36
4 Where Does the Wave Equation Come From?
λ n =
L
n
, ω n = c k n =
2πn
L
c, n = 1, 2, · · · , ∞.
(46)
We insert (46) into (45) and discover the possible states of oscillation s n (x, t) of an
elastic rod of the length L, because we also have the amplitude with B = 1/2 at our
disposal,
s n (x, t) = cos(
2πn
L
x) cos(
2πn
L
ct), n = 1, 2, · · · , ∞.
Free oscillating
rod of the length L
(47)
We consider the oscillations (47) at the positions x i =
L
N
i at those places where we
positioned the single masses m = m/L in the linear chain and find
s n (x, t) = cos( i
N n 2π) cos( 2πn
L c t) , n = 1, 2, · · · , ∞ .
Free oscillating
rod
(47a)
These motions are however identical to the motions (40) of the masses m of
the linear chain for n N if we just dimension them so that their velocity c K
corresponds to the sound velocity c. If the mass m and the length L of the linear
chain and the rod are identical, then ρ e f f = ρ is valid. The N single masses m of
the linear chain are then the centres of inertia of the rod of the N pieces at x i =
L
N
i.
We now only need to dimension the force constant D of our spring at the length L,
so that D · L = E e f f = E and thus c K = c is fulfilled. The oscillations of the linear
chain are then not distinguishable from the oscillations of an elastic rod. We see the
following:
The oscillations of a linear chain consisting of N elastic coupled single masses, where one
wavelength is built up out of n particles, are indistinguishable from the oscillations of a
homogenous elastic rod under the condition that n N .
Is it possible to make a distinction between a linear chain made up out of single
masses and an elastic rod only by means of measurement? Is the continuum, here
the homogenous rod maybe just a mathematical model, with which we can easier
calculate? In the regions of sufficiently high frequencies ω n , the condition (37) for
the linear chain is not fulfilled and the correct formula (43) applies. The linear
chain shows a dispersion. The signal velocity c
(n)
K is dependent according to (44) on
the wavelength λ n (respectively on the wave number k n = 2π/λ n ), the same way
we discussed it above with the help of the formula of approximation (44a). This
dispersion can in fact be measured in the abovementioned example using a metal
rod. We can determine with high-precision experiments that the rod, that behaves
like a continuum when just normally observed, is in fact only a linear chain of single
masses. However, as long we do not have the device to measure this, we cannot
distinguish between them.
Maybe every so-called continuum turns out to be a discontinuous structure if sufficiently
accurate modes of measurements could be applied.
Should we include our physical spacetime continuum into these problems? To
delve deeper into this problem would inevitably lead us to extend our thoughts to
4 Where Does the Wave Equation Come From?
λ n =
L
n
, ω n = c k n =
2πn
L
c, n = 1, 2, · · · , ∞.
(46)
We insert (46) into (45) and discover the possible states of oscillation s n (x, t) of an
elastic rod of the length L, because we also have the amplitude with B = 1/2 at our
disposal,
s n (x, t) = cos(
2πn
L
x) cos(
2πn
L
ct), n = 1, 2, · · · , ∞.
Free oscillating
rod of the length L
(47)
We consider the oscillations (47) at the positions x i =
L
N
i at those places where we
positioned the single masses m = m/L in the linear chain and find
s n (x, t) = cos( i
N n 2π) cos( 2πn
L c t) , n = 1, 2, · · · , ∞ .
Free oscillating
rod
(47a)
These motions are however identical to the motions (40) of the masses m of
the linear chain for n N if we just dimension them so that their velocity c K
corresponds to the sound velocity c. If the mass m and the length L of the linear
chain and the rod are identical, then ρ e f f = ρ is valid. The N single masses m of
the linear chain are then the centres of inertia of the rod of the N pieces at x i =
L
N
i.
We now only need to dimension the force constant D of our spring at the length L,
so that D · L = E e f f = E and thus c K = c is fulfilled. The oscillations of the linear
chain are then not distinguishable from the oscillations of an elastic rod. We see the
following:
The oscillations of a linear chain consisting of N elastic coupled single masses, where one
wavelength is built up out of n particles, are indistinguishable from the oscillations of a
homogenous elastic rod under the condition that n N .
Is it possible to make a distinction between a linear chain made up out of single
masses and an elastic rod only by means of measurement? Is the continuum, here
the homogenous rod maybe just a mathematical model, with which we can easier
calculate? In the regions of sufficiently high frequencies ω n , the condition (37) for
the linear chain is not fulfilled and the correct formula (43) applies. The linear
chain shows a dispersion. The signal velocity c
(n)
K is dependent according to (44) on
the wavelength λ n (respectively on the wave number k n = 2π/λ n ), the same way
we discussed it above with the help of the formula of approximation (44a). This
dispersion can in fact be measured in the abovementioned example using a metal
rod. We can determine with high-precision experiments that the rod, that behaves
like a continuum when just normally observed, is in fact only a linear chain of single
masses. However, as long we do not have the device to measure this, we cannot
distinguish between them.
Maybe every so-called continuum turns out to be a discontinuous structure if sufficiently
accurate modes of measurements could be applied.
Should we include our physical spacetime continuum into these problems? To
delve deeper into this problem would inevitably lead us to extend our thoughts to