The cut-off frequency can be determined from the cut-off wavelength. From the sampling theorem, we know that for wavelengths smaller than 2 a , or twice the sampling distance, every mode is a repeat of a mode with wavelength larger than 2 a , so the cut-off wavelength should be at λ λ D = 2 a . This results again in k D = 2 π π λ λ D = π π / a , rendering ω ω D = π π v s a .