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

Constructive and Destructive Interference

  • 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

applet

Wave Interference and spatial patterns

  • 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

Interference pattern

  • 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

Interferometry - the most precise observations in physics

  • 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

Beating

  • 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)
  • Beating frequency fbeating = f2 - f1