If we have waves in a one-dimensional box, such that the wave displacement _(x, t) _ 0 f...

If we have waves in a one-dimensional box, such that the wave displacement _(x, t) _ 0 for x _ 0 and x _ L, where L is the length of the box, and

show that the solutions are of the from

and a(t) satisfies the (harmonic-oscillator) equation

where is the angular frequency 2 πf.