How Does Water Depth Influence Wave Speed in Shallow Water?

[cyspope]

Homework Statement

The speed of waving in shallow water depend only on the acceleration of gravity g, a quantity with dimensions L/T^2, and on the water depth h.

Which of the following formulas for the wave speed v could be correct?

a) v=1/2gh^2 b)v=√gh

Homework Equations

L = \frac{n\lambda_{n}}{2}

v^{2} = \frac{T}{\frac{m}{l}}

v = f\lambda

The Attempt at a Solution

I think I have to use one of above equations, but I don't know where to start.

 

[AtticusFinch]

I think you are over complicating things. It's a simple dimensional analysis problem. Your a) equation comes out in units of m³/s² while your b) equation comes out in m/s. Velocity has units of m/s, therefore equation b) is correct.

Actually deriving this equation for a water wave involves the inverse hyperbolic tangent function.

 

[cyspope]

could you explain how the equation a) comes out to be m3/s2 because I don't understand the process.

 

[AtticusFinch]

So for a) you have 0.5gh²

Water Depth is measured in SI units of meters. When you square meters it becomes meters squared or more generally length squared.

"g" or the acceleration due to gravity has units of meters per second squared. Or more generally Length per unit time squared (written L/T^2)

0.5 does not have units so you can leave it out in a dimensional analysis problem.

Now multiply the units from h and g together.

Square meters times meters gives you cubic meters just like x² times x gives you x³.

Seconds squared stays seconds squared.

More generally it comes out length cubed over time squared.