2.2 Interference of waves from coherent sources

1.1 Principle of superposition

2.1 Relation between phase difference $\left( \phi \right)$ and path difference $\left( {\Delta x} \right)$
2.2 Interference of waves from coherent sources
2.3 Interference of waves from incoherent sources
2.4 Reflection and transmission of a wave
2.5 Motion of wave during reflection
2.6 Expression for the reflection and transmission of wave

3.1 Transverse stationary wave on a stretched string
3.2 Vibrations in a stretched string
3.3 Melde's Experient
3.4 Resonance

4.1 Open organ pipe
4.2 Closed organ pipe
4.3 End correction
4.4 Resonance tube
4.5 Energy in a stationary wave

Two sources are said to be coherent if they have,

Consider two coherent sources $S_1$ and $S_2$ which oscillate with angular frequency $\omega $. A point $P$ is situated at a distance $x$ and $\left( {x + \Delta x} \right)$ from source $S_1$ and $S_2$ respectively. So, the path difference between two waves reaching point $P$ from source $S_1$ and $S_2$ is $\Delta x$.

The wave equation at point $P$ due to the motion of two waves are described by, $$\begin{equation} \begin{aligned} {y_1} = {A_1}\sin \left( {kx - \omega t} \right) \\ {y_2} = {A_2}\sin \left[ {\left( {kx - \omega t} \right) + \phi } \right] \\\end{aligned} \end{equation} $$ Phase difference, $$\phi = \left( {\frac{{2\pi }}{\lambda }} \right)\Delta x$$

Resultant wave at point $P$ is given by, $$\begin{equation} \begin{aligned} y = {y_1} + {y_2} \\ y = {A_1}\sin \left( {kx - \omega t} \right) + {A_2}\sin \left[ {\left( {kx - \omega t} \right) + \phi } \right] \\ y = {A_1}\sin \left( {kx - \omega t} \right) + {A_2}\sin \left( {kx - \omega t} \right)\cos \phi + {A_2}\cos \left( {kx - \omega t} \right)\sin \phi \\ y = \left( {{A_1} + {A_2}\cos \phi } \right)\sin \left( {kx - \omega t} \right) + {A_2}\sin \phi \cos \left( {kx - \omega t} \right) \\ y = A\cos \theta \sin \left( {kx - \omega t} \right) + A\sin \theta \cos \left( {kx - \omega t} \right) \\ y = A\sin \left( {kx - \omega t + \theta } \right) \\\end{aligned} \end{equation} $$ where, $$\begin{equation} \begin{aligned} A\cos \theta = {A_1} + {A_2}\cos \phi \quad ...(i) \\ A\sin \theta = {A_2}\sin \theta \quad ...(ii) \\\end{aligned} \end{equation} $$ Dividing equation $(ii)$ by $(i)$ we get, $$\tan \theta = \left( {\frac{{{A_2}\sin \theta }}{{{A_1} + {A_2}\cos \phi }}} \right)$$ Squaring and adding equation $(i)$ & $(ii)$ we get, $${A^2} = A_1^2 + A_2^2 + 2{A_1}{A_2}\cos \phi $$ As, $I \propto {A^2}$. So, $$I = {I_1} + {I_2} + 2\sqrt {{I_1}{I_2}} \cos \phi $$ Vector representation of waves and its resultant is shown below,

Condition for constructive interference,

Phase difference, $\phi = 2n\pi $ where $n = 0,1,2....\;$

Relation between phase difference and path difference is, $$\begin{equation} \begin{aligned} \phi = \left( {\frac{{2\pi }}{\lambda }} \right)\Delta x \\ \Delta x = n\lambda \quad (As,\phi = 2n\pi ) \\\end{aligned} \end{equation} $$ So, the path difference for constructive interference is $\Delta x = n\lambda $.

At constructive interference, $$\begin{equation} \begin{aligned} A_{\max }^2 = A_1^2 + A_2^2 + 2{A_1}{A_2}\cos (2n\pi ) \\ A_{\max }^2 = A_1^2 + A_2^2 + 2{A_1}{A_2}\quad \left[ {\cos (2n\pi ) = 1} \right] \\ A_{\max }^2 = {\left( {{A_1} + {A_2}} \right)^2} \\ {A_{\max }} = \left( {{A_1} + {A_2}} \right) \\\end{aligned} \end{equation} $$ Similarly,$${I_{\max }} = {\left( {\sqrt {{I_1}} + \sqrt {{I_2}} } \right)^2}$$

Condition for destructive interference,

Phase difference, $\phi = (2n+1)\pi $ where $n = 0,1,2....\;$

Relation between phase difference and path difference is, $$\begin{equation} \begin{aligned} \phi = \left( {\frac{{2\pi }}{\lambda }} \right)\Delta x \\ \Delta x = (2n + 1)\frac{\lambda }{2}\quad [As,\phi = (2n + 1)\pi ] \\\end{aligned} \end{equation} $$ So, the path difference for constructive interference is $\Delta x = (2n + 1)\frac{\lambda }{2}$

At destructive interference, $$\begin{equation} \begin{aligned} A_{\min }^2 = A_1^2 + A_2^2 + 2{A_1}{A_2}\cos [(2n+1)\pi ] \\ A_{\min }^2 = A_1^2 + A_2^2 - 2{A_1}{A_2}\quad \left[ {\cos [(2n+1)\pi ] = -1} \right] \\ A_{\min }^2 = {\left( {{A_1} - {A_2}} \right)^2} \\ {A_{\min }} = \left( {{A_1} - {A_2}} \right) \\\end{aligned} \end{equation} $$ Similarly,$${I_{\min }} = {\left( {\sqrt {{I_1}} - \sqrt {{I_2}} } \right)^2}$$