I did excersice 10.1 and found that the sound velocity in a diatomic chain is v=a*sqrt(k/ (2*sqrt(Mm)) ).
This seems to suggest that the density is
p = 2*sqrt(Mm)/a, i.e. the effective mass is twice the geometric mean of the masses.
Why do we use the geometric mean and not just the arithmetic mean, i.e. p=(M+m)/a ?