Undamped One-Dimensional Wave Equation:
Vibrations of an Elastic String
Consider a piece of thin flexible string of length
L
, of negligible weight.
Suppose the two ends of the string are firmly secured (“clamped”) at some
supports so they will not move.
Assume the set-up has no damping.
Then,
the vertical displacement of the string, 0 <
x
<
L
, and at any time
t
> 0, is
given by the displacement function
u
(
x
,
t
).
It satisfies the
homogeneous one-
dimensional undamped wave equation
:
a
2
u
xx
=
u
tt
Where the constant coefficient
a
2
is given by the formula
a
2
=
T
/
ρ
, such that
a
= horizontal propagation velocity of the wave motion,
T
= force of tension
exerted on the string,
ρ
= mass density (mass per unit length).
It is subjected
to the homogeneous boundary conditions
u
(0,
t
) = 0,
and
u
(
L
,
t
) = 0,
t
> 0.
The two boundary conditions reflect that the two ends of the string are
clamped in fixed positions.
Therefore, they are held motionless at all time.
The equation comes with 2 initial conditions, due to the fact that it contains
the second partial derivative term
u
tt
.
The two initial conditions are the
initial (vertical) displacement
u
(
x
,
0), and the initial (vertical) velocity
u
t
(
x
,
0), both are arbitrary functions of
x
alone.
(Note that the string is
merely the medium for the wave, it does not itself move horizontally, it only
vibrates, vertically, in place.
The resulting wave form, or the wave-like
“shape” of the string, is what moves horizontally.)