112
12 The Measurement of the Critical Velocity
This makes J. D. Eshelby’s [19] result plausible, the result that the parameters of
the crystal lattice really lead to a remarkable correspondence between the critical
velocity c o of the sine-Gordon equation and the transversal sound velocity c T . In this
sense, we will use this conformity as a model presumption,
c o ≈ c T .
(125)
Factually, the validity of the hypothesis c T ≈ c o is of no real importance for our
further considerations. We can however use this fact to illustrate our point,
1 and we
will arbitrarily refer, due to reasons of linguistic simplicity, to the signal moving
through the crystal with the velocity c o as the ‘sound signal’. By definition, c T is a
velocity that we measure in our preferred frame o , that is, in the inertial system
in which an outside observer would see that the crystal as a whole would be static.
Here, we measure in fact one and the same signal velocity c T in every direction of
our, a crystal approximating three-dimensional continuum. We consider from this
isotropic continuum only one single space dimension described by the choice of the
x-axis. Our linear chain lies in this x-direction, which defines our dislocation line.
We then need the transition to a continuum for this one direction. We only take the
periodical order of the surrounding lattice atoms of the crystal into account. This
led us in Chap. 8 to the sine-Gordon equation with c o ≈ c T . However, isotropy is
also for one-dimensional problems a property of extreme importance and will force
us to make a few explanations further along in this chapter. It has the meaning of
constancy for the critical velocity of both opposite directions. Firstly, we can state:
Our one-dimensional continuum is isotropic in the preferred frame o .
This one-dimensional isotropy of space becomes transparent for the sine-Gordon
equation (88), which is the basis of our one-dimensional considerations, as follows:
Whilst keeping the time t, we change the orientation of the x-axis; in other words,
we consider a space inversion, so that all the positions can be described using new
coordinates ¯
x according to
¯
x = −x , ¯
t = t .
Space inversion
(126)
The transversal displacement q = q(x, t) remains unchanged, in other words,
¯
q( ¯
x, ¯
t) = q(x, t) = q(− ¯
x, ¯
t ) .
(127)
Due to the fact that only the second derivatives appear in the sine-Gordon equation,
using (126) and (127) we get from (88) immediately
1 Though, the transversal sound velocity c T does in fact take the part of the relativistic critical
velocity inside of an isotropic medium for straight screw dislocations (and only for these), and
it even takes this part for the interactions between dislocations and elastic deformations of the
surrounding lattice which is ignored here. In Chap. 26, we will illustrate this with an example.
12 The Measurement of the Critical Velocity
This makes J. D. Eshelby’s [19] result plausible, the result that the parameters of
the crystal lattice really lead to a remarkable correspondence between the critical
velocity c o of the sine-Gordon equation and the transversal sound velocity c T . In this
sense, we will use this conformity as a model presumption,
c o ≈ c T .
(125)
Factually, the validity of the hypothesis c T ≈ c o is of no real importance for our
further considerations. We can however use this fact to illustrate our point,
1 and we
will arbitrarily refer, due to reasons of linguistic simplicity, to the signal moving
through the crystal with the velocity c o as the ‘sound signal’. By definition, c T is a
velocity that we measure in our preferred frame o , that is, in the inertial system
in which an outside observer would see that the crystal as a whole would be static.
Here, we measure in fact one and the same signal velocity c T in every direction of
our, a crystal approximating three-dimensional continuum. We consider from this
isotropic continuum only one single space dimension described by the choice of the
x-axis. Our linear chain lies in this x-direction, which defines our dislocation line.
We then need the transition to a continuum for this one direction. We only take the
periodical order of the surrounding lattice atoms of the crystal into account. This
led us in Chap. 8 to the sine-Gordon equation with c o ≈ c T . However, isotropy is
also for one-dimensional problems a property of extreme importance and will force
us to make a few explanations further along in this chapter. It has the meaning of
constancy for the critical velocity of both opposite directions. Firstly, we can state:
Our one-dimensional continuum is isotropic in the preferred frame o .
This one-dimensional isotropy of space becomes transparent for the sine-Gordon
equation (88), which is the basis of our one-dimensional considerations, as follows:
Whilst keeping the time t, we change the orientation of the x-axis; in other words,
we consider a space inversion, so that all the positions can be described using new
coordinates ¯
x according to
¯
x = −x , ¯
t = t .
Space inversion
(126)
The transversal displacement q = q(x, t) remains unchanged, in other words,
¯
q( ¯
x, ¯
t) = q(x, t) = q(− ¯
x, ¯
t ) .
(127)
Due to the fact that only the second derivatives appear in the sine-Gordon equation,
using (126) and (127) we get from (88) immediately
1 Though, the transversal sound velocity c T does in fact take the part of the relativistic critical
velocity inside of an isotropic medium for straight screw dislocations (and only for these), and
it even takes this part for the interactions between dislocations and elastic deformations of the
surrounding lattice which is ignored here. In Chap. 26, we will illustrate this with an example.