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*

(3)

where constant *y*0 and *v*0 represent steady uniform flow in a channel with constant slope

+ *v*0 1 + *y*0 1 = 0

(4)

2*v*

+ *v*0 1 + *g * 1 + *gS*0 1 - 1 = 0.

(5)

*v*0

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:

2*v*

+ *S*0 1 - 1 = 0.

(6)

*v*0

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 + *c*k φx ) = 0

φtt + 2*v*0 φxt + *v*0 - *c*0 (v0 + *c*0 ) φxx +

(7)

η

where the dependent variable φ represents either *v*1 or *y*1, subscripts *x *and *t *indicate differ-

entiation with respect to those variables, c0 = * gy*0 is the celerity of a disturbance in still

water, *c*k = 3*v*0/2 is the kinematic wave celerity, and η = *v*0/2*gS*0. 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

(8)

=

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 *- *c*kt)

(10)

3