22
4 Where Does the Wave Equation Come From?
F 1
F 1
F 2
F 2
F 3
F 3
F = F 1 + F 2 + F 3
X M
x 1
x 2
x 3
Fig. 4.2 Forces F 1 , F 2 and F 3 acting on the masses m 1 at x 1 , m 2 at x 2 and m 3 at x 3 respectively
can be added up vectorially to a total force F, which then acts at the centre of mass X M . This
centre of mass then moves according to Eq. (11),
d
dt
M
dX M
dt
= F, where M = m 1 + m 2 + m 3
and M X M = m 1 x 1 + m 2 x 2 + m 3 x 3
Newtonian Axiom and has also become known under the term ‘actio = reactio’. To
be more precise, if the mass m 1 acts on the mass m 2 with the force F, then the mass
m 2 acts on the mass m 1 with the force −F. The interaction forces that influence each
other are of the same size but have the opposite direction. We will concern ourselves
with this axiom later on.
We will now consider a mass m that can only move along the x-axis where it
possesses a position of equilibrium x o and where no total resulting force acts on it.
A small deflection s = x − x o out of the position of equilibrium x o should result in a
force F = −D s with the so-called force constant D > 0, which drives the mass back
into the position x o . We also take as granted that the mass m stays constant throughout
the whole state of motion, m = const. With the velocity v =
d
dt
x =
d
dt
(x o + s) =
˙
s, p =
d
dt
(m x) = m ˙
s is then valid for the momentum p. According to Newton’s
law (11), the motion of the mass m along the x-axis follows the law of harmonic
oscillation,
m ¨
s = −D s.
(13)
Out of the general solution of the equation of oscillation (13),
s = s o cos(ω t + φ),
(14)
4 Where Does the Wave Equation Come From?
F 1
F 1
F 2
F 2
F 3
F 3
F = F 1 + F 2 + F 3
X M
x 1
x 2
x 3
Fig. 4.2 Forces F 1 , F 2 and F 3 acting on the masses m 1 at x 1 , m 2 at x 2 and m 3 at x 3 respectively
can be added up vectorially to a total force F, which then acts at the centre of mass X M . This
centre of mass then moves according to Eq. (11),
d
dt
M
dX M
dt
= F, where M = m 1 + m 2 + m 3
and M X M = m 1 x 1 + m 2 x 2 + m 3 x 3
Newtonian Axiom and has also become known under the term ‘actio = reactio’. To
be more precise, if the mass m 1 acts on the mass m 2 with the force F, then the mass
m 2 acts on the mass m 1 with the force −F. The interaction forces that influence each
other are of the same size but have the opposite direction. We will concern ourselves
with this axiom later on.
We will now consider a mass m that can only move along the x-axis where it
possesses a position of equilibrium x o and where no total resulting force acts on it.
A small deflection s = x − x o out of the position of equilibrium x o should result in a
force F = −D s with the so-called force constant D > 0, which drives the mass back
into the position x o . We also take as granted that the mass m stays constant throughout
the whole state of motion, m = const. With the velocity v =
d
dt
x =
d
dt
(x o + s) =
˙
s, p =
d
dt
(m x) = m ˙
s is then valid for the momentum p. According to Newton’s
law (11), the motion of the mass m along the x-axis follows the law of harmonic
oscillation,
m ¨
s = −D s.
(13)
Out of the general solution of the equation of oscillation (13),
s = s o cos(ω t + φ),
(14)