Consider a simple harmonic oscillator that has friction, then the equations of
motion must be changed with the addition of a friction term. So we write
m
ⅆ
2
x
ⅆ
t
2
=
−
k
x
−
b
ⅆ
x
ⅆ
t
m
ⅆ
2
x
ⅆ
t
2
=
−
k
x
−
b
ⅆ
x
ⅆ
t
where
b
ⅆ
x
ⅆ
t
b
ⅆ
x
ⅆ
t
is the friction term. Rearranging we obtain:
m
ⅆ
2
x
ⅆ
t
2
+
b
ⅆ
x
ⅆ
t
+
k
x
=
0
m
ⅆ
2
x
ⅆ
t
2
+
b
ⅆ
x
ⅆ
t
+
k
x
=
0
or
ⅆ
2
x
ⅆ
t
2
+
γ
ⅆ
x
ⅆ
t
+
ω
0
2
x
=
0
ⅆ
2
x
ⅆ
t
2
+
γ
ⅆ
x
ⅆ
t
+
ω
0
2
x
=
0
Where
γ
=
b
m
γ
=
b
m
and
ω
0
2
=
k
m
ω
0
2
=
k
m
Assume a solution of form
x
=
A
e
i
(
p
t
+
α
)
x
=
A
e
i
(
p
t
+
α
)
substitute
into equation and get
(
−
p
2
+
i
p
γ
+
ω
0
2
)
A
e
i
(
p
t
+
α
)
=
0
(
−
p
2
+
i
p
γ
+
ω
0
2
)
A
e
i
(
p
t
+
α
)
=
0
so
−
p
2
+
i
p
γ
+
ω
0
2
=
0
−
p
2
+
i
p
γ
+
ω
0
2
=
0
p
p
must have real and imaginary parts, so rewrite:
p
=
ω
+
i
s
p
=
ω
+
i
s
p
2
=
ω
2
+
2
i
ω
s
−
s
2
p
2
=
ω
2
+
2
i
ω
s
−
s
2
So
the equation
−
p
2
+
i
p
γ
+
ω
0
2
=
0
−
p
2
+
i
p
γ
+
ω
0
2
=
0
becomes
upon substitution:
−
ω
2
−
2
i
ω
s
+
s
2
+
i
ω
γ
−
s
γ
+
ω
0
2
=
0
−
ω
2
−
2
i
ω
s
+
s
2
+
i
ω
γ
−
s
γ
+
ω
0
2
=
0
This
equation implies that the real and imaginary parts are each zero.Separate
the real and imaginary partsImaginary parts give:
−
2
ω
s
+
ω
γ
=
0
s
=
γ
2
−
2
ω
s
+
ω
γ
=
0
s
=
γ
2
From
Real parts get
−
ω
2
+
s
2
−
s
γ
+
ω
0
2
=
0
−
ω
2
+
s
2
−
s
γ
+
ω
0
2
=
0
or
−
ω
2
+
γ
2
4
−
γ
2
γ
+
ω
0
2
=
0
−
ω
2
+
γ
2
4
−
γ
2
γ
+
ω
0
2
=
0
−
ω
2
−
γ
2
4
+
ω
0
2
=
0
−
ω
2
−
γ
2
4
+
ω
0
2
=
0
Which
rearranges to
ω
2
=
ω
0
2
−
γ
2
4
ω
2
=
ω
0
2
−
γ
2
4
Thus
the solution becomes
x
=
A
e
−
γ
t
/
2
e
i
(
ω
t
+
α
)
x
=
A
e
−
γ
t
/
2
e
i
(
ω
t
+
α
)
where
ω
=
ω
0
2
−
γ
2
4
ω
=
ω
0
2
−
γ
2
4
Note that this has assumed a frictional damping force. For a more complicated
damping force, the result would be different.

Comments:"This book covers second year Physics at Rice University."