Highest frequency of sound waves in a thin aluminium rod

[aednat]

Homework Statement

Estimate the highest possible frequency (in Hertz) and the smallest possible wavelength, of a sound wave in aluminium due to the discrete atomic structure of this material. The mass density, Young's modulus, and atomic weight of aluminium are 2.7x103kg m-3, 6x1010 N m-2, and 27 respectively.

Homework Equations

Second partial of Ψ(x,t) WRT t = second partial of Ψ(x,t) WRT x multiplied by (Young's modulus / mass density)

The Attempt at a Solution

I've obtained the expression v = sqrt(Y/p), but v=frequency x wavelength
so v = 4714.045 = frequency x wavelength.
I understand that the minimum wavelength occurs at wavelength = the interatomic spacing between the aluminium atoms, but I am unsure as to how to obtain an expression for it's value.

The maximum frequency occurs at the minimum wavelength so I can just algebraically sub in the min wavelength to find the max frequency.

I'm assuming I have to use the atomic weight of aluminium somewhere in the question, but I can't seem to find the next step.

I have a feeling I may have strayed in the wrong direction ; perhaps I could use some boundary conditions and solve the wave equation? I don't know if I'm oversimplifying the problem.

Any help is massively appreciated.

 

[mfb]

The maximum frequency occurs at the minimum wavelength so I can just algebraically sub in the min wavelength to find the max frequency.

Right.

You have 2700kg/m³ and every atom has a weight of 27 u. How many aluminium atoms do you expect per m³? What does that tell you about the distances between those atoms?