2.6 Expression for the reflection and transmission of wave

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

Let us understand the reflection and transmission of wave mathematically.

The equation of an incident wave can be written as, $${y_i} = {A_i}\cos ({k_2}x - \omega t)$$ The equation of a reflected wave can be written as, $${y_r} = {A_r}\cos ({k_1}x + \omega t)$$ Similarly, the equation of transmitted wave is, $${y_t} = {A_t}\cos ({k_2}x - \omega t)$$

The wave must be continious. So, at the boundary (junction), the resultant wave in medium 1 should be equal to the resultant wave in medium 2.

Mathematically, $${y_i} + {y_r} = {y_t}$$ $$\begin{equation} \begin{aligned} {y_i} + {y_r} = {y_t} \\ {A_i}\cos \left( {{k_1}x - \omega t} \right) + {A_r}\cos \left( {{k_1}x + \omega t} \right) = {A_t}\cos \left( {{k_2}x - \omega t} \right) \\\end{aligned} \end{equation} $$ Applying boundary condition $(x=0)$, $$\begin{equation} \begin{aligned} {A_i}\cos \left( { - \omega t} \right) + {A_r}\cos \left( {\omega t} \right) = {A_t}\cos \left( { - \omega t} \right) \\ {A_i} + {A_r} = {A_t}\quad ...(i) \\\end{aligned} \end{equation} $$

Slope at the boundary must be equal, $$\begin{equation} \begin{aligned} \frac{{\partial {y_i}}}{{\partial x}} + \frac{{\partial {y_r}}}{{\partial x}} = \frac{{\partial {y_t}}}{{\partial x}} \\ - {A_i}{k_1}\sin \left( {{k_1}x - \omega t} \right) - {A_r}{k_1}\sin \left( {{k_1}x + \omega t} \right) = - {A_t}{k_2}\sin \left( {{k_2}x - \omega t} \right) \\\end{aligned} \end{equation} $$ Applying boundary condition $(x=0)$, $$\begin{equation} \begin{aligned} - {A_i}{k_1}\sin \left( { - \omega t} \right) - {A_r}{k_1}\sin \left( {\omega t} \right) = - {A_t}{k_2}\sin \left( { - \omega t} \right) \\ {A_i}{k_1} - {A_r}{k_1} = {A_t}{k_2}\quad ...(ii) \\\end{aligned} \end{equation} $$ From equation $(i)$ and $(ii)$ we get, $${A_r} = \left( {\frac{{{k_1} - {k_2}}}{{{k_1} + {k_2}}}} \right){A_i}\quad \& \quad {A_t} = \left( {\frac{{2{k_1}}}{{{k_1} + {k_2}}}} \right){A_i}\quad ...(iii)$$ As we know, $${k_1} = \frac{\omega }{{{v_1}}}\quad and\quad {k_2} = \frac{\omega }{{{v_2}}}$$ So, equation $(iii)$ can be written as, $${A_r} = \left( {\frac{{{v_2} - {v_2}}}{{{v_1} + {v_2}}}} \right){A_i}\quad \& \quad {A_t} = \left( {\frac{{2{v_2}}}{{{v_1} + {v_2}}}} \right){A_i}$$