5—Fourier Series 1145.6 Return to Parseval
When you have a periodic wave such as a musical note, you can Fourier analyze it. The boundary
conditions to use are naturally the periodic ones, Eq. (5.20) or (5.33), so that
f(t) =
∑∞
−∞aneinω^0 t
If this represents the sound of a flute, the amplitudes of the higher frequency components (thean)
drop off rapidly withn. If you are hearing an oboe or a violin the strength of the higher components
is greater.
If this function represents the sound wave as received by your ear, the power that you receive
is proportional to the square off. Iffrepresent specifically the pressure disturbance in the air, the
intensity (power per area) carried by the wave isf(t)^2 v/Bwherevis the speed of the wave andB
is the bulk modulus of the air. The key property of this is that it is proportional to the square of the
wave’s amplitude. That’s the same relation that occurs for light or any other wave. Up to a known
factor then, the power received by the ear is proportional tof(t)^2.
This time average of the power is (up to that constant factor that I’m ignoring)〈f^2
〉
= lim
T→∞1
2 T
∫+T
−Tdtf(t)^2
Now put the Fourier series representation of the sound into the integral to get
lim
T→∞1
2 T
∫+T
−Tdt
[∞
∑
−∞aneinω^0 t
] 2
The soundf(t)is real, so by problem5.11,a−n=a*n. Also, using the result of problem5.18the time
average ofeiωtis zero unlessω= 0; then it’s one.
〈
f^2
〉
= lim
T→∞1
2 T
∫+T
−Tdt
[
∑
naneinω^0 t
][
∑
mameimω^0 t
]
= lim
T→∞1
2 T
∫+T
−Tdt
∑
n∑
maneinω^0 tameimω^0 t
=
∑
n∑
manam lim
T→∞1
2 T
∫+T
−Tdtei(n+m)ω^0 t
=
∑
nana−n
=
∑
n|an|^2 (5.39)
Put this into words and it says that the time-average power received is the sum of many terms, each
one of which can be interpreted as the amount of power coming inat that frequency nω 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. (10.15).