Question

The adjacent diagram shows three different modes of vibrations $P,Q$ and $R$ of the same string.
State the ratio of wavelengths of $P$ and $R$.

Answer 1

A wavelength is a measure of distance between two identical peaks (crests and troughs) in a wave. Here, we can consider $R$ is having one crest or one trough between the ends of the string. So, its having a wavelength ($\lambda$) of $\frac{\lambda }{2}$. Similarly, $P$ is having two crests and one trough (or one crest and two troughs) which are happening at the same time. Thus its wavelength($\lambda$) is $\frac{3\lambda }{2}$.

Thus, ${\lambda }_R=\frac{\lambda }{2}$

and ${\lambda }_P=\frac{3\lambda }{2}$

Hence ratio of wavelength of $P$ and $R$ i.e., ${\lambda }_P:{\lambda }_R=\frac{3\lambda }{2}:\frac{\lambda }{2}=3:1$.