4 Where Does the Wave Equation Come From?
27
s 1 (t) =
v o
2ω
(sin ωt +
1
2
sin 2ωt) ,
s 2 (t) =
v o
2ω
(sin ωt −
1
2
sin 2ωt)
with the velocities
˙
s 1 (t) =
1
2
v o (cos ωt + cos 2ωt) ,
˙
s 2 (t) =
1
2
v o (cos ωt − cos 2ωt) .
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(25)
We can read out of Eq. (25) the stated wave-like energy propagation in our mechanical
oscillation system for both masses.
Both masses have the distance (x o2 − x o1 ) in their position of equilibrium. At
time t = 0, only the first mass moves. The second mass is at rest,
s 1 (0) = 0 , ˙
s 1 (0) = v o , s 2 (0) = 0 , ˙
s 2 (0) = 0 .
(26)
According to (26), the first mass possesses (e.g. through an elastic blow) a starting
velocity v o and has at time t = 0 the total system energy of U =
1
2
mv
2
o in the form
of kinetic energy. After one half of an oscillation, after time t = T /2 = π/ω, the
first mass completely stops moving.
Now, only the second mass moves precisely in the opposite direction,
s 1 (
π
ω
) = 0 , ˙
s 1 (
π
ω
) = 0 , s 2 (
π
ω
) = 0 , ˙
s 2 (
π
ω
) = −v o .
(27)
The total energy U =
1
2
mv
2
o is at time t = T /2 = π/ω completely at the second
mass in the form of kinetic energy. Due to system limitations, i.e. the walls, the energy
cannot pass on. It flows back to the first mass and is there at time t = T = 2π/ω.
The process then restarts from the beginning.
The system made up out of two masses and three springs produces a transportation
of energy from the first mass to the second and back again. This energy transport
occurs with the velocity c = covered distance/needed time, so with a distance L
between both walls,
c =
2L
T
= L
ω
π
= L
1
π
D
m
.
(28)
Here, the distance L is just as much a material parameter of our oscillation device,
as is the frequency ω =
√
D/m. The parameter c is the characteristic velocity for
the propagation of a wave front through our medium, which is represented here by
the two masses coupled together by two elastic springs; c is thus the critical signal
velocity. The energy is transported with this velocity through our medium. This result
deserves to be especially brought forward:
The transportation of energy, as we understand it from the fields of electromagnetic or
acoustic waves, is already realised as an elementary property of a mechanical system made
up of only two elastic coupled masses. The signal velocity c is a system parameter that we
27
s 1 (t) =
v o
2ω
(sin ωt +
1
2
sin 2ωt) ,
s 2 (t) =
v o
2ω
(sin ωt −
1
2
sin 2ωt)
with the velocities
˙
s 1 (t) =
1
2
v o (cos ωt + cos 2ωt) ,
˙
s 2 (t) =
1
2
v o (cos ωt − cos 2ωt) .
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(25)
We can read out of Eq. (25) the stated wave-like energy propagation in our mechanical
oscillation system for both masses.
Both masses have the distance (x o2 − x o1 ) in their position of equilibrium. At
time t = 0, only the first mass moves. The second mass is at rest,
s 1 (0) = 0 , ˙
s 1 (0) = v o , s 2 (0) = 0 , ˙
s 2 (0) = 0 .
(26)
According to (26), the first mass possesses (e.g. through an elastic blow) a starting
velocity v o and has at time t = 0 the total system energy of U =
1
2
mv
2
o in the form
of kinetic energy. After one half of an oscillation, after time t = T /2 = π/ω, the
first mass completely stops moving.
Now, only the second mass moves precisely in the opposite direction,
s 1 (
π
ω
) = 0 , ˙
s 1 (
π
ω
) = 0 , s 2 (
π
ω
) = 0 , ˙
s 2 (
π
ω
) = −v o .
(27)
The total energy U =
1
2
mv
2
o is at time t = T /2 = π/ω completely at the second
mass in the form of kinetic energy. Due to system limitations, i.e. the walls, the energy
cannot pass on. It flows back to the first mass and is there at time t = T = 2π/ω.
The process then restarts from the beginning.
The system made up out of two masses and three springs produces a transportation
of energy from the first mass to the second and back again. This energy transport
occurs with the velocity c = covered distance/needed time, so with a distance L
between both walls,
c =
2L
T
= L
ω
π
= L
1
π
D
m
.
(28)
Here, the distance L is just as much a material parameter of our oscillation device,
as is the frequency ω =
√
D/m. The parameter c is the characteristic velocity for
the propagation of a wave front through our medium, which is represented here by
the two masses coupled together by two elastic springs; c is thus the critical signal
velocity. The energy is transported with this velocity through our medium. This result
deserves to be especially brought forward:
The transportation of energy, as we understand it from the fields of electromagnetic or
acoustic waves, is already realised as an elementary property of a mechanical system made
up of only two elastic coupled masses. The signal velocity c is a system parameter that we