One can also consider the case of two oscillations with the same phase but
different
frequencies:
x
1
=
A
1
e
i
(
ω
1
t
)
x
2
=
A
2
e
i
(
ω
2
t
)
x
=
x
1
+
x
2
=
A
1
e
i
(
ω
1
t
)
+
A
2
e
i
(
ω
2
t
)
x
=
e
i
(
ω
1
t
)
[
A
1
+
A
2
e
i
(
ω
2
−
ω
1
)
t
]
x
1
=
A
1
e
i
(
ω
1
t
)
x
2
=
A
2
e
i
(
ω
2
t
)
x
=
x
1
+
x
2
=
A
1
e
i
(
ω
1
t
)
+
A
2
e
i
(
ω
2
t
)
x
=
e
i
(
ω
1
t
)
[
A
1
+
A
2
e
i
(
ω
2
−
ω
1
)
t
]
In an acoustical system, this gives beats, which is more easily seen if we
take the case where
A
1
=
A
2
≡
A
A
1
=
A
2
≡
A
,
then:
x
=
x
1
+
x
2
=
A
e
i
(
ω
1
t
)
+
A
e
i
(
ω
2
t
)
=
A
e
i
(
ω
1
+
ω
2
2
+
ω
1
−
ω
2
2
)
t
+
A
e
i
(
ω
1
+
ω
2
2
−
ω
1
−
ω
2
2
)
t
=
A
e
i
(
ω
1
+
ω
2
2
)
t
[
e
i
(
ω
1
−
ω
2
2
)
t
+
e
−
i
(
ω
1
−
ω
2
2
)
t
]
=
2
A
e
i
(
ω
1
+
ω
2
2
)
t
cos
[
(
ω
1
−
ω
2
2
)
t
]
x
=
x
1
+
x
2
=
A
e
i
(
ω
1
t
)
+
A
e
i
(
ω
2
t
)
=
A
e
i
(
ω
1
+
ω
2
2
+
ω
1
−
ω
2
2
)
t
+
A
e
i
(
ω
1
+
ω
2
2
−
ω
1
−
ω
2
2
)
t
=
A
e
i
(
ω
1
+
ω
2
2
)
t
[
e
i
(
ω
1
−
ω
2
2
)
t
+
e
−
i
(
ω
1
−
ω
2
2
)
t
]
=
2
A
e
i
(
ω
1
+
ω
2
2
)
t
cos
[
(
ω
1
−
ω
2
2
)
t
]
Where
the last step used
cos
θ
=
e
i
θ
+
e
−
i
θ
2
cos
θ
=
e
i
θ
+
e
−
i
θ
2
So
in an acoustical system we will get a dominant sound that has the average of
the two frequencies and and envelope of amplitude that slowly oscillates.
This will be looked at more closes in the context of mechanical waves.

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