Question

A wave is travelling in string 1 with velocity $v_1$. It is then both reflected and transmitted from a joint to string 2 where the velocity is $v_2$. If $A_i$ represents the incident amplitude, What are the reflected and transmitted amplitudes?

• A. $A_r=\left(\frac{v_2-v_1}{v_1+v_2}\right)A_i$ and $A_t=\left(\frac{2v_2}{v_1+v_2}\right)A_i$
• B. $A_r=\left(\frac{2v_2}{v_1+v_2}\right)A_i$ and $A_t=\left(\frac{v_2-v_1}{v_1+v_2}\right)A_i$
• C. $A_r=\frac{A_i}{2}$ and $A_t=\frac{A_i}{2}$
• D. None of these

Answer 1

The correct option is A

$A_r=\left(\frac{v_2-v_1}{v_1+v_2}\right)A_i$ and $A_t=\left(\frac{2v_2}{v_1+v_2}\right)A_i$

By conservation of energy, the average incident power equals the average reflected power plus the average transmitted power

or $P_i=P_r+P_t$

$\therefore \frac{1}{2}{\mu }_1{\omega }^2{A}_1^2{v}_1=\frac{1}{2}{\mu }_1{\omega }^2{A}_r^2{v}_1+\frac{1}{2}{\mu }_2{\omega }^2{A}_r^2{v}_2$

or $\left(\frac{T}{v_1^2}\right){\omega }^2{A}_i^2{v}_1=\left(\frac{T}{v_1^2}\right){\omega }^2{A}_r^2{v}_1+\left(\frac{T}{v_2^2}\right){\omega }^2{A}_t^2{v}_2$

or $\frac{A_i^2}{v_1}=\frac{A_r^2}{v_1}+\frac{A_t^2}{v_2}$ ............(i)

Further $A_i+A_r=A_t$ [from boundary conditions] ............(ii)

Solving these two equations for $A_r$ and $A_t$, we get

$A_r=\left(\frac{v_2-v_1}{v_1+v_2}\right)A_i$ and $A_t=\left(\frac{2v_2}{v_1+v_2}\right)A_i$

Hence proved