Question

A harmonic source $\left(S\right)$ is driving a taut string. The other end of the string is tied to a wall that is not so rigid. It is observed that standing waves are formed in the string with ratio of amplitudes at the antinodes to that at the nodes equal to $8$. What percentage of wave energy is transmitted to the wall?

• A. $18\%$
• B. $24\%$
• C. $39\%$
• D. $53\%$

Answer 1

The correct option is C $39\%$

From the data given in the question,

$\frac{A_{max}}{A_{min}}=\frac{A_i+A_r}{A_i-A_r}=8$

Where $A_i\to$amplitude of the incident wave.
$A_r\to$ amplitude of the reflected wave.

By using, componendo -dividendo rule we get, $A_i=\frac{9}{7}{A}_r$

Since, $I\propto A^2$ , we can write that,

$\frac{I_r}{I_i}={\left(\frac{A_r}{A_i}\right)}^2=\frac{49}{81}$

Percentage of energy transmitted,

$\left(\frac{I_i-I_r}{I_i}\right)\times 100=(1-\frac{49}{81})\times 100=39.5\%$

Percentage of wave energy transmitted to the wall is $39.5\%\approx 39\%$

Thus, option (c) is the correct answer.