Write the given relation

The wave speed is given by $v = \lambda^a g^b \rho^c$

where:

• $v$ = speed of wave
• $\lambda$ = wavelength
• $g$ = acceleration due to gravity
• $\rho$ = density of water

Write dimensions of each quantity

• Speed: $[v] = LT^{-1}$
• Wavelength: $[\lambda] = L$
• Acceleration due to gravity: $[g] = LT^{-2}$
• Density: $[\rho] = ML^{-3}$

Substitute dimensions into the equation

$[v] = [\lambda]^a [g]^b [\rho]^c$

So, $LT^{-1} = (L)^a (LT^{-2})^b (ML^{-3})^c$

Expanding, $LT^{-1} = M^c L^{a+b-3c} T^{-2b}$

Compare powers of $M$, $L$, and $T$

• For mass $M$: $c = 0$

• For time $T$: $-2b = -1 \Rightarrow b = \frac{1}{2}$

• For length $L$: $a + b - 3c = 1$ Since $c=0$ and $b=\frac{1}{2}$, $a + \frac{1}{2} = 1$ $a = \frac{1}{2}$

Final values

$a = \frac{1}{2}, \quad b = \frac{1}{2}, \quad c = 0$

Check options

This matches: Option D: $\left(\frac{1}{2}, \frac{1}{2}, 0\right)$