3.2 Motion of a Continuum
55
The displacement of an arbitrary particle from its initial position is
u = x − X.
(3.3)
Substituting Eq. (3.1) for x yields
u(X, t) = αXt
2 ,
(3.4)
whereas substituting Eq. (3.2) for X gives
u(x, t) =
αxt 2
1 + αt 2 .
(3.5)
These equations describe the same particle displacement field (distribution) in two
different ways. Equation (3.4) gives the displacement of the particle located initially
at X and tracks how the displacement of that same particle changes with time. In
contrast, Eq. (3.5) gives the displacement of the particle that is located at a fixed
point x at time t. As time passes, different particles are instantaneously located at
point x.
For example, consider the special case for α = 1 (ignoring units). For the particle
initially located at X = 5, Eq. (3.4) gives u = 5t 2 as plotted in Fig. 3.2c (upper
curve). If a tiny observer (Wilma) with a stopwatch glues the free end of a digital
tape measure to the point X = 5 on the ground, then hops on the particle at t = 0
and starts her watch (Fig. 3.2a), this plot represents the displacement vs. time that
she would record (beginning with u = 0) as she travels through space and the tape
unwinds.
Suppose now that a second observer (Fred) watches Wilma’s device while
standing on the ground next to her starting point at X = 5 (Fig. 3.2b). Unfortunately,
however, Fred is extremely near-sighted, and the only reading he sees on Wilma’s
tape measure is u = 0 before she moves away (at a speed much less than the speed
of light), and he records this value while setting his own clock to t = 0. On the
other hand, Fred alertly notices that other particles follow Wilma’s, each containing
an observer holding a tape measure. He records the reading from each (or as many as
he can) as they pass him, along with the corresponding time on his clock. Since Fred
records x = 5 for each passing particle, his data yield the curve u = 5t 2 /(1 + t 2 ),
as given by Eq. (3.5) and shown in Fig. 3.2c.
Clearly, the two plots are different. The first represents the displacement of
Wilma’s particle as it increases quadratically with time. The second represents the
instantaneous displacement of all particles, beginning with Wilma’s, as they pass
Fred. Note the peculiar form of Fred’s curve. The first point, u = 0 at t = 0, comes
from Wilma’s gauge, the next from a gauge just behind Wilma’s, and so on; hence,
Fred’s curve initially follows Wilma’s curve. However, his recorded displacements
approach the value u = 5 as t → ∞. The reason is that the left end of the bar is
fixed so it never reaches Fred, while a particle near that end eventually passes Fred
after moving through a distance of nearly five units. Particles closer to Wilma (for
small t) have moved relatively shorter distances when they pass Fred.
55
The displacement of an arbitrary particle from its initial position is
u = x − X.
(3.3)
Substituting Eq. (3.1) for x yields
u(X, t) = αXt
2 ,
(3.4)
whereas substituting Eq. (3.2) for X gives
u(x, t) =
αxt 2
1 + αt 2 .
(3.5)
These equations describe the same particle displacement field (distribution) in two
different ways. Equation (3.4) gives the displacement of the particle located initially
at X and tracks how the displacement of that same particle changes with time. In
contrast, Eq. (3.5) gives the displacement of the particle that is located at a fixed
point x at time t. As time passes, different particles are instantaneously located at
point x.
For example, consider the special case for α = 1 (ignoring units). For the particle
initially located at X = 5, Eq. (3.4) gives u = 5t 2 as plotted in Fig. 3.2c (upper
curve). If a tiny observer (Wilma) with a stopwatch glues the free end of a digital
tape measure to the point X = 5 on the ground, then hops on the particle at t = 0
and starts her watch (Fig. 3.2a), this plot represents the displacement vs. time that
she would record (beginning with u = 0) as she travels through space and the tape
unwinds.
Suppose now that a second observer (Fred) watches Wilma’s device while
standing on the ground next to her starting point at X = 5 (Fig. 3.2b). Unfortunately,
however, Fred is extremely near-sighted, and the only reading he sees on Wilma’s
tape measure is u = 0 before she moves away (at a speed much less than the speed
of light), and he records this value while setting his own clock to t = 0. On the
other hand, Fred alertly notices that other particles follow Wilma’s, each containing
an observer holding a tape measure. He records the reading from each (or as many as
he can) as they pass him, along with the corresponding time on his clock. Since Fred
records x = 5 for each passing particle, his data yield the curve u = 5t 2 /(1 + t 2 ),
as given by Eq. (3.5) and shown in Fig. 3.2c.
Clearly, the two plots are different. The first represents the displacement of
Wilma’s particle as it increases quadratically with time. The second represents the
instantaneous displacement of all particles, beginning with Wilma’s, as they pass
Fred. Note the peculiar form of Fred’s curve. The first point, u = 0 at t = 0, comes
from Wilma’s gauge, the next from a gauge just behind Wilma’s, and so on; hence,
Fred’s curve initially follows Wilma’s curve. However, his recorded displacements
approach the value u = 5 as t → ∞. The reason is that the left end of the bar is
fixed so it never reaches Fred, while a particle near that end eventually passes Fred
after moving through a distance of nearly five units. Particles closer to Wilma (for
small t) have moved relatively shorter distances when they pass Fred.