The dependency of speed of water surface waves (capillary waves) on the density of water ( ρ ) their wavelength ( λ ) and surface tension ( γ ) is -

Question

TestHub · Accepted Answer

Velocity of surface waves depends on density of water, its own wavelength of surface tension.

$v \propto ρ^{a} λ^{b} γ^{c}$

$[L T^{- 1}] \propto {[M L^{- 3}]}^{a} {[L]}^{b} {[\frac{M L T^{- 2}}{L}]}^{c}$

$[L T^{- 1}] \propto M^{a + c} L^{- 3 a + b} T^{- 2 c}$

$a + c = 0 ..........(i)$

$- 3 a + b = 1 .............(ii)$

$- 2 c = - 1 .............(iii)$

Solving
$a = - \frac{1}{2}$ $, b = - \frac{1}{2}$ $, c = \frac{1}{2 ,}$

$v \propto ρ^{- \frac{1}{2}} λ^{- \frac{1}{2}} γ^{\frac{1}{2}}$

$v \propto \sqrt{\frac{γ}{ρ λ}}$

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