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WOLFRAM|DEMONSTRATIONS PROJECT

Complex Addition of Harmonic Motions and the Phenomenon of Beats

​
harmonic motion 1
a
1
ω
1
δ
1
harmonic motion 2
a
2
ω
2
δ
2
time
2
4
6
8
10
play
two independent harmonic motions
complex addition of two harmonic motions
Harmonic motion can be represented by either trigonometric or complex exponential functions. Any vector
X
in the
x
-
y
plane can be written as a complex number:
X
=A(cosθ+isinθ)
, where
A
is the amplitude (also called the norm) and
θ
is the phase (also called the argument). The projection onto the
y
axis of such a rotating vector is the imaginary part of
X
. According to Euler's formula, a complex number can be expressed as
X
=A
iθ
e
=A
iωt
e
, where
ω
denotes the frequency of rotation in the counterclockwise direction and
t
is time. Multiplying by
iδ
e
shifts the phase.
This Demonstrations shows the addition of two independent harmonic motions in terms of complex numbers. The sum of two harmonic motions with nearly equal frequencies exhibits a phenomenon known as beats.
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