Liquid Mechanics: Wavelength & Variables

[pisiks]

Homework Statement

The wavelength of $$\lambda$$ of waves on the surface of a water on a cup has three variables:
surface tension of the water $$\sigma$$, water density $$\rho$$, and frequency of vibration f. Deduce the relationship of these three variable to the wavelength.

Would the same relationship hold if you were on the moon?

Homework Equations

$$\lambda$$ = $$\lambda$$($$\sigma$$, $$\rho$$, f)

The Attempt at a Solution

 

[rock.freak667]

I think you can use the Buckingham-pi theorem here but I don't quite know how it works

But normally I'd do this

$$\lambda = k \sigma^a \rho^b f^c$$

then write everything in tersm of the fundamental units and equate the units on the LHS and RHS

 

[tomcue]

The relationship between the wavelength and the three variables can be described by the equation \lambda = \sqrt{\frac{\sigma}{\rho f}}. This equation shows that as the surface tension of the water increases, the wavelength decreases. Similarly, as the water density increases, the wavelength also decreases. Finally, as the frequency of vibration increases, the wavelength decreases. This relationship can be explained by the fact that higher surface tension and density make it more difficult for the water molecules to move, resulting in shorter wavelengths. Similarly, a higher frequency of vibration means more waves are passing through a given point in a given time, resulting in shorter wavelengths.

This relationship may not hold on the moon due to the differences in surface gravity and atmospheric conditions. The moon has a lower surface gravity than Earth, which could affect the surface tension and density of water, resulting in a different relationship between the variables and the wavelength. Additionally, the moon has no atmosphere, which could also impact the behavior of waves on the surface of water. Further research and experiments would be needed to determine the exact relationship between these variables and the wavelength on the moon.