Mathematical Tools for Physics

5—Fourier Series 131

All that I have to do now is to solve for an inhomogeneous solution one term at a time and add the results. Take
one term alone on the right:

m

d^2 x

dt^2

+b

dx

dt

+kx=einωet

This is what I just finished solving a few lines ago, Eq. ( 23 ), only withnωeinstead of simplyωe. The inhomoge-
neous solution is the sum of the solutions from each term.

xinh(t) =

∑∞

n=−∞

an

1

−m(nωe)^2 +binωe+k

eniωet (27)

Suppose for example that the forcing function is a simple square wave.

Fext(t) =

{

F 0 ( 0 < t < T/ 2 )

−F 0 (T/ 2 < t < T)

and Fext(t+T) =Fext(t) (28)

The Fourier series for this function is one that you can do in problem 12. The result is

Fext(t) =F 0

2

πi

∑

nodd

1

n

eniωet (29)

The solution corresponding to Eq. ( 27 ) is now

xinh(t) =F 0

1

2 πi

∑

nodd

1

(

−m(nωe)^2 +binωe+k

)

1

n

eniωet (30)

A real force ought to give a real result; does this? Yes. For every positiven in the sum, there is a
corresponding negative one and the sum of those two is real. You can see this because everynthat appears is
either squared or is multiplied by an “i.” When you add then= +5term to then=− 5 term it’s adding a
number to its own complex conjugate, and that’s real.
What peculiar features does this result imply? With the simply cosine force the phenomenon of resonance
occurred, in which the response to a small force at a frequency that matched the intrinsic frequency

√

k/m
produced a disproportionately large response. What other things happen here?