Chapter 25
Elastic Displacements and Waves
We now sketch out the case of the three-dimensional ideal lattice. Our starting point
is once again Newton’s equations, (53) from which we already derived the wave
equation (61), and which we will now formulate for small masses m i . We will
once again ignore the external forces f a for the sake of simplicity. We will also,
once again, begin with the supposition that there is a simultaneous position of equilibrium for all masses of the solid, of such a type that every solid’s strain state
different from this position can in fact be achieved by a simultaneous displacement
of all the masses of this solid. We will also discover further on that exactly this
supposition considerably restricts the number of possible physical stress states. We
therefore move towards the so-called classical theory of elasticity. For a detailed
analysis of this theory, we refer the reader to the textbook of L. D. Landau [54] and
E. M. Lifschitz. The spatial displacements out of this position of equilibrium are
called s i , therefore
d
dt
m i
d
dt
s i
=
k =i
f ki ,
f ik = −f ki .
⎫
⎪ ⎬
⎪ ⎭
(312)
The time-dependent displacements s i (t) we replace with the function s(x, t) in the
limit to the continuum. We have to note for this limit that the term of inertia on the
left-hand side of the Newtonian equations describes the acceleration of a certain mass
m, which according to definition changes its position. In other words, the volume
V m occupied by this mass changes its position as well as its form. The possibly large
number of masses in the volume V m we replace with a spatial density ρ
not to be
mistaken with the density ρ o which we constructed from our linear chain according
to (84)
, thus
m i −→ ρV m , s i (t) −→ s(x, t) ,
(313)
where x = (x, y, z) = (x 1 , x 2 , x 3 ) .
© The Editor(s) (if applicable) and The Author(s), under exclusive
license to Springer Nature Singapore Pte Ltd. 2020
H. Günther, Elementary Approach to Special Relativity,
https://doi.org/10.1007/978-981-15-3168-2_25
261
Elastic Displacements and Waves
We now sketch out the case of the three-dimensional ideal lattice. Our starting point
is once again Newton’s equations, (53) from which we already derived the wave
equation (61), and which we will now formulate for small masses m i . We will
once again ignore the external forces f a for the sake of simplicity. We will also,
once again, begin with the supposition that there is a simultaneous position of equilibrium for all masses of the solid, of such a type that every solid’s strain state
different from this position can in fact be achieved by a simultaneous displacement
of all the masses of this solid. We will also discover further on that exactly this
supposition considerably restricts the number of possible physical stress states. We
therefore move towards the so-called classical theory of elasticity. For a detailed
analysis of this theory, we refer the reader to the textbook of L. D. Landau [54] and
E. M. Lifschitz. The spatial displacements out of this position of equilibrium are
called s i , therefore
d
dt
m i
d
dt
s i
=
k =i
f ki ,
f ik = −f ki .
⎫
⎪ ⎬
⎪ ⎭
(312)
The time-dependent displacements s i (t) we replace with the function s(x, t) in the
limit to the continuum. We have to note for this limit that the term of inertia on the
left-hand side of the Newtonian equations describes the acceleration of a certain mass
m, which according to definition changes its position. In other words, the volume
V m occupied by this mass changes its position as well as its form. The possibly large
number of masses in the volume V m we replace with a spatial density ρ
not to be
mistaken with the density ρ o which we constructed from our linear chain according
to (84)
, thus
m i −→ ρV m , s i (t) −→ s(x, t) ,
(313)
where x = (x, y, z) = (x 1 , x 2 , x 3 ) .
© The Editor(s) (if applicable) and The Author(s), under exclusive
license to Springer Nature Singapore Pte Ltd. 2020
H. Günther, Elementary Approach to Special Relativity,
https://doi.org/10.1007/978-981-15-3168-2_25
261