What frequencies allow for the long wavelength limit in solid state physics?

[Yu-Ting]

In the "Introduction to Solid State Physics" by C. Kittel, there is a long wavelength limit in chapter 4 -Phonons I.

When Ka << 1 we can expand cos Ka ≡ 1 - ½ (Ka)²

the dispersion relation will become ω² = (C/M) K² a²

Does anyone know what frequencies can allow this long wavelength limit to hold?

 

[RUber]

I am not sure that there is a standard, since it likely depends on your application.
This is simply the Taylor approximation of the cosine function:
$\cos x = 1 - \frac12 x^2 + \frac{1}{24}x^4 - \frac{1}{720}x^6 ...$
Therefore, you can cut off the higher order terms whenever you feel they are small enough to be insignificant.
For example, when x = .1, your third term is less than .00001. Maybe this is small enough to disregard for your application. Maybe you need your error to be less than 10^-12. Then you should only apply the approximation when x is on the order of 10^-3.