As with the simple harmonic oscillator, the great power of having the
general solution is that we now can find the particular solution for
any problem without solving complicated differential equations but by
solving for the adjustable parameters in the general solution. The
basic strategy is the same as with the harmonic oscillator: ``(1)
write each condition in terms of the general solution, (2) solve the
resulting set of equations for the adjustable parameters, and (3)
write down the general solution while substituting the particular
values found for the adjustable parameters.''
Although the strategy is the same, following this procedure is more challenging in the case of waves, particularly because the ``adjustable parameters'' are actually now unknown functions, f() and g(). We therefore shall work through two different examples: first, when we are given the initial position and initial velocity of every chunk of the system, and, second, when a pulse collides with a boundary.