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`.

Explanation

According to the provided information ,
`v prop lambda^(a) rho^(b) g^( c )`
`implies v = k lambda^(a) rho ^(b) g^( c )` ….(i)
where `k` is constant of proportionality.
Using principle of homogeneity
`[ M^(0) L^(1) T^(-1)] = [ L]^(a) [ M^(-1) L^(-3) T^(0) ]^(b) [M^(0) L^(1) T^(-2)]^(c )`
`= [ M^(b) L^(a - 3b + c )T^(-2c )]`
Comparing powers of like quantities on both the sides , we have ,
` b = 0 (ii) a - 3b + c = 1 (iii) -2c = -1 (iv)`
Using (ii) , (iii), and (iv) , we have , `a = (1)/(2) , b = 0 , c = (1)//(2)`
Using these values in (i) , we have ` v = k . lambda^(1//2) . rho^(0) . g(1//2)`
`implies v = k sqrt( lambda g)` which is the required relation .