This a learning object to show how to work with two different types of interference, which are constructive and destructive

1. 2D Wave Interference
By: Rick Grootes

2. What is it?
 Interference happens when two waves moving along the
same medium meet.
 There are two different types of interference:
- Constructive Interference
- Destructive Interference
 Now, let’s get into the nitty gritty of this concept!

3. Constructive Interference
 This occurs when the two pulses have the same shape, and
their amplitudes add up to form an amplitude twice the
height of the original.
 Let’s look at two point sources that are in phase and have the
same frequency and wavelengths.
 If source 1 has a path length d1 and source 2 has a path length
d2:
 d1=mλ where m=1,2,3,…
 d2=nλ where n=1,2,3,…
 Δd=d2-d1=(n-m)λ=pλ where p=0,±1,±2,±3,…

4. Where do we find this?
 When two waves are propagating in slightly different
directions, the constructive interference occurs where
two waves are perfectly in phase.

5. Destructive Interference
 This occurs when the two pulses cancel each other out
as the pulses are in the opposite direction.
 Let’s again look at two point sources that are in phase
and have the same frequency and wavelengths.
 If source 1 has a path length d1 and source 2 has a path
length d2:
 Δd=d2-d1=[(2n-1)λ]/2=(n+1/2)λ where n=0,±1,±2,±3,…

6. Where do we find this?
 When two waves are propagating in slightly different
directions, the destructive interference occurs where
two waves are perfectly out of phase.

7. Let us try an Example (1)
 Please take a look at this image and find the question
on the next slide:

8. Ex) Cont.
 The black lines on the image are the peaks, and the
lighter lines are the troughs.
 Now, please answer the below questions:
 a) which coloured dots are constructive interference?
 b) which coloured dots are destructive interference?
 c) which letters are neither?
 p.s. A and B are to depict source A and source B

9. Interference in 2-Dimensions
 Waves don’t only happen on strings, so let’s think of
another wave we know all too well…
 SOUND! EXACTLY!
 Sound waves travel in spheres away from the source,
and create spherical pulses of sound.
 Let’s look closer at this…

10. The wave function of Spherical Waves
 We can write it as:
s(r,t)=sm(r)cos(kr-ωt+Φ)
 This function is very close to that of a wave function in 1-
Dimension, except the x value has been replaced by r
(radius), as spheres are calculated with the radius.
 As radius (r) increases, the amplitude decreases.

11. Mathematically
 Let’s do some more math:
 If two waves are in phase, we can take the difference of the
two waves:
 (kd2-ωt)-(kd1-ωt)=k(d2-d1)=n2π
 d2-d1=2πn/k=nλ where n=0,±1,±2,±3,…
 From this, we can tell that the two waves must differ by an
integer of 2π.
 When d2=d1=d:
S(d,t)=sm(d)cos(kd-ωt+Φ)+sm(d)cos(kd-ωt+Φ)
=2sm(d)cos(kd-ωt+Φ)

12. Answer to Example 1

13. Sources Cited
 http://hyperphysics.phy-
astr.gsu.edu/hbase/sound/imgsou/int5.gif
 http://cnx.org/resources/a4005719a108fcb0bab88b300
6667605/PG12C6_007.png