Evans SPH 4U1

Unit 5: Waves

Note 3: Wave Interference in 2D

Reference: Chapter 9.3

Demo in ripple tank.

Application:

AM radio stations often have more than one point source (i.e have more than one antenna). They use constructive interference to target certain high population areas.

Two point sources vibrating in phase in a ripple tank may produce a characteristic symmetrical standing wave pattern with the following features:

1. Along the perpendicular bisector of the line joining the 2 sources we see regions of constructive interference. (crest on crest - supercrest - extra bright, trough on trough - supertrough - extra dark).

2. On either side of the central axis are regions of continuous total destructive interference. In these regions, named nodal lines, the 2 sets of waves always arrive in opposite phase - crest on trough.  The result is no displacement in the water surface. These "regions of calm" neither converge nor diverge the light. Hence these regions are seen as regions of normal brightness.

3. These nodal lines are hyperbolic in shape. Far away from the sources these lines straighten out but curve sharply near the source.

4. Nodal lines are numbered from the central axis.

5. If the distance between the sources increases, # of nodal lines increases

6. If the frequency of the 2 sources is increased (wavelength decreases), #of nodal lines increases.

7. If the sources are 180 degrees out of phase destructive interference will be on the right bisector.

Two point interference pattern applets:

Show 2 point interference pattern

On overhead show the path difference for any point on the nth nodal line:

for nodal lines (destructive interference)

The following relationships are based on one additional condition imposed on the standing wave pattern from the 2 points vibrating in phase which is:

PS1, PS2>>>>>>>>>S1S2 (i.e. when the Point P is far away)

Show:

·  d is distance between sources.

sometimes hard to find if wavelength and d are very small so need another way to solve………………

We will prove:

is the perpendicular distance from right bisector to point on nodal line

L is the distance from the point P to the midpoint between the 2 sources

is the nodal line number

·     NOTE: make sure you understand that  is the angle between the right bisector and L so you know how to APPLY this proof and remember that it is only for two point sources in phase.

Example 1:

A ripple tank has two identical point sources 4.0 cm apart, operating in phase at a frequency of 10.0 Hz. It produces an interference pattern where a point on the second nodal line is located 15.0 cm from one source and 18.0 cm from the other. What is

a) the wavelength of the waves? (2.0 cm)

b) the speed of the waves? ( 20.0 cm/s)

Example 2:

Two sources are vibrating in phase, and set up waves in a ripple tank. A point P on the second nodal line is 12.0 cm from source A and 20.0 cm from source B. When the sources are started, it takes 2.0s for the first wave to reach the edge of the tank, 30 cm from the source. Find the velocity, wavelength and frequency of the wave. (15 cm/s, 5.3 cm, 2.8 Hz)

Suggested Text Questions ( students do page 460 #5 in class):

Page 459 #'s 1-3

Page 460 #'s 5,6,8,9