after the investigation showing how in a typical case they come in time to dominate 

 the scene, must now be described in detail. 



19. Kinematic waves 



Although such flood waves were discovered 98 years ago by Kleitz (unpub- 

 lished), also treated very ably by Boussinesq [3] in 1877 and greatly illuminated by 

 James Seddon [29] in 1900 with his studies of the Mississippi and the Missouri, Dr. 

 Whitham and I felt that the subject was still sufficiently little-known, and contained a 

 sufficient number of doubts and ambiguities, to warrant a new treatment [22]. In this 

 treatment we considered such flood waves as a special case of a general class of wave 

 motions which we ventured to call "kinematic waves." 



Kinematic waves exist in any one-dimensional flow system if, to a sufficient 

 approximation, there is a functional relationship between 



(i) the flow q (quantity passing a given point in unit time) , 



(ii) the concentration k (quantity per unit distance), and 

 (iii) the position x. 

 Thus we assume that 



q = q(k,x) or k = k(q,x) . (19) 



Note that qjk is the mean velocity of flow, V say, so that the assumption also fixes V 

 as a function of flow or of concentration at every place in the system. 



In the application to river waves, "quantity" is taken as volume of water; q is 

 volume flow rate, and k is the cross-sectional area of the water. For any river there 

 is a series of possible steady regimes, each with uniform flow rate q all along the river, 

 and k (or V) is a definite observed function of x for each regime; if this is known for 

 every value of q we have our function k{q,x). I shall have more to say about the 

 mechanisms that govern the relationship between k and q at a given position x, but a 

 common one is of course the balance between gravity and friction. In a flow in which 

 q is varying there is obviously a time lag before the changes in A: at a point catch up 

 with the changes in q, but for sufficient slowly-varying flows it is reasonable to neglect 

 this time lag. 



On this basic assumption, the properties of kinematic waves follow from the 

 equation of continuity alone, 



dk dq 



— + — = (20) 



dt dx 



It is for this reason that the name "kinematic" is suggested, in contrast to the classical 

 wave motions which we would call "dynamic," depending as they do on Newton's 

 second law of motion. Another contrast is that kinematic waves possess only one wave 

 velocity at each point, while dynamic waves possess at least two (forwards and back- 

 wards relative to the medium). This is because equation (20) is a first-order equation. 

 On multiplication by 



C = (—) (21) 



\dk/ x = constant 



it becomes 



dq dq 



h C— = (22) 



dt dx 



This means that the flow q is constant on waves travelling past the point with velocity 

 C Mathematically, the equation has one system of "characteristics" (given by 

 dx — C dt), and along each of these q is constant. 



32 



