Question

What is the wavelength for a third harmonic standing wave on a string with fixed ends if the two ends are $2.4\text{ m}$ apart?

Answer 1

Hint: If it is said the standing wave is in its third harmonic then it means that the wave would have three anti-nodes which means that there would be three loops. The wavelength of the wave would be two-third of the total length of the string. Hence, to solve this problem it is necessary to calculate the wavelength of the wave.

Complete step by step answer:
To solve this question we would use the relation between the wavelength of the standing wave and the length of the string. The wavelength of a particular wave also depends on the number of anti-nodes present in the wave. As stated above the wavelength of the wave would be two-third of the total length of the string, hence if we write it in terms of a formula then it would be:
$\lambda =\left( \dfrac{2}{3} \right)L$
Here $\lambda $ is the wavelength of the third harmonic standing wave.
And in the question, we are given that the fixed ends of the string are $2.4\text{ m}$ apart, hence the value of $L=2.4\text{ m}$. On substituting the values, we get:
$\begin{align}
  & \lambda =\left( \dfrac{2}{3} \right)\times 2.4 \\
 & \therefore \lambda =1.6\text{ m} \\
\end{align}$
Thus, the wavelength for a third harmonic standing wave on a string with fixed ends if the two ends are $2.4\text{ m}$ apart is $1.6\text{ m}$.

Note: The number of nodes is greater than, we can conclude that the number of antinodes. Hence in this case where the number of antinodes in the wave of the third harmonic is three, then the number of nodes present in the third harmonic will always be greater than three. The number of nodes present in it is four.