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and the consequences
p 1 (x,t) = - B u 1 (x,t)/ x = - (P s /2) e i
(t +x/c)
(1.178)
p 2 (x,t) = - B u 2 (x,t)/ x = + (P s /2) e i
(t -x/c)
(1.179)
Thus, we have proved that half of the P s signal propagates to the left, with the
remaining half propagating to the right.
Note that we had dropped our delta function formalism, and used an
“undergraduate approach” based on Equation 1-175 instead, which equivalently
prescribes a nonzero jump in the first spatial derivative of u. Two functions u 1
and u 2 were required to carry out the matching in Equations 1-174 and 1-175,
and the elegance provided by delta functions and transform methods was lost.
1.9 First-Order Partial Differential Equations
In problems governed by high-order equations, solutions are sometimes
obtained by solving equivalent sets of first-order differential equations. Here we
develop recipes useful to solving first-order partial differential equations. These
ideas are covered more fully in Courant and Hilbert (1989). In order to avoid
any confusion, index notation and summation conventions are avoided, in favor
of x, y and z terminology. This section is relevant to three-dimensional
kinematic wave theory and geophysical applications discussed later.
The solution for (x,y,z,t) satisfying specific equations, for instance, / t
+ a / x + b / y + c / z = x + y, can be handled straightforwardly. But
in general, we need not focus on any one particular equation, since it is just as
convenient to consider the representation
H(
,
/ x,
/ y ,
/ z ,
/ t , x , y, z, t) = 0
(1.180)
Using subscript notation for our partial derivatives, we have
H(
,
x , y , z , t , x , y, z, t) = 0
(1.181)
in which case our example becomes H = t + a x + b y + c z - x - y.
Equation 1.181 may be linear or nonlinear in and its derivatives; for
instance, z - t = x i s linear , while ( z ) 2 - t = x i s nonlinear . The explicit
presence of x, y, z and t in the argument of H indicates the existence of variable
coefficients. Now, the total differential for any function (x,y,z,t) satisfies
d = x dx + y dy + z dz + t dt (1.182)
irrespective of H. The idea behind “parameterization” is simple (e.g., the circle
x 2 + y 2 = 1 can be expressed as x = sin , y = cos , where 0 < < 2 ); it will
be useful to develop the general case in which is parametrically represented.
Thus, the change in
as a function of any parameter along a spatial trajectory
is expressible using
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