energy gradient obtained from the Chezy equation with a constant conveyance coefficient,
Lighthill and Whitham (1955) obtained a linear form of eq 1 and 2 by substituting for y
and v
y = y0 + y1
v = v0 + v1
(3)
where constant y0 and v0 represent steady uniform flow in a channel with constant slope
y1
y
v
+ v0 1 + y0 1 = 0
(4)
t
x
x
y
2v
y
v1
v
+ v0 1 + g 1 + gS0 1 - 1 = 0.
(5)
y0
v0
t
x
x
The momentum equation (eq 5) describes linear dynamic waves. If we assume that the
inertia terms of eq 5 are small relative to the other terms, the momentum equation for linear
diffusion waves is obtained:
y
2v
y1
+ S0 1 - 1 = 0.
(6)
y0
v0
x
The momentum equation for linear kinematic waves can be obtained from eq 6 by assum-
ing the depth gradient is small relative to the bed slope and can be neglected.
Following Lighthill and Whitham (1955) we combine eq 4 and 5 to eliminate either the
depth or velocity derivatives. The resulting second-order linear equations for depth or vel-
ocity are the same, and equivalent to the original system of first-order equations:
(
)
1
(φt + ck φx ) = 0
φtt + 2v0 φxt + v0 - c0 (v0 + c0 ) φxx +
(7)
η
where the dependent variable φ represents either v1 or y1, subscripts x and t indicate differ-
entiation with respect to those variables, c0 = gy0 is the celerity of a disturbance in still
water, ck = 3v0/2 is the kinematic wave celerity, and η = v0/2gS0. Mendoza (1995) provided
an historical perspective on the development and solution of this hyperbolic equation,
which we call the linear dynamic wave equation. Higher-order dynamic waves travel
along two sets of characteristics described by
dx v0 + c0 = c+
(8)
=
dt v0 - c0 = c-
where the product of c and c+, the dynamic wave celerities in the upstream () and down-
stream (+) directions for subcritical flow, appears as a coefficient in eq 7.
If the lower-order terms of eq 7 were absent, corresponding to large η, the general solu-
tion would have the form
φ = φ1(x - c+t) + φ2 (x - c-t)
(9)
with a structure totally dependent on the dynamic waves. On the other hand, if the higher-
order terms were absent, corresponding to η → 0, the general solution would have the form
φ = φ0 (x - ckt)
(10)
3
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