The dispersion relation becomes ω ω i = v s , i | k | , with i = 1 , 2 , 3 , each corresponding to one of the three polarizations. The cut-off frequency ω ω D , however, does not depend on i . We can write the total number of modes as ∑ ∑ i ∑ ∑ m o d e s 1 , which is again equal to 3 N . Here the summation over the modes is now dependent on i .