2D Wave InterferenceBy: Rick Grootes

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!

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,…

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.

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,…

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.

Let us try an Example (1)

Please take a look at this image and find the question on the next slide:

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

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…

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This function is very close to that of a wave function in 1-Dimension, except the x value has been replaced by r (radius), as spheres are calculated with the radius.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.

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)

Answer to Example 1

Sources Cited

http://hyperphysics.phy-astr.gsu.edu/hbase/sound/imgsou/int5.gif

http://cnx.org/resources/a4005719a108fcb0bab88b3006667605/PG12C6_007.png