I was going though the derivation of intensity of waves from coherent sources for constructive and destructive interference:

Suppose you have two sources that are at the same frequency and have the same amplitude and phase but are at different locations. One source might be a distance $x$ away from you and the other a distance $x+\Delta x$ away from you. The waves from these two sources add like: $\displaystyle s(x,t)$ $\textstyle =$ $\displaystyle s_0 \sin(k x - \omega t) + s_0 \sin(k (x + \Delta x) - \omega t)$

The resultant wave at any point is given by


Where $A^2$=${A1}^2$+${A2}^2$+2A1A2cosϕ.

Now as I(Intensity) ∝ $A^2$,This equaion can be written as


EDIT: Coherent sources have same frequency but they can have varying wavelength so why is wavelength assumed equal here?

  • $\begingroup$ Isn't the wave velocity constant? $\endgroup$ – Farcher Mar 31 '18 at 9:17
  • $\begingroup$ I think the Op is asking about why coherent sources have same velocity as shown $\endgroup$ – user184271 Mar 31 '18 at 9:57
  • $\begingroup$ If the waves are overlapping at the same point in space then the speed of the waves and hence their wavelength will be the same if the waves have the same frequency. $\endgroup$ – Farcher Mar 31 '18 at 10:35
  • $\begingroup$ @Farcher That could explain this $\endgroup$ – user184272 Mar 31 '18 at 15:49
  • $\begingroup$ @Farcher If two waves interfere at the same point in the space with different frequency then will the converse of above happen $\endgroup$ – user184272 Mar 31 '18 at 16:23

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