Editing Debye model (section)

==== One-dimensional chain in 3D space ====
The summation over the modes is rewritten
:<math> \sum_{i}\sum_{\rm modes} 1 = \sum_i \frac {L}{\pi v_{s,i}} \int_0^{\omega_{\rm D}} d \omega_i = 3 N .</math>

The result is
:<math> \frac {L \omega_{\rm D}}{\pi} (\frac {1}{v_{s,1}} + \frac {1}{v_{s,2}} + \frac {1}{v_{s,3}}) = 3 N .</math>

Thus the Debye frequency is found
:<math> \omega_{\rm D} = \frac{ \pi N}{L} \frac{3}{\frac {1}{v_{s,1}} + \frac {1}{v_{s,2}} + \frac {1}{v_{s,3}}} = \frac {3 \pi N}{L} \frac {v_{s,1} v_{s,2} v_{s,3}}{v_{s,2} v_{s,3} + v_{s,1} v_{s,3} + v_{s,1} v_{s,2}} = \frac{\pi N}{L} v_{\mathrm{eff}}\,. </math>

The calculated effective velocity <math> v_{\mathrm{eff}} </math> is the harmonic mean of the velocities for each polarization. By assuming the two transverse polarizations to have the same phase speed and frequency,
:<math> \omega_{\rm D} = \frac {3 \pi N}{L} \frac {v_{s,t}v_{s,l}}{2v_{s,l} + v_{s,t}} .</math>

Setting <math> v_{s,t} = v_{s,l} </math> recovers the expression previously derived under the assumption that velocity is the same for all polarization modes.