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