Geology Reference
In-Depth Information
du/dt = u/ t + dx/dt u/ x (6.83)
If we compare Equations 6.81 and 6.83, we find that du/dt = 0 provided we
identify dx/dt = u; in other words, u(x,t) does not change if we follow the ray
path dx/dt = u(x,t). This can be succinctly expressed by the equivalent
functional statement
u(x,t) = G(x-ut) (6.84)
which emphasizes that u(x,t) must be a function of “x-ut.” G can be any
function, of course. If, in the problem at hand, we are to enforce the initial
condition
u(x,0) = F(x) (6.85)
where F(x) is given, then it is clear that the choice G = F solves the problem.
Hence, we have the general solution
u(x,t) = F{x-u(x,t),t} (6.86)
6.6.2 Singularities in the low-order model.
As in the case of Equation 6.82, Equation 6.86 may or may not lead to
shock formation, infinities, or singularities in the first derivatives. To see how
these may arise, let us differentiate Equation 6.86 with respect to x; using the
chain rule, we obtain u/ x = F'{x-u(x,t)t}{1 - t u/ x}. If we solve for u/ x,
we find that
u/ x = F'{x-u(x,t)t}/{1 + tF'} (6.87)
If the initial condition u(x,0) = F(x) is such that F' > 0, it follows that the
denominator 1 + tF' > 0 is positive, and u/ x is well behaved. On the other
hand, if F' is negative, shockwave solutions with infinite values of u/ x form in
a finite amount of time. These shocks are mathematically analogous to the
saturation discontinuities encountered in reservoir flow and the pressure jumps
obtained in the theory of gas-dynamic shocks.
6.6.3 Existence of the singularity.
We have shown that Equation 6.86 embodies a class of solutions
containing shockwaves. But do these exist in reality? Although shocks can
form as a mathematical consequence of Equation 6.86, it is often the case that
Equation 6.81 arises only as a rough model to a more accurate problem
formulation containing more detailed information. Fortunately, for us, exact
solutions to two higher-order equations whose low-order terms are identical to
Equation 6.81 are available; these are, namely, Burger' s equation and the
Korteweg de Vries equation. “Burger' s equation,” which arises in the
description of gasdynamic shocks in high speed aerodynamics, is given by
u/ t + u u/ x =
2 u/ x 2
(6.88)
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