Question

The speed of ocean waves depends on their wavelength \(\lambda\) (measured in meters) and the gravitational field strength \(g\) (measured in $\mathrm{m} / \mathrm{s}^{2}$ ) in this way: $$ v=K \lambda^{p} g^{q} $$ where \(K\) is a dimensionless constant. Find the values of the exponents \(p\) and \(q\)

Short answer

Answer: The values of the exponents are p = 1/2 and q = 1/2.

Step-by-step solution

Identify the dimensions of each variable

First, let's figure out the dimensions of each variable in the given equation. The dimensions are as follows: - \(v\) (speed) has dimensions of \(L T^{-1}\), where \(L\) represents length and \(T\) represents time. - \(\lambda\) (wavelength) has dimensions of \(L\). - \(g\) (gravitational field strength) has dimensions of \(L T^{-2}\).

Write the given equation with dimensions

Next, let's rewrite the given equation with dimensions instead of variables. This will help us find the values of \(p\) and \(q\): $$ [L T^{-1}] = K [L]^p \times [L T^{-2}]^q $$

Equate the dimensions on both sides

Now, let's equate the dimensions on both sides of the equation to determine the values of \(p\) and \(q\). Equating the dimensions of length, we get: $$ p + q = 1 $$ Equating the dimensions of time, we get: $$ -2q = -1 $$

Solve the system of equations

Now, we have a system of two linear equations with the variables \(p\) and \(q\). Let's solve for \(p\) and \(q\): $$ -2q = -1 \\ q = 1/2 $$ Substituting this into the first equation: $$ p + 1/2 = 1 \\ p = 1 - 1/2 \\ p = 1/2 $$ So, \(p\) = 1/2 and \(q\) = 1/2, which means the given equation is: $$ v=K \lambda^{1/2} g^{1/2} $$