Could you please show me how to do the problem attached? You don't have to do the first part (proving solutions to the wave equation by a separation of variables) as I know how to do that. Please start where it asks what is a normal mode, etc.
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The solution is attached below in two files identical in content but differ in format (MS-word and pdf) so you can choose the more suitable one for your needs.
A normal mode of a system is a specific motion of the system where all its components are moving with the same frequency and the same phase is a sinusoidal motion.
Since the normal modes are orthogonal to each other, they form an eigenvector basis for the system, so the general motion of any oscillating system can be described as a linear combination (superposition) of an infinite number of the system's normal modes.
Since the normal modes are pure sinusoidal waves, the energy contained in each mode is simple proportional to its amplitude.
When a system is forced to oscillate ...
The solution shows how to find normal modes for a wave equation and discusses properties of modes in an oscillating system, as well as discussing displacement and ratio of energies.