Question

A sinusoidal wave on a string is described by the wave function
$y=0.3\sin (1.6x-100t)$
where $x$ and $y$ are in $\text{metre}$ and $t$ is in $\text{second}$. The mass per unit length of this string is $10\text{g/m}.$ Determine the power transmitted to the wave.

• A. $260\text{J}$
• B. $360\text{J}$
• C. $281\text{J}$
• D. $320\text{J}$

Answer 1

The correct option is C $281\text{J}$

Given,
wave function is
$y=0.3\sin (1.6x-100t)$

Comparing this with the general wave equation,
$y=A\sin (kx-\omega t)$

we have ,
Amplitude$\left(A\right)=0.3\text{m}$
Angular frequency$\left(\omega \right)=100\text{rad/s}$
Angular wave number$\left(k\right)=1.6{\text{m}}^{-1}$

wave speed $\left(v\right)$ for the given wave is
$v=\frac{\omega }{k}=\frac{100}{1.6}=62.5\text{m/s}$
and
$\mu =10\text{g/m}=10\times {10}^{-3}\text{kg/m}=0.01\text{kg/m}$

The average power carried by the wave is
$P=\frac{1}{2}\mu {\omega }^2{A}^{2}v$

Substituting the data,

$P=\frac{1}{2}\times 0.01\times (100{)}^2\times (0.3{)}^2\times 62.5=281.25\text{J}\approx 281\text{J}$

Hence, option (c) is the correct answer.