Question

Distance between the trough and crest of a wave is $1.5\text{m}$ while distance between two crests is $5\text{m}$. Which of the following wavelengths is possible?

• A. $\frac{1}{2},\frac{1}{4},\frac{1}{6},...$
• B. $1,2,3,...$
• C. $1,\frac{1}{3},\frac{1}{5},...$
• D. $1,3,5,...$

Answer 1

The correct option is C $1,\frac{1}{3},\frac{1}{5},...$

Distance between the trough and crest is an odd integral multiple of half wavelength ($\lambda /2$).
$\Rightarrow 1.5=(2n_1+1)\frac{\lambda }{2}....\left(1\right)$
where $n_1=0,1,2,\mathrm{3..}$

Distance between two crests is an integral multiple of wavelength.
$\Rightarrow 5=n_2\lambda ....\left(2\right)$
where $n_2=1,2,\mathrm{3..}$

From $\left(1\right)$ and $\left(2\right)$,
$\frac{1.5}{5}=\frac{2n_1+1}{2n_2}$
$\Rightarrow 3n_2=10n_1+5$$....\left(3\right)$

From $\left(2\right)$ and $\left(3\right)$,
If, $n_1=1,n_2=5and\lambda =1\text{m}$
If, $n_1=4,n_2=15and\lambda =\frac{1}{3}\text{m}$
If, $n_1=7,n_2=25and\lambda =\frac{1}{5}\text{m}$

Why this question? Concept involved: Distance between two successive troughs/crests and between successive crest and trough. Importance in $JEE$ : Asked in Main $2020$.