WATER RESOURCES RESEARCH, VOL. 20, NO. 12, PAGES 1883-1890, DECEMBER 1984 A Model of Flood Wave Speed-Discharge Characteristics of Rivers TONY H. F. WONG AND ERIC M. LAURENSON Department of Civil Engineering, MonashUniversity, Clayton, Victoria,Australia Recent wavespeed-discharge investigations cardedout on six Australian fiver reaches show consistent variations of wave speed with discharge and provide new knowledge in this area with applications in both flood routingand runoff routing.A modeldeveloped to define the wavespeed-discharge relationis described. The model is basedon concepts of the physical factorsaffecting the wave speed-discharge characteristics of natural channels. It provides for overbank discharge as well as main channel flow. The applicability of the modelis demonstrated using data from the six reaches. INTRODUCTION Storage flood routing methods haveproved to be very suc- cessful in simulating the passage of a flood wavealong a river channel.While most of the parametersof such methods are regarded as empirical, they do have physical interpretations related to the channel shape and flow retardance.Central to all flood routing methods is the travel time of the flood wave through the reach. This characteristic is often expressed through the length of reach and the wave speed. In storage routing it is implicit in the storage-discharge relation for the reach. Typically, the storage S-discharge Q relation is ex- pressed as S- kQ '• (1) where k and m are constants. Traveltimeis defined as the time taken by a given hydro- graph feature, usually the peak discharge,to propagate through the reach. In general, travel time varies nonlinearly with discharge. Wave speed-discharge investigations carried out by Wong and Laurenson [1983a] on six Australian river reaches, defined in Table 1, show consistent variations of wave; speed with discharge and provide empirical confirmation of the form of relation first postulated by Price [1973]. Wave speeds were determined as reach length divided by travel timeof flood peak.These speeds are plotted against the average of the peak discharges at thetwoends of thereach in Figures 1-6. To better define the low dischargeends of the relations, travel times of commencement of rise were also used in several cases. The main feature of the empirical relation is that the wave speed first increases rapidly with discharge, th en decreases sharply, andfinally increases at a slower rate.All 'thereaches except for the Mitta Mitta reach (MA1) have distinct ch/tnnels and extensiveflood plain flow area, while MA1 has a steep confined channel with noflood plain forhalf :its length and a distinct but narrow flood plain for the other half. Power func- tions were fitted to the low and high discharge ranges of the , wave speed-discharge curves. The bankfull discharges are assumedto lie between two measures of bankfulldischarge, Qb• and .Oh2 (see Figures 1-6). The discharge at which warnings of local flooding are first issued to landholders, Qo•, can be regarded as the lower limit, while the average discharge, determined from rating curves, for a waterleveleqqalto the bank levelat the upstream and downstreamgauging 'stations, Qo2, can be regarded as an upper limit. Bank level was identifiedby an abrupt change in the surface width versus depth relation. Copyright 1984 by the American Geophysical Union. Paper number4W 1102. 0043-1397/84/004W- 1102505.00 It is interesting to note that for all cases, the wave speed deviates from the initial power function and enters a rapidly decreasing phaseat a discharge substantially lower than the bankfull discharge.The transition between the two power functions commences at about half bankfull discharge and extendsto above bankfull, encompassing the two flow con- ditions of channel and flood plain flow. While wave speed- discharge relations defined by curves eye-fitted to these data can be usedgraphically, it is desirable to find somegeneral formfor the relatio n to helpclarify the processes involved and to facilitate flood routing.Investigations have,therefore, been carried out to definea model, with parameters as far as possi- ble related to physical characteristics, that fits the complicated form ofthe wave speed-discharge data in Figures 1-6. EQUATION FOR THE KINEMATIC WAVE SPEED Sinceflood waves in natural channels travel approximately as kinematic waves [Henderson, 1•966], kinematic wave theory provides a starting point for the modeling of wave speed- discharge relations in such channels. At a given cross section the kinematic wave speedc is related to the surface width B and the inverse slope of the rating curve dQ/dy by the equa- tion 1 dO c = - (2) B dy Evaluating the derivative in (2) from the Manning formula gives (AI__ 2dR/dy dn/dyh c =Q +3 BR Bn ,] (3) where A is the cross-sectional area, R is the hydraulic radius, and n is the Manning roughness coefficient. If (3) is to describe the c(Q)data of Figures!-6, its various termsmust be regard- ed as representativevalues for an irregular channel reach. Also, the term in parentheses must vary with Q in a manner generally similar to the data. The sharp decreasein wave speedoccurringwell below bankfull discharge in Figures 1-6 cannot be explainedby variation of the geometrical terms of (3). If (3) is to describe the observed changes in wave speed, then dn/dy, and consequently dn/dQ, must be positive at dis- charges well below bankfull, contrary to the accepted notion of decreasing roughness with increasing discharge. To resolve thisapparent anomaly, it is postulated that flow retardance increases due to increased roughness arising from the effectsof bank vegetationand also due to the delaying effect of localized flood plain storage. We no longerrely on (3), nor on the Manning formula upon which it is based, with their dependence on geometricand roughness characteristics alone, to model the observed data for the natural channels 1883