Answer to Question #156543 in Physics for Kaosarat

Question

 The speed c of ocean wave is proportional to the acceleration due to gravity g, wavelengthλ, and densityρof the wave such that c =kg^xλ^yρ^z,  where k is a dimensionless constant. What is the correct equation for the speed of the ocean?

Answer 1

The dimension of the acceleration due to gravity is:

$[g] = \dfrac{m}{s^2}$

The dimention of the wavelength is:

$[\lambda] = m$

The dimention of density is:

$[\rho] = \dfrac{kg}{m^3}$

The dimention of the speed is:

$[c] = \dfrac{m}{s}$

Subsituting these values into the formula (and omitting $k$ as a dimensionless constant), obtain:

$\dfrac{m}{s} = \dfrac{m^x\cdot m^y\cdot kg^z}{s^{2x}\cdot m^{3z}}$

Comparing the right and left hand side, obtain:

$z = 0\\ 2x = 1\\ x+y =1$

Solving this system, obtain:

$x = 0.5\\ y = 0.5$

Then the equation is:

$c = k\sqrt{g\lambda}$

Answer. $c = k\sqrt{g\lambda}$ .