The speed of a water wave v depends on its wa | vedclass

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(C) Let the relationship be $v = k g^a \lambda^b \rho^c$,where $k$ is a dimensionless constant.Writing the dimensions of each quantity:$[v] = [L T^{-1

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$v^2 \propto g \lambda$

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(C) Let the relationship be $v = k g^a \lambda^b \rho^c$,where $k$ is a dimensionless constant.Writing the dimensions of each quantity:$[v] = [L T^{-1}]$$[g] = [L T^{-2}]$$[\lambda] = [L]$$[\rho] = [M L^{-3}]$Substituting these into the equation: $[L T^{-1}] = [L T^{-2}]^a [L]^b [M L^{-3}]^c$$[L^1 T^{-1} M^0] = [M^c L^{a+b-3c} T^{-2a}]$Comparing the powers of $M, L,$ and $T$ on both sides:For $M$: $c = 0$For $T$: $-2a = -1 \implies a = 1/2$For $L$: $a + b - 3c = 1 \implies 1/2 + b - 0 = 1 \implies b = 1/2$Thus,$v \propto g^{1/2} \lambda^{1/2} \rho^0 \implies v \propto \sqrt{g \lambda} \implies v^2 \propto g \lambda$.

The speed of a water wave $v$ depends on its wavelength $\lambda$,the density of water $\rho$,and the acceleration due to gravity $g$. Determine the relationship between these quantities.

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