 # Coherent and Inconherent Addition of Waves

• No two Light rays are coherent
• The difference in the paths traversed by the two Light waves at the time is called path difference (Δ x).
• The difference in the phase angles expressed in radians between the two waves at the time of arrived at a point is called phase difference (Δx).
• The relation between phase difference and path difference is \tt \Delta \phi=\frac{2\pi}{\lambda}\left(\Delta x\right)
• According to the principle of superposition \tt \overline{Y}=\overline{Y_{1}}+\overline{Y_{2}}
• Expression for resultant Amplitude \tt A=\sqrt{A_1^2+A_2^2+2A_{1}A_{2}\cos\phi}
• If \tt A_{1}=A_{2}=a\ \Rightarrow\ A=2a\ \cos\frac{\phi}{2}
• Expression for resultant Intensity \tt I=I_{1}=I_{2}+2\sqrt{I_{1}I_{2}}\ \cos\ \phi
• If \tt I_{1}=I_{2}=I_{0}\ then\ I =4I_{0}\cos^{2}\frac{\phi}{2}
• Maximum Amplitude in superposition Amax = A1 + A2
• Maximum Intensity is Superposition \tt I_{max}=\left(\sqrt{I_{1}}+\sqrt{I_{2}}\right)^2=\left(A_{1}+A_{2}\right)^2
• Minimum Amplitude Amin = A1 – A2
• Minimum Intensity in superposition \tt I_{min}=\left(\sqrt{I_{1}}-\sqrt{I_{2}}\right)^2=\left(A_{1}-A_{2}\right)^2
• \tt \frac{I_{max}}{I_{min}}=\frac{\left(\sqrt{I_{1}}+\sqrt{I_{2}}\right)^{2}}{\left(\sqrt{I_{1}}-\sqrt{I_{2}}\right)^{2}}=\frac{\left(A_{1}+A_{2}\right)^{2}}{\left(A_{1}-A_{2}\right)^{2}}
• Interference is the modification in the distribution of intensity of light in the region of superposition of waves is called interference.
• The phenomenon of interference is explained by Huygens wave theory.
• Constructive interference is formed when two waves reach with a phase difference of 0, 2π, 4π ……….
• Destructive interference is formed when two waves each with a phase difference of π, 3π, 5π …………
• Interference obeys law of conservation of energy.
• Sustained interference is the interference in which the intensity of maxima and minima are visible,
• Condition for project bright fringe in young's Double slit Experiment with path difference = \tt 2n\left(\frac{\lambda}{2}\right) {n=0,1,2,3.....}
• Phase difference for bright fringe = 2nπ . {n = 0, 1, 2, 3 …………}
• The Distance of nth maxima from central maxima is \tt y=\frac{n\lambda D}{d} {n=0,1,2,3.....}
• Path difference for Dark fringe = \tt \left(2n-1\right)\frac{\lambda}{2} {n= 1,2,3.....}
• Phase difference for Dark fringe = (2n − 1)π {n = 1, 2, 3 ………..}
• The Distance of nth minima from central maxima is \tt y=\left(2n-1\right)\frac{\lambda D}{d} {n = 1, 2, 3 ………..}

### View the Topic in this video From 00:15 To 05:05

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1. When two waves superimpose the resultant intensity at a point is given by
I = I_{1} + I_{2} + 2\sqrt{I_{1}I_{2}}

2. Ratio of maximum and minimum intensities :
\tt \frac{I_{max}}{I_{min}} = \frac{(a_{1} + a_{2})^{2}}{(a_{1} - a_{2})^{2}} = \frac{\left[1 + \frac{a_{2}}{a_{1}}\right]^{2}}{\left[1 - \frac{a_{2}}{a_{1}}\right]^{2}} =\frac{\left[1+\frac{\sqrt{I_2}}{I_1}\right]^2}{\left[1-\frac{\sqrt{I_2}}{I_1}\right]^2}= \frac{\left(\sqrt{I_{1}} + \sqrt{I_{2}}\right)^{2}}{\left(\sqrt{I_{1}} - \sqrt{I_{2}}\right)^{2}}