Question

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

• A. $\sqrt{\frac{\gamma_{a}}{\rho}}$
• B. $\sqrt{\frac{\gamma}{\rho\lambda}}$
• C. $\left( \frac{\gamma}{\rho\lambda} \right)^{1/3}$
• D. $\frac{\gamma}{\rho\lambda}$

Answer 1

Answer: (B)

$\sqrt{\frac{\gamma}{\rho\lambda}}$

Answer 2

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

v ∝ ρ^a λ^bγ^c

[LT^–1] ∝ [ML^–3]^a [L]^b $\left\lbrack \frac{MLT^{- 2}}{L} \right\rbrack^{c}$

[LT^–1] ∝ M^a+c L^–3a+b T^–2c

a + c = 0 c = ½

–3a + b = 1 a = – ½

–2c = –1 b = 1 + 3a ⇒  b = 1 – 3/2

b = –1/2

v ∝ ρ^–1/2λ^–1/2γ^1/2; v ∝ $\sqrt{\frac{\gamma}{\rho\lambda}}$