Question

Experiments reveal that the velocity $v$ of water waves may depend on their wavelength $\lambda$, density of water $\rho$, and acceleration due to gravity g. Establish a possible relation between $v$ and $\lambda ,g,{\rho }^0.$.

Answer 1

According to the provided information,
$v\propto {\lambda }^a{\rho }^b{g}^c$
$\Rightarrow v=k{\lambda }^a{\rho }^b{g}^c$
where k is constant of proportionality.
Using principle of homogeneity
$\left[M^0{L}^1{T}^{-1}\right]$

$=\left[L^a\right]\left[M^1{L}^{-3}{T}^0{]}^b\right[M^0{L}^1{T}^{-2}{]}^c$
Comparing powers of like quantities on both the sides, we have,
$b=0\left(ii\right)a-3b+c=1\left(iii\right)-2c=-1\left(iv\right)$
Using (ii), (iii) and (iv), we have, $a=\frac{1}{2},b=0,c=1/2$
Using these values in (i), we have $v=k.{\lambda }^{1/2}.{\rho }^0.g^{1/2}$
$\Rightarrow v=k\sqrt{\lambda g}$
which is the required relation.