodal line 2

Point E, on the nodal line 2

PD = | P2S1– P2S2| = | 3.5–5 | = 1.5


If you do this for a while

If you do this for a while…


Two point source interference pattern

Blue: Nodal Lines

Red: Antinodal Lines


General relationship

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


Path difference for distant points

Path Difference for Distant Points


A bit of geometry

A bit of geometry…


Nodal line analysis

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

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

What angle does q measure?


Two point source interference pattern

Make note of what each variable means

using the diagram to the right.


Example 1

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

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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