In deriving the motion of a string under tension we came up with an equation:
∂
2
y
∂
x
2
=
1
v
2
∂
2
y
∂
t
2
∂
2
y
∂
x
2
=
1
v
2
∂
2
y
∂
t
2
which is known as the wave equation. We will show that this leads to waves
below, but first, let us note the fact that solutions of this equation can be
added to give additional solutions.
Say you have two waves governed by two equations Since they are traveling in
the same medium,
v
v
is the
same
∂
2
f
1
∂
x
2
=
1
v
2
∂
2
f
1
∂
t
2
∂
2
f
1
∂
x
2
=
1
v
2
∂
2
f
1
∂
t
2
∂
2
f
2
∂
x
2
=
1
v
2
∂
2
f
2
∂
t
2
∂
2
f
2
∂
x
2
=
1
v
2
∂
2
f
2
∂
t
2
add
these
∂
2
f
1
∂
x
2
+
∂
2
f
2
∂
x
2
=
1
v
2
∂
2
f
1
∂
t
2
+
1
v
2
∂
2
f
2
∂
t
2
∂
2
f
1
∂
x
2
+
∂
2
f
2
∂
x
2
=
1
v
2
∂
2
f
1
∂
t
2
+
1
v
2
∂
2
f
2
∂
t
2
∂
2
∂
x
2
(
f
1
+
f
2
)
=
1
v
2
∂
2
∂
t
2
(
f
1
+
f
2
)
∂
2
∂
x
2
(
f
1
+
f
2
)
=
1
v
2
∂
2
∂
t
2
(
f
1
+
f
2
)
Thus
f
1
+
f
2
f
1
+
f
2
is a solution to the wave equation
Lets say we have two functions,
f
1
(
x
−
v
t
)
f
1
(
x
−
v
t
)
and
f
2
(
x
+
v
t
)
f
2
(
x
+
v
t
)
.
Each of these functions individually satisfy the wave equation. note that
y
=
f
1
(
x
−
v
t
)
+
f
2
(
x
+
v
t
)
y
=
f
1
(
x
−
v
t
)
+
f
2
(
x
+
v
t
)
will
also satisfy the wave equation. In fact any number of functions of the form
f
(
x
−
v
t
)
f
(
x
−
v
t
)
or
f
(
x
+
v
t
)
f
(
x
+
v
t
)
can be added together and will satisfy the wave equation. This is a very
profound property of waves. For example it will allow us to describe a very
complex wave form, as the summation of simpler wave forms. The fact that
waves add is a consequence of the fact that the wave equation
∂
2
f
∂
x
2
=
1
v
2
∂
2
f
∂
t
2
∂
2
f
∂
x
2
=
1
v
2
∂
2
f
∂
t
2
is
linear, that is
f
f
and its derivatives only appear to first order. Thus any linear combination of
solutions of the equation is itself a solution to the equation.
Any well behaved (ie. no discontinuities, differentiable) function of the form
y
=
f
(
x
−
v
t
)
y
=
f
(
x
−
v
t
)
is
a solution to the wave equation. Lets define
f
′
(
a
)
=
d
f
d
a
f
′
(
a
)
=
d
f
d
a
and
f
′
′
(
a
)
=
d
2
f
d
a
2
.
f
′
′
(
a
)
=
d
2
f
d
a
2
.
Then
using the chain rule
∂
y
∂
x
=
∂
f
∂
(
x
−
v
t
)
∂
(
x
−
v
t
)
∂
x
=
∂
f
∂
(
x
−
v
t
)
=
f
′
(
x
−
v
t
)
,
∂
y
∂
x
=
∂
f
∂
(
x
−
v
t
)
∂
(
x
−
v
t
)
∂
x
=
∂
f
∂
(
x
−
v
t
)
=
f
′
(
x
−
v
t
)
,
and
∂
2
y
∂
x
2
=
f
′
′
(
x
−
v
t
)
.
∂
2
y
∂
x
2
=
f
′
′
(
x
−
v
t
)
.
Also
∂
y
∂
t
=
∂
f
∂
(
x
−
v
t
)
∂
(
x
−
v
t
)
∂
t
=
−
v
∂
f
∂
(
x
−
v
t
)
=
−
v
f
′
(
x
−
v
t
)
∂
y
∂
t
=
∂
f
∂
(
x
−
v
t
)
∂
(
x
−
v
t
)
∂
t
=
−
v
∂
f
∂
(
x
−
v
t
)
=
−
v
f
′
(
x
−
v
t
)
∂
2
y
∂
t
2
=
v
2
f
′
′
(
x
−
v
t
)
.
∂
2
y
∂
t
2
=
v
2
f
′
′
(
x
−
v
t
)
.
We
see that this satisfies the wave equation.
Lets take the example of a Gaussian pulse.
f
(
x
−
v
t
)
=
A
e
−
(
x
−
v
t
)
2
/
2
σ
2
f
(
x
−
v
t
)
=
A
e
−
(
x
−
v
t
)
2
/
2
σ
2
Then
∂
f
∂
x
=
−
2
(
x
−
v
t
)
2
σ
2
A
e
−
(
x
−
v
t
)
2
/
2
σ
2
∂
f
∂
x
=
−
2
(
x
−
v
t
)
2
σ
2
A
e
−
(
x
−
v
t
)
2
/
2
σ
2
and
∂
f
∂
t
=
−
2
(
x
−
v
t
)
(
−
v
)
2
σ
2
A
e
−
(
x
−
v
t
)
2
/
2
σ
2
∂
f
∂
t
=
−
2
(
x
−
v
t
)
(
−
v
)
2
σ
2
A
e
−
(
x
−
v
t
)
2
/
2
σ
2
or
∂
2
f
(
x
−
v
t
)
∂
t
2
=
v
2
∂
2
f
(
x
−
v
t
)
∂
x
2
∂
2
f
(
x
−
v
t
)
∂
t
2
=
v
2
∂
2
f
(
x
−
v
t
)
∂
x
2
That is it satisfies the wave equation.
To find the velocity of a wave, consider the wave:
y
(
x
,
t
)
=
f
(
x
−
v
t
)
y
(
x
,
t
)
=
f
(
x
−
v
t
)
Then can see that if you increase time and x by
Δ
t
Δ
t
and
Δ
x
Δ
x
for a point on the traveling wave of constant amplitude
f
(
x
−
v
t
)
=
f
(
(
x
+
Δ
x
)
−
v
(
t
+
Δ
t
)
)
.
f
(
x
−
v
t
)
=
f
(
(
x
+
Δ
x
)
−
v
(
t
+
Δ
t
)
)
.
Which is true if
Δ
x
−
v
Δ
t
=
0
Δ
x
−
v
Δ
t
=
0
or
v
=
Δ
x
Δ
t
v
=
Δ
x
Δ
t
Thus
f
(
x
−
v
t
)
f
(
x
−
v
t
)
describes a wave that is moving in the positive
x
x
direction. Likewise
f
(
x
+
v
t
)
f
(
x
+
v
t
)
describes a wave moving in the negative x direction.
Lots of students get this backwards so watch out!
Another way to picture this is to consider a one dimensional wave pulse of
arbitrary shape, described by
y
′
=
f
(
x
′
)
y
′
=
f
(
x
′
)
,
fixed to a coordinate system
O
′
(
x
′
,
y
′
)
O
′
(
x
′
,
y
′
)
Now let the
O
′
O
′
system, together with the pulse, move to the right along the x-axis at uniform
speed v relative to a fixed coordinate system
O
(
x
,
y
)
O
(
x
,
y
)
.
As
it moves, the pulse is assumed to maintain its shape. Any point P on the pulse
can be described by either of two coordinates
x
x
or
x
′
x
′
,
where
x
′
=
x
−
v
t
x
′
=
x
−
v
t
.
The
y
y
coordinate is identical in either system. In the stationary coordinate
system's frame of reference, the moving pulse has the mathematical form
y
=
y
′
=
f
(
x
′
)
=
f
(
x
−
v
t
)
y
=
y
′
=
f
(
x
′
)
=
f
(
x
−
v
t
)
If
the pulse moves to the left, the sign of v must be reversed, so that we may
write
y
=
f
(
x
±
v
t
)
y
=
f
(
x
±
v
t
)
as
the general form of a traveling wave. Notice that we have assumed
x
=
x
′
x
=
x
′
at
t
=
0
t
=
0
.
Waves carry momentum, energy (possibly angular momentum)
but not matter
We will often use a sinusoidal form for the wave. However we can't use
y
=
A
sin
(
x
−
v
t
)
y
=
A
sin
(
x
−
v
t
)
since
the part in brackets has dimensions of length. Instead we use
y
=
A
sin
2
π
λ
(
x
−
v
t
)
.
y
=
A
sin
2
π
λ
(
x
−
v
t
)
.
Notice
that
y
(
x
=
0
,
t
)
=
y
(
x
=
λ
,
t
)
y
(
x
=
0
,
t
)
=
y
(
x
=
λ
,
t
)
which
gives us the definition of the wavelength
λ
λ
.
Also note that the frequency is
ν
=
v
λ
.
ν
=
v
λ
.
The
angular frequency is defined to be
ω
≡
2
π
Vibrations on a String
