Common partial differential equations of computational hydraulics 29
known initial distribution f(x, t = 0). The parameter N expresses the rate at
which the diffusion is realized; faster for large and slower for small values
of N. In order for the diffusion equation to be well-posed, both boundary
conditions and initial conditions for the entire solution domain should be
known.
For hyperbolic PDEs (Δ > 0), describing one-dimensional time-dependent
problems, the equation in terms of the x-t variables takes the form
∂
∂
=
∂
∂
2
2
2
2
2
f
t
c
f
x
o
(3.6)
or for a two-dimensional space (variables x, y, t)
∂
∂
=
∂
∂
+
∂
∂






2
2
2
2
2
2
2
f
t
c
f
x
f
y
o
(3.7)
where c o is known as celerity, phase velocity or speed of propagation.
Equations 3.6 and 3.7 are known as wave or telegrapher’s equations, and
describe the propagation of a signal over time, in both positive and negative directions along the x-axis (and y-axis), moving from its original position (f(x, t = 0)) with celerity c o . This equation applies to various types of
water waves including gravity waves, tsunamis, astronomical tides, and
elastic waves.
3.2 PARTIAL DIFFERENTIAL EQUATIONS
AND CHARACTERISTIC DIRECTIONS
A much better appreciation of the physical significance of elliptic, parabolic
and hyperbolic equations can be provided by the introduction of the concept
of characteristic directions. Since the classification of PDEs depends only
on the coefficients of the second-order derivatives, for simplicity Equation
3.1 can be condensed to
A
f
x
B
f
x y
C
f
y
H
∂
∂
+
∂
∂ ∂
+
∂
∂
+ =
2
2
2
2
2
0
(3.8)
where H contains all of the remaining terms in Equation 3.1. If we consider
within the solution domain a curve S on which the variable f(x,y) and all of