Question

A wave is represented by the equation
$y=10\sin \left[2\pi \right(2t-3x\left)\right]$. The tension in the string on which wave is set is $10\text{N}.x,y$ are in $\text{cm}$. Which of the following is/are correct?

• A. Average power of the wave is $12\text{W}$.
• B. Average energy per wave is $18\text{J}$.
• C. Average power of wave is $24\text{W}$.
• D. The velocity of the wave is $\frac{2}{3}{\text{ms}}^{-1}$.

Answer 1

Answer: (A), (B), (D)

The velocity of the wave is $\frac{2}{3}{\text{ms}}^{-1}$.

Comparing the given wave equation with standard wave equation, $y=A\sin (\omega t-kx)$, we get

$A=10\text{cm}=0.1\text{m}$
$\omega =4\pi$
$k=6\pi$

$T=10\text{N}$ (Given)

Average power of wave is given,

$P=\frac{1}{2}T{A}^2\omega k$

$P=\frac{1}{2}\times 10\times (0.1{)}^2\times 4\pi \times 6\pi$

$\Rightarrow P\approx 12\text{W}$

Average energy per wave is given by,

$E=\frac{1}{2}\mu {\omega }^2{A}^2$

Velocity of wave $V_w=\frac{\omega }{k}=\frac{4\pi }{6\pi }=\frac{2}{3}{\text{ms}}^{-1}$

Also,

$V_w=\sqrt{\frac{T}{\mu }}$

$\Rightarrow \frac{2}{3}=\sqrt{\frac{10}{\mu }}$

$\Rightarrow \frac{4}{9}=\frac{10}{\mu }$

$\Rightarrow \mu =\frac{90}{4}$

$\Rightarrow E=\frac{1}{2}\times \frac{90}{4}\times (4\pi {)}^2\times (0.1{)}^2$

$\Rightarrow E\approx 18\text{J}$

Hence, option $A,B\text{and}D$ are correct.

Why this question? This question gives you understanding about power and energy of wave. It helps you in remembering formula, $P=\frac{1}{2}T{A}^{2}wk,E=\frac{1}{2}\mu w^2{A}^2/\text{wave}$