The harmonic oscillator

The harmonic oscillator is described by a Hooke's law force, F(x) = -kx. Thus, Newton's second law becomes

or

Note that the ratio k/m has units of inverse time squared. Thus, let us define a frequency, so that we may write

This simple second order differential equation can be solved by taking as an ansatz the general form

Substituting in to the differential equation gives

which can only be satisfied if

or

Therefore, we can have solutions of the form and . It should also be clear that if we take an arbitrary linear combination of these two, we also get a solution

The reason we need to take such a linear combination is that we still have to apply the initial conditions, and which requires that we have two constants in the final solution that we determine by the initial conditions. The above linear combination has the required constants. Thus, applying the initial conditions, we find

This gives two equations in the two unknowns A and B. The solution for A and B is easily seen to be

Substituting these into the solution x(t) gives

A plot of the positions, x(t) vs. t is

  
Figure 2:

Note that the space between the crests of the oscillating function is just the period .

Home:Top