2d wave interference

InterferenceChapter 15-5

L02

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Why Interference again?We already studied interference but that was only in 1D

The phase does not vary with time or position

In 3D interference: Waves propagate in different directions so their relative phase varies with position

We will be focusing on 2D interference

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Constructive/Destructive

Remember:

When two waves with same wavelength and frequency are in phase, they are constructive.

The amplitudes add at these points

When two waves with same wavelength and frequency are out of phase they are destructive

Amplitude usually decreases

Constructive interference

Two points have to be in phase, regardless of time

Constructive interference means that both sources are at the peak positive amplitude (when both sources are at the peak positive amplitude)

Then, there is an integer number of wavelengths between each source and the point under consideration

Constructive interference

Any point that is an integer multiple of wavelength from both sources will undergo continuous constructive interference http://blog.ocad.ca/wordpress/gdes3b78-fw201203-01/files/2013/03/wave.jpg

The Math of Constructive Interference

If d1 = path length from source 1 and d2 = path length from source 2 the condition for constructive interference is :

d1 = mλ, m=1,2,3, …

d2 = nλ, n=1,2,3, …

The difference between the distances from the two sources to the point of constructive interference is given by:

Δd = d2 -d1 = (n-m)λ = pλ where p = 0, ±1, ±2, ±3, …

Condition for Constructive Interference

Path difference condition: The path difference between the two sources must be an integer multiple of the wavelength http://www.physicsclassroom.com/Class/light/u12l3b11.gif

BUT WAIT

What we just did demands that both paths individually be integer multiples of wavelength. But this does not have to be true…

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The path difference condition

Remember the path difference condition? That condition might be sufficient enough to produce constructive interference

https://coherence.files.wordpress.com/2011/10/waves.png

Spherical wavesAs spherical waves travel away from its source, it oscillates in space and time

The amplitude is constant over any spherical surface centred on the source

Spatial variations are described as a function of r, the distance from the source. The wave function becomes:

s(r,t) = sm(r)cos(kr-ωt+ɸ)

The functions(r,t) = sm(r)cos(kr-ωt+ɸ) may look familiar to you

Recall: s(x,t) = sm(r)cos(kx-ωt+ɸ)

The only difference is that our new equation replaces x with r. This is because the wave spreads out over a larger area as it propagates outwards

The Math of Constructive Interference, Again…

For two waves to be in phase, the arguments of the cosine function must differ by an integer multiple of 2∏.

Both waves are in phase so:

(kd2-ωt) - (kd1-ωt) = k(d2-d1) = n2∏

Therefore: (d2-d1) = n(2∏/k) = nλ , n = 0, ±1, ±2, ±3

Confusion?Remember how in our initial condition we got:

Δd = d2 -d1 = (n-m)λ = pλ where p = 0, ±1, ±2, ±3, …

But both paths individually had to be integer multiples of wavelength.

We just demonstrated that this does not always have to be the case since:

(d2-d1) = n(2∏/k) = nλ , n = 0, ±1, ±2, ±3 http://fc01.deviantart.net/fs70/f/2012/023/3/f/unagi_by_co__existance-d4ndy09.png

Constructive Interference Formula

Constructive interference occurs whenever the path difference is an integer multiple of the wavelength.

In a special case where d2= d1= d, we can add the two waves together to find a resultant wave equation:

s(d,t) = 2sm(d)cos(kd-ωt+ɸ)

Destructive Interference Occurs when one path is an integer number of wavelengths and the other is a half-integer multiple multiple

Therefore: The path difference is a half-integer multiple of the wavelength (odd number of half wavelengths)

Equation: Δd = d2 -d1 = ((2n+1)/2)λ = (n+1/2)λ where n = 0, ±1, ±2, ±3

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Tips for 2-D InterferenceThe equations discussed may look complex. Try and understand what each individual variable represents

Remember, 2D interference is still interference so if you get confused try and remember 1D interference. It might help clarify certain concepts for you.

Interference: Two or more waves combining to produce a resultant wave