Correct Answer - Option 2 : 2a cos
\(\left ( \frac{\phi}{2} \right )\)
CONCEPT:
Coherent addition of waves:
- The phenomenon in which two or more waves superpose to form a resultant wave of greater, lower, or the same amplitude is called interference.
- The interference is based on the superposition principle according to which at a particular point in the medium, the resultant displacement produced by a number of waves is the vector sum of the displacements produced by each of the waves.
- Let two sources of wave S1 and S2 are coherent.
- Let the intensity of the waves are Io.
For constructive interference:
- If at any point the waves emerging from the source S1 and S2 are in phase then we will have constructive interference and the resultant intensity will be 4Io.
⇒ Path difference = nλ and n = 0,1,2,3,...
For destructive interference:
- If at any point the waves emerging from the source S1 and S2 are in opposite phase then we will have destructive interference and the resultant intensity will be zero.
⇒ Path difference = \(\left ( n+\frac{1}{2} \right )\)λ and n = 0,1,2,3,...
Where λ = wavelength
- Let at any point the phase difference between the displacements of the two waves is ϕ.
- So the displacement of the two waves is given as,
⇒ y1 = a.cos(ωt)
⇒ y2 = a.cos(ωt + ϕ)
- When the waves are superimposed at this point the resultant displacement is given as,
\(\Rightarrow y=2acos\left ( \frac{\phi}{2} \right )cos\left ( \omega t+ \frac{\phi}{2} \right )\)
The amplitude of the resultant displacement is given as,
\(\Rightarrow A=2acos\left ( \frac{\phi}{2} \right )\)
The intensity at that point is given as,
\(\Rightarrow I=4I_ocos^2\left ( \frac{\phi}{2} \right )\)
EXPLANATION:
Given y1 = a.cos(ωt), and y2 = a.cos(ωt + ϕ)
- We know that if the two waves are superimposed, then the amplitude of the resultant displacement is given as,
\(\Rightarrow A=2acos\left ( \frac{\phi}{2} \right )\)
- Hence, option 2 is correct.
