1. Key wave geometry facts

For a sinusoidal transverse wave:

• Distance between two successive crests $= \lambda$
• Distance between a crest and the nearest trough $= \dfrac{\lambda}{2}$

More generally:

• Distance between any two crests can be $n\lambda$, where $n$ is an integer.
• Distance between a crest and any trough can be $\frac{(2m+1)\lambda}{2}, \qquad m=0,1,2,\dots$

2. Use the given crest-to-crest distance

Given one distance between two peaks is $5\,\text{m}$. So, $5=n\lambda$ for some positive integer $n$. Hence, $\lambda=\frac{5}{n}$ So possible wavelengths from this condition are: $5,\; \frac{5}{2},\; \frac{5}{3},\; \frac{5}{4},\dots$

3. Use the given crest-to-trough distance

Given one crest-to-trough distance is $1.5\,\text{m}$. So, $1.5=\frac{(2m+1)\lambda}{2}$ which gives $\lambda=\frac{3}{2m+1}$ Thus possible wavelengths from this condition are: $3,\;1,\;\frac{3}{5},\;\frac{3}{7},\dots$

4. Find common values

The wavelength must satisfy both conditions: $5=n\lambda \quad \text{and} \quad 1.5=\frac{(2m+1)\lambda}{2}$

Equating: $\frac{5}{n}=\frac{3}{2m+1}$ $5(2m+1)=3n$

Since $n$ must be an integer, $5(2m+1)$ must be divisible by $3$. This happens when $(2m+1)=3,9,15,\dots$ So, $\lambda=\frac{3}{3}=1,\quad \frac{3}{9}=\frac{1}{3},\quad \frac{3}{15}=\frac{1}{5},\dots$

Therefore the possible wavelengths are: $1,\;\frac{1}{3},\;\frac{1}{5},\dots$

5. Match with options

This corresponds to Option B.

6. Comparison with stored answer

Stored correct answer: B

My derived answer: B

They agree.