Problem: Vibrations of a Square Lattice

   A simple, monoatomic, two-dimensional square lattice is modeled by balls and springs. The lattice spacing is a, the mass of the balls is m, the spring constant of the nearest-neighbor springs (along the edges of the squares) is , and the spring constant between the next-nearest neighbors (along the diagonal of the squares) is . All other interactions are negligible.

What is the frequency of the mode oscillations? (Warning: The complete solution of the equations of motion is time-consuming and not necessary for the answer.)