Analysis of Linear and Monoclinal River Wave Solutions
MICHAEL G. FERRICK AND NICHOLAS J. GOODMAN
Much effort over several years has gone into the development and application of numer-
ical models for unsteady river flow problems. In particular, one-dimensional numerical
models that solve either the nonlinear dynamic wave or diffusion wave equations have been
applied over wide ranges of river and flow conditions. Criteria for choosing a model from
among these and other alternatives for given conditions have been obtained from analyses
of linearized equations (Ponce and Simons 1977, Ponce et al. 1978, Menendez and Norscini
1982, Kundzewicz and Dooge 1989). Dimensionless parameters of the nonlinear equations
have also been used for model selection (Woolhiser and Liggett 1967, Ferrick 1985). Several
authors have treated the diffusion wavedynamic wave modeling decision as a choice be-
tween simplicity and accuracy. However, Lighthill and Whitham (1955) argued that
dynamic waves are subordinated when the flow is "well subcritical," making the character-
istics of the dynamic wave system unsuitable as a basis for computation. Numerical
dynamic wave and diffusion wave models cannot be readily used to resolve relative accura-
cy issues or to identify optimal model selection criteria.
The dynamic wave equations include flow inertia terms, and form a second-order hyper-
bolic system with two sets of characteristics that trace the paths of dynamic waves on the
x-t plane. The diffusion wave equations neglect the inertia terms as small, resulting in a para-
bolic system that models a diffusing "mass wave." As the magnitude of the wave diffusion
term decreases, this system approaches a zero-diffusion limit, the kinematic wave equation.
This first-order hyperbolic equation has a single set of characteristics, the subcharacteristics
of the dynamic wave equations, that trace the paths of kinematic waves on the x-t plane.
Dynamic waves and kinematic waves are both present during unsteady river flow, and it is
difficult to conceptualize their respective roles in the dynamic wave and diffusion wave
models. An improved understanding of these models would be an important step toward
resolution of relative accuracy and model selection questions.
Linearized forms of the dynamic wave and diffusion wave equations have been solved
analytically to obtain approximate river flows (Dooge and Harley 1967, Hayami 1951). With
variable coefficients treated as constants, linear solutions are strictly valid only for small
flow disturbances. However, these solutions are valuable because of their common structure
with corresponding nonlinear solutions. The linear dynamic wave solution is the most gen-
eral and provides a standard for comparison with simpler linear solutions, but systematic
comparisons have not been developed. Potential benefits include better definition of the
correspondence between models, and resolution of the time and distance scales where dif-
ferences are important. Relationships between dynamic wave and kinematic wave celerities
and the downstream translation of linear wave profiles have not been quantified because
equations for the celerity of points along these profiles are not available. Such celerity rela-
tions would clarify the roles of characteristics and subcharacteristics and provide insight
into the structure of each solution.