Question

experiment indicates that for long wave lengths that the speed c is effectively independent of amplitude and surface, suppose c=kg^xλp^z where k is a dimensionless constant, show by dimensional analysis that c is √kgλ?

Answer 1

Gives

$$c=kg^x\lambda p^z\\ L.H.S. demension \\c=[M^0L^1T^{-1}]\\R.H.S dimensions \\=[M^0L^1T^{-2}]^{x}[M^0L^1T^0][ML^{-1}T^{-2}] ^z$$

$$[M^0L^1T^{-2}]=[M]^{z}[L]^{x+1-z}[T]^{-2x-2z}$$

Comparison x y ,z pawor

$1=x-1-z$

$z=0$

$-1=-2x-2z$

$x=\frac{1}{2}$

Put x ,y value

$c=\sqrt{kg\lambda}$

Hence proved