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2D Wave Interference. Constructive and Destructive Interference :. When waves overlap, their displacements can CANCEL or ADD UP. Out of phase- ½  Delay. In phase- 0  Delay. Result: Constructive Interference Destructive Interference. 1-D interference.

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2d wave interference

Constructive and Destructive Interference:

When waves overlap, their displacements can CANCEL or ADD UP.

Out of phase- ½  Delay

In phase- 0  Delay

Result: Constructive Interference Destructive Interference

1-D interference

Complete destructive interference occurs when the phase delay between the waves is : ½ , 3/2 , 5/2 , 7/2  ……. Etc.

Or

The points of destructive interference are called NODES.

Or


2d wave interference

2-D interference

2-D interference simulation

Interference of Waves in Two Dimensions :

In two dimensions, interfering waves from two sources with the same wavelength produce stationary NODAL LINES:

Constructive Interference

Destructive Interference

Constructive Interference

Stationary

Nodal Lines

Source Separation, d


2d wave interference

General Pattern:

n=1

● Nodal lines have a hyperbolic shape but appear STRAIGHT at a distance

n=1

n=2

n=2

● Nodal line number depends on the wavelength and source separation

, #

d , #


2d wave interference

Determining Wavelength:

Waves from S1 and S2 arriving at ANY point P on the first nodal line are out of phase by

PS1

PS2

Path difference for first nodal line, n=1:

For second nodal line, n=2:

For third nodal line, n=3:

Equation 1:

General formula for the nth nodal line:

where n=1,2,3…..


2d wave interference

II.Angle Dependence of Nodal Lines:

At large distances

PS1 || PS2

Angles X 90

P

Path Difference

A

A

x

x

n

S1

d

S1

d

S2

S2

Sin n = AS1

d

Path Difference=AS1

AS1= dsin n

At large distances from the sources, the path

difference becomes equal todsin n


2d wave interference

We can write this as: |PS1 – PS2|=dsin n

Combine with Eqn: 1:

We get a second equation with the angle of the

nth nodal line:

Equation 2:

Where n is the nodal line number

 is the wavelength

d is the source separation

is the angle of the nodal line

Sample Question 3


2d wave interference

P

III. Cases wheren difficult to measure:

In some cases (e.g. light interference), the angles of the nodal lines are not easily measured.

x

B

Centre

line

C

S1

We’ll now identify a way find the angle from distances measured on the interference pattern. nodal pattern

L

A

n

n

From Triangle BCP we can see:

S2

midpoint

We will now combine this with equation 2:


2d wave interference

Equation 3:

Where d- source separation * All distances in metres!

 - wavelength

n = nodal line number

L- distance measured from the centre of S1S2 to nodal point P

X- the perpendicular distance from the centre line of the pattern

to point P

Try Sample question 4


2d wave interference

Light Interference: Young’s Double Slit Experiment

● prior to 1802, interference of light was NOT observed

Why not?

● incandescent light sources

emit incoherent light (random phase)

●  very small, so nodal line spacing very small

1802- Young developed the DOUBLE SLIT experiment

● this was the deciding evidence for WAVE model of light


2d wave interference

Young’s Experiment:

Screen

Interference

Fringe

Pattern

Incident sunlight

Coherent

Spreading

wavefronts

Bright Bands-constructive interference

-maxima

Dark Bands-destructive interference

-minima

x2

Fringe Pattern:

x1

Central maximum

*From this pattern the easiest measurement is the node to node spacing x

n=1

x