Question

Two waves propagating in a given medium are represented by the curve shown below.

Find the ratio of average potential energy per unit length of wave $1$ is to wave $2$. All measurements are in $\text{S.I}$ unit.

• A. $1$
• B. $1/2$
• C. $1/4$
• D. $1/8$

Answer 1

The correct option is A $1$

For wave $1$
Amplitude $\left(A\right)=4\text{m}$ and time period $=T$
For wave $2$
Amplitude $\left(A^{\prime }\right)=\frac{A}{2}=2\text{m}$
and time period $=\frac{T}{2}$
Average potential energy per unit length of wave is,
$<\frac{dU}{dx}>=\frac{1}{4}\mu {\omega }^2{A}^2$
$\Rightarrow \frac{<\frac{d{U}_1}{dx}>}{<\frac{d{U}_2}{dx}>}=\frac{{\omega }^2{A}^2}{{\omega }^{\prime 2}{A}^{\prime 2}}$
$\Rightarrow \frac{<\frac{d{U}_1}{dx}>}{<\frac{d{U}_2}{dx}>}=\frac{{\left(\frac{2\pi }{T}\right)}^2{A}^2}{{\left(\frac{2\pi }{T/2}\right)}^2{\left(\frac{A}{2}\right)}^2}=1$
$\therefore$Option $\left(a\right)$ is the correct answer.