5 The Wave Equation and the Third Axiom
45
We insert (59) and (60) in (58) and receive, based alone on Newtonian mechanics,
starting out from our equation for oscillations (31), for a linear chain with the periodical boundary conditions in the passing to the limit N −→ ∞ the d’Alembertian
wave equation (1) and also the explanation for the sound velocity c from mechanical
parameters,
∂
2
∂x 2 s(x, t) −
1
c 2
∂
2
∂t 2 s(x, t) = 0 ,
c =
E
ρ
.
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
(61)
With (59) and (60) for ρ and E, we find our formula (39) for the sound velocity
which was introduced in the linear chain
c =
D L
m/L
= L
D
m
.
(62)
We can therefore state.
The wave equation can be fully understood alone out of the postulates of Newton’s point
mechanics.
We still have not achieved much with this knowledge. Sound waves can be comprehended without the help of Special Relativity. We, of course, wish for more and
think here of the vision of E. Mach, cf. Thiele [93], ‘Light somewhat like sound.
Sound somewhat like light’, wherein a sense yet to be determined, the dynamics of
light, as well as sound are supposedly equivalent.
3 In spite of everything, the Lorentz
transformation, the centrepiece of the Special Theory of Relativity was discovered
in 1887 by W. Voigt [96] long before the Special Theory of Relativity, from the
‘differential equations for the oscillations of an incompressible medium’, thus from
the wave equation, without however having received much attention. In his paper,
on the Doppler effect (cf. also Chap. 13 and the fourth footnote on Chap. 18) Voigt
primarily had electromagnetic waves in mind based on an elastic ether model. He
also expressively had acoustic waves in mind, a ‘ringing bell’, see Voigt [96], so
that we in fact first come across the Lorentz transformation in the field of acoustics.
(Here Voigt’s equations differ by a common coefficient from the ‘correct’ Eq. (151)
on for Lorentz transformation, which however is of no great importance). In 1908
Voigt [97] verified, ‘ . . . already then [in 1887] some results were formed which
latter were obtained from the electromagnetic theory’.
3 A mechanical theory of light is in this view as absurd as an electromagnetic theory of sound. A
Special Theory of Relativity for the elastic disturbances on a lattice, for sound, really cannot exist
as we will see in the following chapter. There is however a second possibility of motion on a lattice
that of plastic displacement; see Chap. 7. The analysis of this will inevitably lead us to relativistic
concepts including all details as far as to the secondary relativistic effects (e.g. pair creation which
we will bring to attention in Chap. 26).
45
We insert (59) and (60) in (58) and receive, based alone on Newtonian mechanics,
starting out from our equation for oscillations (31), for a linear chain with the periodical boundary conditions in the passing to the limit N −→ ∞ the d’Alembertian
wave equation (1) and also the explanation for the sound velocity c from mechanical
parameters,
∂
2
∂x 2 s(x, t) −
1
c 2
∂
2
∂t 2 s(x, t) = 0 ,
c =
E
ρ
.
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
(61)
With (59) and (60) for ρ and E, we find our formula (39) for the sound velocity
which was introduced in the linear chain
c =
D L
m/L
= L
D
m
.
(62)
We can therefore state.
The wave equation can be fully understood alone out of the postulates of Newton’s point
mechanics.
We still have not achieved much with this knowledge. Sound waves can be comprehended without the help of Special Relativity. We, of course, wish for more and
think here of the vision of E. Mach, cf. Thiele [93], ‘Light somewhat like sound.
Sound somewhat like light’, wherein a sense yet to be determined, the dynamics of
light, as well as sound are supposedly equivalent.
3 In spite of everything, the Lorentz
transformation, the centrepiece of the Special Theory of Relativity was discovered
in 1887 by W. Voigt [96] long before the Special Theory of Relativity, from the
‘differential equations for the oscillations of an incompressible medium’, thus from
the wave equation, without however having received much attention. In his paper,
on the Doppler effect (cf. also Chap. 13 and the fourth footnote on Chap. 18) Voigt
primarily had electromagnetic waves in mind based on an elastic ether model. He
also expressively had acoustic waves in mind, a ‘ringing bell’, see Voigt [96], so
that we in fact first come across the Lorentz transformation in the field of acoustics.
(Here Voigt’s equations differ by a common coefficient from the ‘correct’ Eq. (151)
on for Lorentz transformation, which however is of no great importance). In 1908
Voigt [97] verified, ‘ . . . already then [in 1887] some results were formed which
latter were obtained from the electromagnetic theory’.
3 A mechanical theory of light is in this view as absurd as an electromagnetic theory of sound. A
Special Theory of Relativity for the elastic disturbances on a lattice, for sound, really cannot exist
as we will see in the following chapter. There is however a second possibility of motion on a lattice
that of plastic displacement; see Chap. 7. The analysis of this will inevitably lead us to relativistic
concepts including all details as far as to the secondary relativistic effects (e.g. pair creation which
we will bring to attention in Chap. 26).