Final answer:
The Bloch wave method, not mentioned in Goldstein's textbook, is used for finding wave equations in periodic systems by expressing solutions through waves modulated by periodic functions, and is advantageous for complicated boundary conditions.
Step-by-step explanation:
When dealing with the continuous limit of coupled harmonic oscillators in a graduate-level mechanics course, the Bloch wave method is an alternative to the determinant method for finding the wave equation. Unfortunately, it's not mentioned in Goldstein's classical mechanics textbook.
The Bloch wave method takes advantage of the fact that in a periodic system, solutions to the wave equation can be expressed as waves modulated by a periodic function - capturing the translational symmetry of the system. This method simplifies the analysis of wave behavior in periodic systems such as crystal lattices where oscillations can be treated quantum mechanically.
Using the Bloch wave method, the displacement of the oscillators can be described by a function that has the same periodicity as the crystal lattice. This approach is particularly advantageous for its ease in dealing with the boundary conditions intrinsic to the problem, which can be cumbersome with the determinant method.
Studying oscillatory motion and waves, as noted in the question, involves addressing various principles such as simple harmonic motion, superposition, and interference.
These principles also pertain to the analysis of more complex oscillating systems using quantum mechanics, which expands beyond the classic formulation of a harmonic oscillator such as a mass on a spring.