54
6 Lattice and the Continuum
F i+1 i ≡ τ (x i , t) = σ sin α = σ tan α ,
hence taking the approximation (69) into account, tan α = q i (t)//x, in the passing
to the limit of continuous quantities,
τ (x, t) = σ
∂s(x, t)
∂x
, ρ = const.
(70)
This is however nothing else than one of the basic assumptions (68) of the linearised
theory of elasticity.
For transversal oscillations, we only get an additional statement telling us, that
the modulus of elasticity is identical to the base tension σ of the line. Finally, we
arrive once again at our wave equation (61). In case (b) of transversal oscillations, c
stands for the transversal sound velocity, and the following is valid,
Modulus of elasticity E = line tensionσ .
(71)
We have finally found a very simple procedure, starting from the Newtonian equations
(57) (or rather (63)), moving to the system of ordinary differential equations (31) of
a linear chain and then to the wave equation (61) of the one-dimensional continuum,
for which we had to do a great deal of calculation in the last chapter.
Transition from single masses to a continuum:
With the passing to the limit (66), we move over from Newtonian point mechanics (63) to
the continuum and assume the condition (68) of a linearised theory of elasticity.
Such a simplified transition from single masses to an approximation of our solid
by a continuum, which we have seen to be valid for both longitudinal and transversal
displacements, will come to save us a lot of work. This allows us to include all the
phenomena on a lattice elegantly in a mathematical form, which could otherwise not
be so simply shown. A typical example would be the transportation of energy by a
wave motion in a lattice. This wave motion is very easily described in the linearised
continuum theory, as shown in the appendix, in Chap. 25. Every mechanical process
in the continuum picture has an equivalent on the lattice, where it however has to be
mathematically described with a lot of effort as seen in Chap. 4 when we dealt with
the linear chain.
Equation (61) is a wave equation for the displacement s of continuous distributed
masses. To be able to measure (61), we have to determine the momentary positions
and the equilibrium positions of the masses involved. Equation (61) has a background
of moveable single masses as a condition, bound to equilibrium positions. (We do
however know that these are only the centres of mass coordinates). We can free
ourselves from this fixed background if we ask for the equation of the relative strain
ε = ∂s/∂x. Then, partial differentiation of (61) with respect to x results in
∂
2
∂x 2 ε(x, t) −
1
c 2
∂
2
∂t 2 ε(x, t) = 0 .
(72)
6 Lattice and the Continuum
F i+1 i ≡ τ (x i , t) = σ sin α = σ tan α ,
hence taking the approximation (69) into account, tan α = q i (t)//x, in the passing
to the limit of continuous quantities,
τ (x, t) = σ
∂s(x, t)
∂x
, ρ = const.
(70)
This is however nothing else than one of the basic assumptions (68) of the linearised
theory of elasticity.
For transversal oscillations, we only get an additional statement telling us, that
the modulus of elasticity is identical to the base tension σ of the line. Finally, we
arrive once again at our wave equation (61). In case (b) of transversal oscillations, c
stands for the transversal sound velocity, and the following is valid,
Modulus of elasticity E = line tensionσ .
(71)
We have finally found a very simple procedure, starting from the Newtonian equations
(57) (or rather (63)), moving to the system of ordinary differential equations (31) of
a linear chain and then to the wave equation (61) of the one-dimensional continuum,
for which we had to do a great deal of calculation in the last chapter.
Transition from single masses to a continuum:
With the passing to the limit (66), we move over from Newtonian point mechanics (63) to
the continuum and assume the condition (68) of a linearised theory of elasticity.
Such a simplified transition from single masses to an approximation of our solid
by a continuum, which we have seen to be valid for both longitudinal and transversal
displacements, will come to save us a lot of work. This allows us to include all the
phenomena on a lattice elegantly in a mathematical form, which could otherwise not
be so simply shown. A typical example would be the transportation of energy by a
wave motion in a lattice. This wave motion is very easily described in the linearised
continuum theory, as shown in the appendix, in Chap. 25. Every mechanical process
in the continuum picture has an equivalent on the lattice, where it however has to be
mathematically described with a lot of effort as seen in Chap. 4 when we dealt with
the linear chain.
Equation (61) is a wave equation for the displacement s of continuous distributed
masses. To be able to measure (61), we have to determine the momentary positions
and the equilibrium positions of the masses involved. Equation (61) has a background
of moveable single masses as a condition, bound to equilibrium positions. (We do
however know that these are only the centres of mass coordinates). We can free
ourselves from this fixed background if we ask for the equation of the relative strain
ε = ∂s/∂x. Then, partial differentiation of (61) with respect to x results in
∂
2
∂x 2 ε(x, t) −
1
c 2
∂
2
∂t 2 ε(x, t) = 0 .
(72)