Lecture 20: Interference of Waves, Beats

Wave Interference

Fundamental principle for the linear waves is superposition principle

y(x,t) = y₁(x,t) + y₂(x,t) + y₃(x,t) ...
Interference , generally, is outcome of superposition of several waves. Standing wave is an example of interference

applet

The most interesting phenomena occur when the waves of the same or nearly the same frequency interfere
For those waves, the oscillations in medium can be enhanced or (almost) completely eliminated in particular regions of space

When waves interfere to add up to a wave of higher amplitude, we speak about constructive interference	When waves interfere to cancel each other we speak about destructive interference

When we speak about interference we primarily concerned with spatial patterns of the combined wave.
Let us consider inerference pattern between two spherical waves produced by two coherent sources
coherent means that the sources have the same frequency, and, in our example, oscillate in phase

One source	Two sources at the same position

Two source separated by d = λ	Two source separated by d = 2 λ

As we see interference pattern which is stationary, although the waves are traveling (and the combined wave is traveling as well, this is not a standing wave ! )
The pattern depends on the distance between the source
- When two sources are at the same position, the interference is constructive everythere, everywhere we have a wave of twice the amplitude.
- When source are one wavelength apart, there are two rays along which the interference is destructive - there is no perturbations in the medium ! In between the interference is constructive.
- When the distance increases to 2 λ , we see 4 direction of destructive interference.
What is going on ? We shall study this on a board.
And derive that for two coherent point source
- The destructive interference occurs when the distance two waves travel differ by the odd number of half wavelength destructive: d₂ - d₁ = (2n+1) λ/2
- The constructive interference occurs when the distance two waves travel differ by the even number of half wavelength (or just integer number of full wavelengths) constructive: d₂ - d₁ = 2n λ/2 = n λ
We can also look at the Mastering Physics Interference applets

Interferometric pattern carries wealth of information about the source (and the medium)
By using interferometry one can detect the shift in phase between waves with extremely high precision
By mapping interferometric pattern some of the most precise observations in physics are done.
But one needes to retain phase information about the wave, not just measure the energy of the wave. Longer the wavelength, easier it is to retain phase information

In astronomy: first radio interferometers (for example VLA on the left and CBI on the right )
and now infrared and even optical interferometers are able to achieve the very high angluar resolution. (Large the separation between the parts of interferometer, higher the resolution is). Recently resolved planetary disc around a young star

What if the frequencies of the waves in interference are not the same ?
Not much interesting if they are widely different
But interesting phenomenom of beating occurs when they are close to each other (click on image for applet and choose case 8)

For sound waves, our ear percieve the beat as amplitude modulation if beat frequency is below 7 Hz or so.
If beat frequency is in audible range above 7 Hz or so, we hear a tone that corresponds to this frequency. Example is famous London Police whistle
Again, hearing for beat, allows to match two wave frequency very precisely, for example in musical instrument tuning
One can see beats of spherical waves here