Mathematical Tools for Physics

5—Fourier Series 133

This time average of the power is (up to that constant factor that I’m ignoring)

〈

f^2

〉

= lim

T→∞

1

2 T

∫+T

−T

dtf(t)^2

Now put the Fourier series representation of the sound into the integral to get

lim

T→∞

1

2 T

∫+T

−T

dt

[∞

∑

−∞

aneinω^0 t

] 2

The soundf(t)is real, so by problem 11 ,a−n=a*n. Also, using the result of problem 18 the time average of
eiωtis zero unlessω= 0; then it’s one.

〈

f^2

〉

= lim

T→∞

1

2 T

∫+T

−T

dt

[

∑

n

aneinω^0 t

][

∑

m

ameimω^0 t

]

= lim

T→∞

1

2 T

∫+T

−T

dt

∑

n

∑

m

aneinω^0 tameimω^0 t

=

∑

n

∑

m

anam lim

T→∞

1

2 T

∫+T

−T

dtei(n+m)ω^0 t

=

∑

n

ana−n

=

∑

n

|an|^2 (31)

Put this into words and it says that the time-average power received is the sum of many terms, each one of
which I can interpret as the amount of power coming inat that frequencynω 0. The Fourier coefficients squared
(absolute-squared really) are then proportional to the part of the power at a particular frequency. The “power
spectrum.”

Other Applications
In section10.2Fourier series will be used to solve partial differential equations, leading to equations such as
Eq. (10.14).