Question

The speed c of ocean wave is proportional to the acceleration due to gravity g, wavelength λ, and density ρ of 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

Write the equation:

$$c=kg^x\lambda^y\rho^z.$$

Write the units instead of quantities:

[\text{m/s}]=1\cdot[\text{m/s}^2]^x[\text{m}]^y[\text{kg/m}^3]^z,\\ [\text{m/s}]=[\text{m}^x/\text{}{s}^{2x}][\text{m}^y][\text{kg}^z/\text{m}^{3z}].

We see that the left part (m/s) does not contain kilograms, thus, $z=0$.

Since we have seconds in the power of 1, $x=1/2$.

Since in the left part we have meters in the power of one and in the right part we have $m^{1/2}$, the product with $m^y$ must result in $m^1$, or

$$1/2+y=1,\\ y=1/2.$$

Thus, the equation is

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

Indeed, the equation used to calculate wave velocity is

$$c=\frac{1}{\sqrt{2\pi}}\cdot\sqrt{g\lambda}.$$