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Two Point Source Interference Pattern

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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  1. Two Point Source Interference Pattern A Mathematical Analysis

  2. 2 Point Interference Pattern

  3. 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

  4. Point A, on antinodal line 1 PD = | S1A - S2A | = | 5 - 6 | = 1

  5. Point B, on antinodal line 1 PD = | S1B - S2B | = | 3 - 4 | = 1

  6. Point C, on antinodal line 2 PD = | S1C - S2C | = | 4 - 6 | = 2

  7. 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

  8. Point D, on nodal line 1 PD = | P1S1 – P1S2 | = | 5–4.5 | = 0.5

  9. Point E, on the nodal line 2 PD = | P2S1– P2S2| = | 3.5–5 | = 1.5

  10. If you do this for a while…

  11. Blue: Nodal Lines Red: Antinodal Lines

  12. 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

  13. Path Difference for Distant Points

  14. A bit of geometry…

  15. 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

  16. 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

  17. What angle does q measure?

  18. Make note of what each variable means using the diagram to the right.

  19. 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]

  20. 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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