Estimating Debye Frequency from Dispersion Curves: What's the Proper Approach?

[gomboc]

This is a short question from an old final exam I'm studying from.

If you're given the graph of the dispersion curves for some material (I attached an example, but pretend the axes have numerical values on them), how would I go about estimating the Debye frequency?

My initial thought was just to approximate it as the maximum frequency on that graph, and although that gives a fairly close result, my professor says it's not quite the right approach.

Please help!

(NB: the lettering in the image stands for transverse optical, longitudinal optical, transverse acoustic, longitudinal acoustic.)

 

[Nate810]

The proper approach for estimating the Debye frequency from the given graph is to calculate the average speed of sound in the material. The Debye frequency is then equal to the product of the average speed of sound and the density of the material. To calculate the average speed of sound, you need to find the intersection point between the longitudinal and transverse acoustic curves, and then calculate the average of the two velocities at this intersection point. Once you have these two values, the Debye frequency can be calculated using the formula: Debye frequency = Average speed of sound x density.