What Is the Maximum Frequency of Sound in Aluminium?

[BringBackF77]

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

Assuming the mode will follow the form

Ψ(x,t) = Acos(kx)cos(ωt - φ)

then the second partial WRT t will be

Ψ''(x,t) = -ω²Acos(kx)cos(ωt - φ)

and the second partial WRT x will be

Ψ''(x,t) = -k²Acos(kx)cos(ωt - φ)

Plugging into wave equation I get

-ω²Acos(kx)cos(ωt - φ) = -c²k²Acos(kx)cos(ωt - φ)

--> ω² = κ²(Y/ρ)

--> ω = k(Y/ρ)^1/2

--> $$2\pi f$$ = k(Y/ρ)^1/2

--> f = $$\frac{k}{2\pi}$$ $$\sqrt{\frac{Y}{\rho}}$$

Have no clue where to go from here. This may not even be the way to go about doing it. I guess I technically have the Young's modulus and mass density for the problem but I do not know how to calculate k, and don't understand how this system could vary in frequency to find the highest possible one. Any help would be appreciated, thanks. (Sorry, that I suck a latex btw)

 

[kuruman]

Highest frequency is shortest wavelength. Can the wavelength be shorter than the interatomic distance?

 

[BringBackF77]

Highest frequency is shortest wavelength. Can the wavelength be shorter than the interatomic distance?

Whoops, forgot to reply. Yes, thank you, I understand where I was going wrong now, it was much simpler than I tried to make out to be. Thanks again!