On Kinematic Waves. I.

Flood Movement in Long Rivers

Author(s): M. J. Lighthill and G. B. Whitham
Source: Proceedings of the Royal Society of London. Series A, Mathematical and Physical
Sciences, Vol. 229, No. 1178 (May 10, 1955), pp. 281-316
Published by: The Royal Society
Stable URL: http://www.jstor.org/stable/99768
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 On kinematic waves
I. Flood movement in long rivers

BY M. J. LIGHTHILL, F.R.S. AND G. B. WHITHAM

Department of Mathematics, University of Manchester

(Received 15 November 1954-Read 17 March 1955)

In this paper and in part II, we give the theory of a distinctive type of wave motion, which
arises in any one-dimensional flow problem when there is an approximate functional relation
at each point between the flow q (quantity passing a given point in unit time) and con-
centration k (quantity per unit distance). The wave property then follows directly from the
equation of continuity satisfied by q and k. In view of this, these waves are described as
'kinematic', as distinct from the classical wave motions, which depend also on Newton's
second law of motion and are therefore called 'dynamic'. Kinematic waves travel with the
velocity aq/ak, and the flow q remains constant on each kirematic wave. Since the velocity
of propagation of each wave depends upon the value of q carried by it, successive waves may
coalesce to form 'kinematic shock waves'. From the point of view of kinematic wave theory,
there is a discontinuous increase in q at a shock, but in reality a shock wave is a relatively
narrow region in which (owing to the rapid increase of q) terms neglected by the flow-
concentration relation become important. The general properties of kinematic waves and
shock waves are discussed in detail in ? 1. One example included in ?1 is the interpretation of
the group-velocity phenomenon in a dispersive medium as a particular case of the kinematic
wave phenomenon.
The remainder of part I is devoted to a detailed treatment of flood movement in long
rivers, a problem in which kinematic waves play the leading role although dynamic waves
(in this case, the long gravity waves) also appear. First (?2), we consider the variety of
factors which can influence the approximate flow-concentration relation, and survey the
various formulae which have been used in attempts to describe it. Then follows a more
mathematical section (?3) in which the role of the dynamic waves is clarified. From the full
equations of motion for an idealized problem it is shown, that at the 'Froude numbers'
appropriate to flood waves, the dynamic waves are rapidly attenuated and the main disturb-
ance is carried downstream by the kinematic waves; some account is then given of the
behaviour of the flow at higher Froude numbers. Also in ?3,, the full equations of motion are
used to investigate the structure of the kinematic shock; for this problem, the shock is the
'monoclinal flood wave' which is well known in the literature of this subject. The final
sections (??4 and 5) contain the application of the theory of kinematic waves to the deter-
mination of flood movement. In ?4 it is shown how the waves (including shock waves)
travelling downstream from an observation point may be deduced from a knowledge of the
variation with time of the flow at the observation point; this section then concludes with
a brief account of the effect on the waves of tributaries and run-off. In ?5, the modifications
(similar to diffusion effects) which arise due to the slight dependence of the flow-concentration
curve on the rate of change of flow or concentration, are described and methods for their in-
clusion in the theory are given.

1. INTRODUCTION

In this paper and in part II (Lighthill & Whitham I955), we wish to draw attention
to a class of wave motions physically quite distinct from the classical wave motions
encountered in dynamical systems. They have received some attention already in
connexion with flood movement in long rivers, but no general treatment seems to
have been given.

I9 [ 281 ] Vol. 229. A. (Io May I955)

282 M. J. Lighthill and G. B. Whitham
The waves will be considered only for one-dimensional flow systems. Then they
exist if, to sufficient approximation, there is a functional relationship between
(i) the flow q (quantityt passing a given point in unit time),
(ii) the concentration k (quantity per unit distance), and
(iii) the position x.
On this assumption the wave property follows from the equation of continuity
alone. Accordingly, we suggest that the waves be described as 'kinematic'. The
classical wave motions would in contrast be described as 'dynamic' waves, depend-
ing as they do on Newton's second law of motion, together with some assumption
relating a stress to a displacement (as in gravity waves), to a strain (as in the non-
dispersive longitudinal and transverse waves), or to a curvature (as in capillary
waves and flexural waves).
One important difference is that kinematic waves possess only one wave velocity
at each point, while dynamic waves possess at least two (forwards and backwards
relative to the medium). This is because the equation of continuity, that is, the
conservation law
ak aq
at ax ' (1)
which states that the quantity in a small element of length changes at a rate equal
to the difference between the inflow and outflow, is of the first order only. If we
assume that
q = q(k,x), (2)
then, on multiplying (1) by
c ==( =c(k, x), (3)
kx constant

0.
we obtain
at _cax= (4)

This means that q is constant on waves travelling past the point with velocity c
given by (3). Mathematically, the equation has one system of 'characteristics'
(given by dx = cdt), and along each of these the flow q is constant.
The wave velocity c, by (3), is the slope of the flow-concentration curve for fixed x.
This fact has been referred to in the literature on flood movements (see ? 2 below)
as the Kleitz-Seddon law.
In terms of the mean velocity at a point, which is
v - q/k, (5)
d dv
the wave velocity is c = d (vk) = v + kd (6)

Thus c > v when the mean velocity increases with concentration (as in rivers), while
c < v when it decreases with concentration (as in traffic flow).
Kinematic waves are not dispersive, but they suffer change of form due to non-
linearity: (dependence of the wave velocity c on the flow q carried by the wave)
exactly as do travelling sound waves of finite amplitude. Accordingly, continuous
t For example, volume of water (in a river), number of vehicles (on a road).
t This might be called 'amplitude dispersion', in contrast to 'frequency dispersion'.
 Kinematic waves. I 283
wave forms may develop discontinuities, due to the overtaking of slower waves by
faster ones. We propose to describe these as shock waves, since their process of
formation is exactly that of shock waves in a gas.
The law of motion of kinematic shock waves is derived from conservation con-
siderations, as was the law governing continuous kinematic waves. If the flow and
concentration take the values ql, k1 on one side, and q2, k on the other side, of the
shock wave, which moves with speed U, then the quantity crossing it per unit time
may be written either as ql - Uk1 or as q2 - Uk2. This gives the velocity of the shock
wave as
U= q- (7)
- c1'
Tk2
This is the slope of the chord joining the two points on the flow-concentration curve
(for given x) which correspond to the states ahead of and behind the shock wave
when it reaches x. In the limit when the shock wave becomes a continuous wave,
the slope of the chord becomes the slope of the tangent and the velocity given by
(7) coincides with that given by (3).
It will appear that kinematic shock waves can change strength by absorbing
continuous waves, and can unite with other shock waves to form single shock waves,
exactly like dynamic shock waves in gases.
Now there is probably no system in which the flow, as has been assumed, is
accurately a function of concentration and position. Normally some small time lag
may intervene between adjustments of flow and concentration at a given point; or,
again, the relationship between them may have only statistical validity. Treatment
as a 'kinematic wave' will describe the development of the flow with reasonable
accuracy over times large compared with such a time lag, provided that the diffusive
effects due to it, and to statistical deviations from the mean flow-concentration
relation, are small by comparison with the wave effects. Estimates of accuracy from
such considerations are obtained below and in part II.
In particular, the shock waves will not be perfect discontinuities. They will
have a definite thickness, produced (as with dynamic shock waves in gases) by a
balance between the diffusive effects mentioned abovet and the tendency to
thinning due to the excess wave velocity behind the shock wave over that in front.
However, it may still be convenient to calculate the motion of these shock waves
as if they were discontinuous, bearing in mind their real thickness when the theory
is finally interpreted.
The formation of a kinematic shock wave is illustrated in figures 1, 2 and 3, in the
specially simple and important case when the flow-concentration curve (figure 1)
is independent of the position x, that is, when q is a function of k alone. In this
case, since q is constant along any wave, k and hence c must also be constant along
it, so the wave moves with constant velocity. Thus, in a space-time diagram (figure 2),
the waves are straight lines, parallel to the tangent to the flow-concentration curve.

t The diffusive effects may be represented (? 5) by a second derivative of q inserted in the

equation of motion (4). The thickness of a shock wave is governed by this higher-order term
in the equation, which outside the shock wave is negligible-exactly as in gas dynamics, where
the analogous higher-order term represents the effects of viscosity and heat conduction.
I9-2
284 M. J. Lighthill and G. B. Whitham

FIGURE 1

FIGU:RE2

FIGURE 3
 Kinematic waves. I 285

at the point which corresponds to the values of q and k carried by the wave; this
makes their construction particularly easy.
When the flow-concentration curve changes with x, the waves are no longer
straight lines. The path of the wave carrying a given flow q is still, however, pre-
dictable once for all. Thus, if we express the wave velocity c as a function of q and x,
the path of the wave is

x" dx
t = c(q,- - + constant TX /3k\ dx + constant. (8)
o c(q, x) saq
tx constant

When the integral in (8) has been calculatedt for all values of q, the construction
of the wave pattern presents hardly more difficulty than in the case illustrated
in figure 2.
Figure 2 shows the progress of a 'hump', that is, a region of higher concentration
in the midst of a region of uniform concentration. When the (q, k) curve is concave
upwards, like that in figure 1, then the wave velocity c = dq/dk increases with k,
and hence increases also with x for the waves in the rear of such a hump. Accordingly,
those waves spread out fanwise, getting ever farther apart. In the front of the hump,
however, k and hence also c decrease with x, so that the waves there converge and
finally cross. Obviously one cannot accept such a solution, in which the flow
effectively has two values at some points. Fortunately, it is always possible to
avoid this by fitting in a shock wave, as in figure 3 (which is drawn on a smaller
scale than figure 2, to show the later development of the shock wave). Techniques
for calculating the progress of the shock wave, from the condition that at each
point its velocity is given by (7), in which ql and q2 are the flows carried by two
continuous waves which meet on the shock wave, are given below (? 4). Figure 3
shows how the increase in concentration at the shock wave grows initially, and also
how after a very long time the waves which meet on it are inclined to each other
at a smaller angle again-that is, the increase is reduced. Thus the shock wave,
and with it the hump, ultimately decay, as the shock wave passes farther and
farther into the region of uniform concentration ahead of the hump.
If the curve of q against k is convex upwards (as in the problem of traffic flow
discussed in part II), the wave velocity is reducedin the 'hump', and the shock wave
appears in the rear. But its progress and decay are in other respects similar.
In some applications, including the case of flood waves (see below), kinematic
waves and dynamic waves are both possible together. However, the dynamic
waves have both a much higher wave velocity and also a rapid attenuation. Hence,
although any disturbance sends some signal downstream at the ordinary wave
velocity for long gravity waves, this signal is too weak to be noticed at any con-
siderable distance downstream, and the main signal arrives in the form of a kine-
matic wave at a much slower velocity (? 3).
Now, this situation is so parallel to the familiar behaviour of dynamic waves in
a dispersive medium, where the energy of the vibrations in any narrow frequency
rx
t This might be done most easily by calculating k(q, x) dx for different values of q, and
differentiating with respect to q.
286 M. J. Lighthill and G. B. Whitham
band travels not at the wave velocity but at the group velocity for that frequency
band, that one is impelled to search very carefully for a way in which the behaviour
of flood waves can be regarded as essentially a particular case of the group-velocity
phenomenon. The search is fruitless, however, and the true relationship is different.
The group-velocity phenomenon is itself essentially a particular case of the kine-
matic wave phenomenon, so that it is cognate to, rather than inclusive of, the
behaviour of flood waves.
To understand the behaviour of a travelling wave in a dispersive medium from
the point of view of kinematic wave theory, it is necessary only to choose the
' quantity' whose motion is discussed kinematically to be the number of wave peaks.
Then the flow q (number of peaks passing per unit time) becomes the frequency,
and the concentration k becomes the wave number. A functional relationship
exists, of course, between them. The basic conclusion (4) of kinematic wave theory
then states that the frequency q remains constant for points travelling with the
velocity dq/dk; this is the group velocity if q and k have the meanings mentioned.
This statement describes correctly the process of dispersion (compare the statement
at the beginning of the last paragraph). If the kinematic waves spread out fanwise,
that is, the energy in a narrow frequency band is spread over an increasingly larger
region, then evidently the amplitude must vary as the inverse square root of the
distance between successive kinematic waves (apart, that is, from any damping
due to energy dissipation in that frequency band).
A difference from ordinary kinematic waves arises, however, when the wave
groups cross, as in figure 2 above. There is no physical unreality about this crossing,
since the frequency q can without difficulty take two values at the same place.
For example, there are water waves of two frequencies (a capillary wave and a
gravity wave) corresponding to any given value of the group velocity, and wave
groups with these two frequencies can travel along together. Thus the modification
of figure 2 into figure 3 does not happen in this application; no discontinuities in
frequency can appear.
It is well known, of course, that group velocity can be explained kinematically.
However, such simple physical explanations as have been given previously are
unsatisfactory because the arguments apply only to a wave group with a small total
frequency range. The above argument shows that the essential properties follow
at once for an arbitrary wave train from applying 'conservation of number of wave
peaks'. This conservation is not, of course, accurately true; but once sufficient
dispersion has occurred to render small the frequency change in a single wave-
length, then the appearance or disappearance of peaks through the occurrence of
horizontal points of inflexion must become very rare. The additional argument
bringing in energy shows that Kelvin's asymptotic formula for the travelling wave
resulting from a limited initial disturbance can be deduced by elementary arguments
in every respect except phase.t Further, the kinematic wave approach to the
subject is usefully general; thus, it will show at once how the transmitted waves
redisperse when an established train of waves enters another medium.
t Evidently, the d2q/dk2in Kelvin's formula comes in as the rate of spread dc/dk of kine-
matic waves of different velocities.
 Kinematic waves. I 287
The reader will observe that kinematic wave theory is being advocated not only
as an instrument for research, but also as a demonstrational method for deriving im-
portant results with a minimum of labour. It is in this connexion that we wish to
point out that travelling dynamic waves of longitudinal type can be regarded as
special cases of kinematic waves, and that this may give a conveniently simple way
of deriving their non-linear properties. Thus for plane sound waves, if 'quantity'
signifies mass per unit area perpendicular to the direction of flow, then the
concentration k (quantity per unit length) becomes the ordinary density; the
flow q becomes kv, where v is the fluid velocity. Now, for a wave travelling without
energy dissipation in the direction x increasing, we have at all points Riemann's
relation k
dk
v= a-, (9)

where a is the velocity of sound for density k, and k0 is the density of the undis-
turbed atmosphere. This gives a relationship between q = vk and k, which corre-
sponds to a kinematic wave velocity
dv
c = dq =v+k = v+a. (10)
dlc dlcd
Note that the wave velocity a relative to the medium corresponds to a kinematic
wave velocity (always a velocity in space) v a. It will be seen from (10) that
Riemann's relation is inevitable kinematically if waves are to exist which travel
unchanged with velocity a relative to the medium.
The theory of the formation of shock waves in a gas is then a special case of the
general theory given above. It should be remarked that it will be only approximate,
as equation (9) (the constancy of Riemann's invariant) is not accurately true in
the region behind a shock wave. However, that equation is a very good approxima-
tion for shock waves of moderate strength, and is normally used in all attempts to
calculate shock wave movement.
What has been said applies equally to the behaviour of long gravity waves in
a channel of constant width and horizontal bottom. If 'quantity' signifies volume
of water per unit width, the concentration k becomes the local depth of the water;
equations (9) and (10) are correct with a = 1(gk), and the kinematic shock wave
is now a bore.
But, however convenient such devices may be for developing the theory of a
number of important phenomena from a simple and unified point of view, one must
not forget that in these last two problems the system is only a kinematic wave
system if attention be deliberately restricted to waves travelling in one direction
only. The methods cannot be used to treat reflexion (after which Riemann's relation
(9) ceases to be valid), and, indeed, in a true kinematic wave system no reflexion
of any kind is possible (mathematically, there is only one system of characteristics).
The rest of this paper, and part II, are devoted to kinematic wave systems which
are more 'robust' in that the hypotheses remain essentially valid under a wider
range of states of the system.
288 M. J. Lighthill and G. B. Whitham

2. FLOOD WAVES

Although we believe that the full kinematic wave theory as set out in ? 1, com-
prising the theory of continuous waves, 'shock waves' and the formation of shock
waves out of continuous waves, has not previously been given, parts of it have
been known (though never widely known) for almost a century in their application
to flood movement in rivers.
In particular, several writers independently have given the theory of continuous
kinematic waves, based on equations (1) to (4), as applying to flood movement.
Boussinesq (1877) gives a full treatment, including a derivation of equation (8),
as does Forchheimer (1930) in his invaluable book Hydraulik. They refer to
Kleitz (I858, unpublished), Breton (1867) and Graeff (1875) as pioneers of the
theory.
The earliest account in the English language is by Seddon (900o), who discusses
the problem at length with special reference to the Mississippi and its tributaries.
Seddon was unaware of the earlier work. In some ways, however, his account
(which, conversely, was unknown to Forchheimer (I 930) and has received only rather
perfunctory reference even in the later American literature) is to be preferred. This
is because he shows a greater understanding of the variety of mechanisms which
govern the relationship between flow and concentration, and such understanding
is vital for sound application or improvement of the theory.
In the best known of these mechanisms, a balance is struck between the friction
of the bottom and the component of gravity in a direction which is downstream and
parallel to the free surface of the river.t If the downward slope of the free surface
is S, then the gravitational force per unit length of river is pgSk, where k is the
concentration (volume of water per unit length). The frictional force per unit length
may be expressed as fpv2L, where v is the mean velocity, L the wetted perimeter of
the cross-section, andf a coefficient of friction. Equating these forces, we get

=/JjS (11)

In the alternative form v = C V(RS), where R = klL (area of cross-section divided

by wetted perimeter, or 'hydraulic mean depth') and C = 4(gif), this is the famous
'Chezy formula'. It shows that voc ki, and hence that the wave velocity, by (6), is

,dv
c=v+k k = v, (12)

provided that none of S, f or L vary with k.

In practice at least one of these shows some variation with depth. Thus the
friction coefficient f depends on the ratio of the size of typical roughness elements in
the bed to the hydraulic mean depth R. This relationship, as predicted by turbulence
theory and borne out by the best experiments, is logarithmic, but an approximation
t This direction is appropriate because the force due to hydrostatic pressure gradient has
zero component along it, and so need not be considered.
 Kinematic waves. I 289
to it, reasonably accurate in typical conditions of river flow, is Manning's relation
foc R-i. For constant S and L this gives

vocki, c = v. (13)

Again, the wetted perimeter L increases significantly with k for many shapes of
river cross-section. What may seem a fairly extreme case is a triangular cross-
section (apex downwards), for which Lcc ki. This reduces c from 3-v(equation (12))
to Av iff is a constant, and from 5v (equation (13)) to 4v if fec R-i. Even greater
reductions are possible however; thus, in many reaches of the Mississippi and its
tributaries, even the ratio of width to depth increases with depth, due to the erosion
of narrow channels with convex sides in the river bed; in such extreme cases, c may
exceed v hardly at all.
The French and German writers recognized only the mechanisms cited above,
and regarded S as independent of k, equating it (as far as the work on kinematic
wave theory is concerned) to the mean slope of the bed. It is evident, :however,
that even when the flow q is uniform along the river, the slope S of the free surface
will differ from that of the bed wherever the cross-section is changing with x;
where the river is widening, for example, S must exceed the mean slope of the bed.
Under these circumstances, S might vary with k for fixed x, which would affect
the value of c.
Seddon ( 900) made a more fundamental criticism of any approach to river flow
which is based solely on the Ch6zy formula and extensions to it. Put simply, his
objection is that great rivers, unlike man-made conduits, do not have a uniformly
sloping bed, nor do they in any way approximate to this condition. First, the slope
of the bed exhibits enormous variations (including changes of sign) in the small,
that is, across the width of the river and over distances downstream comparable
with the width. Even more seriously, the large-scale configuration of the bed is
frequently very much like a series of 'pools and bars'. At relatively low water the
flow from one pool to the next is then determined not so much by a velocity-slope
relationship but by the relations governing the flow over a 'submerged weir '-that
is, over one of the bars. Large values of the slope S of the free surface are confined
to the neighbourhood of these bars. The same stretch of river might, however, be
governed by quite a different mechanism at high water. Then, for example, parts of
the 'pools' might become by far the narrowest sections of the river, and control
the flow like an orifice with vertical walls.
Another point observed by Seddon, where alluvial rivers are concerned, is that
the bed is constantly changing with time, since its material is readily handled by the
flow. Variations in depth of 10ft. about its mean at a point are common on the
Lower Mississippi; at the same time the elevation of the surface changed only by
an inch or two for the same value of the flow q. Thus the height of the free surface
above some fixed horizontal plane varies far more smoothly in time, as well as in
space, than the depth of the bottom.
Seddon used the symbol h, and the word 'stage', to denote, at each point on the
river, the height of the free surface above a certain reference plane, fixed as far as
that point is concerned. It is convenient to regard the flow q as primarily a function
290 M. J. Lighthill and G. B. Whitham
of the stage h rather than the area k of the water cross-section-both because such
a relation has more permanence, as we have just seen, and because h is more
easily measurable.t For constant x,
dk = Bdh, (14)
where B is the local breadth of the river.
Hence (3) becomes 1 /(\
c=-
B - (15)
I3~
5B\a0 x constant
and this form for the wave velocity is often far less susceptible to variation with
time, or dependence on the taking of averages, than the standard definition (3).
The two can be reconciled, however, if in ? 1 'quantity' is taken to mean 'volume
of water above the low-water mark'.
Seddon concludes from his long and interesting physical discussion, of which
just the salient points have been mentioned above, that the factors which go to
make up the relationship q = q(h, x), (16)
between flow and stage at different stations on the river, are nearly always too
complicated to make the prediction of this relation a sensible direction in which to
apply scientific method. Rather, this static relationship should be determined by
observation, when, in spite of the endless variety and complication of the processes
involved, it is nevertheless found to have some permanence and reliability. It
may then be used, with a knowledge of the breadth
B = B(h,x) (17)
as a function of stage and position, to predict from equation (15) for the wave
velocity c the still more complicated, dynamic, phenomena involved in flood
movement. Conversely, Seddon has so much confidence in this relation (15) that he
would regard the measured speeds of propagation c of particular values of the flow
q down the river, together with one of the relations (16) and (17), as a reasonable
method of obtaining the other relation! To sum up, equation (15) is the one basic
law to which a river will conform.
We have stated Seddon's views in their original, somewhat exaggerated, form
to draw attention to the danger of concentrating on velocity-slope relations when
dealing with rivers, as opposed to man-made conduits. Our own view is not that such
relations are valueless in all cases, but that a general theory should avoid leaning
heavily on them. Again, we do not claim that the kinematic wave theory gives a
really exact model of flood movement. The literature already contains methods of
improving the approximation. Thus Forchheimer (1907; 1930, p. 299) gives an
expression for the rate of subsidence of the peak of a flood wave, obtained by
applying the Ch6zy formula without neglecting the contribution of stage gradient
(- ah/ax) to the slope S of the free surface. Thomas ( 934, 1940) has devised step-
by-step methods of 'flood routing' based on equations of motion which take this
effect into account together with the (smaller) effect of inertia. Lin (I947) treats
the same equations by the numerical method of characteristics. The characteristics
t He suggests that the reference height h = 0 may be taken as that corresponding to a
particular constant flow q0, the lowest observed on the river; thus q = qo for all x when h = 0.
 Kinematic waves. I 291

here are the paths of the dynamic waves associated with the problem, namely,
long gravity waves.
In ??4 and 5 we give a new procedure for predicting flood movement, which is
bound up more with the kinematic wave as a first approximation, and less with
velocity-slope relations than the methods cited. First, however, in ? 3, we have
thought it desirable to give a mathematical treatment of the 'competition'
between kinematic and dynamic waves in river flow, in order to show how com-
pletely the dynamic waves are subordinated in the case of greatest interest, that is,
when the speed of the river is well subcritical. This demonstrates the unsuitability
of the characteristics of the dynamic wave system as a basis for computation. In
? 3 we show also how the situation is different in supercritical streams, in which the
kinematic and dynamic waves can play equally important parts. The 'roll waves'
observed in mountain streams, as analyzed by Dressler (I949), are a case of this.
Readers interested only in procedures for flood prediction are advised to omit ? 3
at a first reading.
The process by which kinematic waves may steepen into 'shock waves', with a
considerable change in flow occurring in a relatively short distance, has not been
very clearly expressed in the flood-wave literature. However, the possibility of
such a wave progressing down the river, with different, uniform, flows upstream
and downstream of it, has been envisaged as a model of a flood, and its difference
from a bore (the 'dynamic' analogue) clearly seen. Such a wave has been called a
' monoclinal flood wave ', or
'steady profile'. The formula (7) for its velocity is given
by Boussinesq (I877, p. 479). Calculations of the shape of the profile, from the full
friction-slope-inertia equations of motion, have been made by Thomas (I937).t
(See also ? 3 below, and Dressler (1 949), who uses them in Iis theory of roll waves.)
The length of the monoclinal flood wave (or 'shock wave thickness') is found to be
of the order of magnitude h/S, which is the distance downstream in which the river
elevation falls by an amount equal to its depth.
However, it will be seen from the discussion in ? 1, and in particular from figure 3,
that while a kinematic shock wave may play a very important part in the forward
regions of a flood wave, it does not constitute the whole wave. In particular, it
cannot correctly be regarded as remaining uniform in strength, or as having uniform
conditions both upstream and downstream of it. In fact, its growth and decay, due
to interaction with continuous waves on both sides, are an essential part of the
flood-wave phenomenon. The new way of using the kinematic shock wave, which
figure 3 illustrates, is correspondingly an essential part of the method of predicting
flood movement described below in ??4 and 5.

3. COMPETITION BETWEEN KINEMATIC AND DYNAMIC WAVES

The object of this section is to bring out the mathematical relations between
kinematic and dynamic waves, and to demonstrate their relative importance under
various flow conditions in which both are present. For this purpose it is sufficient

t The authors have been unable to consult Thomas's papers, which are very inaccessible.
Accordingly, no reference to their details can be made in ? 3 where the solutions are discussed.
292 M. J. Lighthill and G. B. Whitham
to consider one only of the many possible mechanisms governing the propagation
of kinematic waves which were described in ? 2. In fact, we choose the most straight-
forward of these, namely, the balance between slope and friction, as expressed in
equation (11). However, in order that dynamic waves can be present, we can no
longer neglect the inertia of the fluid, or the dependence of the slope S of the free
surface on the gradient along the river of the stage h. Accordingly the difference of
the gravitational and frictional forces per unit mass of fluid (from ? 2, this difference
is (pgSk-pv2fL)/pk) is set equal to the acceleration of the fluid, to give

vt+vvx = g - , (18)

in place of (11). If the stage h is introduced, we have S = S--hx, where So is the

value of S for uniform stage; hence, since C and R are functions of h, (18) provides
a differential relation between h and v. A second equation for these two quantities
is provided by the continuity equation (1), since q and k can be expressed in terms
of h and v.
There seems little doubt that the general characteristics of the competition
between kinematic and dynamic waves will be reproduced clearly in any formula-
tion of the problem which still retains all the essential features. (Practical methods
of prediction are postponed to ?4.) Accordingly, in this section we make the
following additional simplifications:
(i) It is assumed that by taking the reference surface for the stage as the average
position of the river bed, the hydraulic mean depth R may be approximated by
the stage h.
(ii) The slope So of the reference surface is taken to be constant.
(iii) The Chezy resistance law is used, i.e. C is constant.
(iv) In the undisturbed flow, we suppose that h and v take constant values,
h0 and v0, say.
Then, since q = kv and k = Bh where the breadth B is constant, the equation of
continuity is h + vh+hvx = 0. (19)

Equation (18) becomes v + vv = gS - hx - h (20)

and in the undisturbed flow the values of stage and velocity are related by
v0 = CJ(h0o). (21)
Equations (19) and (20) are often assumed in the literature when theoretical aspects
of river flow are being considered. They follow immediately when the river is
idealized as a uniform rectangular channel with slope So and sufficiently wide for
the hydraulic mean depth to be approximated by the depth h, but the above assump-
tions are rather less severe.
Kinematic waves are obtained by neglecting the derivative terms in (20).
Then voc hi, and from (19), ht+ vhx = 0, (22)

showing that h and v remain constant for waves travelling downstream with velocity
|v. On the other hand, without the terms gSo and v2/C2h,(19) and (20) are the equa-
 Kinematic waves. I 293
tions of the usual theory of long gravity waves. In that case it is well known (see,
for example, Stoker I948) that the solution represents systems of waves moving
upstream and downstream, both with speed /(gh) relative to the flow. In our
terminology, these waves are dynamic and the turbulent bores, which occur if
waves break, are dynamic shocks. Mathematically, the wave property is recognized
from the characteristics of the equations; since (19) and (20) are equivalent to a
second-order equation, they have two systems of characteristic curves, given by
dx/dt = v + (gh) and dx/dt = v- (gh) respectively. Moreover, inclusion of the
additional terms in (20) does not change the characteristics since they are deter-
mined by the derivative terms alone. Hence, dynamic waves always occur. The
additional friction and slope terms can only modify the amplitude of these waves.
Under the conditions appropriate for flood waves, however, they do this to such a
degree that the dynamic waves rapidly become negligible, and it is the kinematic
waves, following at a slower speed, which assume the dominant role.
The decay of the dynamic waves can be demonstrated very simply. Discontinuities
in derivatives of v and h may be taken as typical disturbances. They will propagate
upstream and downstream with the appropriate characteristic velocities (this is in
fact a defining property of characteristics) and the variation in the magnitudes of
the discontinuities can be specified immediately from (19) and (20). The results
give the rate of growth or attenuation of the disturbances carried by the dynamic
waves. The standard procedure is described in Courant & Hilbert (1937, p. 359)
and the results for the present problem have been noted by Masse (I938). Avoiding
reference to the general theory of characteristics, one may simply expand h and v
in power series near the 'wave-front'. (The wave-front is the first disturbance and
propagates with the characteristic velocity appropriate to the undisturbed flow.)
For downstream propagation, the wave-front is r = 0, where Tt-xl(vo
r +-V(gho)),
and the expansions are = Vo+ vl(t) + 2v2(t)
v =-o + rTv(t) + rvT) + +...
...,
h = ho + hl(t) + T2h2(t)+ ....
Here, the first derivatives of v and h are discontinuous, but the argument goes
through with the same result if higher-order derivatives are the first discontinuous
ones. The discontinuities in Ah/at and ahlax at the wave-front are hl(t) and
- hl(t)/(vo + V(gho)), respectively; hence the growth or decay in their magnitudes as
the wave-front travels downstream are determined by hl(t). Substituting the
expansions in (19) and (20), and replacing C from (21), it is found that
dhl 3 2 _So
dt 2ho(l+F)h~ (1 )h, (23)
where F is the Froude number vo/V(gho) (F plays a role analogous to the Mach
number in gas flow). We are interested in the case in which h1 is initially positive.
Then, if F > 2, it is clear from (23) that h1 increases without limit; since hl is pro-
portional to ah/lx, this means that the face of the wave becomes vertical and the
wave breaks into a bore. If F < 2, the sign of the right-hand side of (23) depends on
whether h, is initially greater or less than

K = (S ) (2--F)(1+F). (24)
294 M. J. Lighthill and G. B. Whitham
If h,(O) > K, h, again increases indicating bore formation; if hl(0) < K, h, tends to
zero. The last case is the relevant one for flood waves; we even assume that the flow
is subcritical (F < 1), and values of 3h/at as large as K are never found. Under these
conditions, the solution of (23) is
Kh(0) e-bt
,(t) = K-h)(1e-bt (25)

where b = gSO(l- F)/vo; thus the decay is exponential. The dynamic waves are
rapidly damped out, and bore formation is prevented.
The formation of a bore in other cases does not in itself imply that the dynamic
waves are any more important; the strength of the bore may decrease just as rapidly.
We shall indicate later that this is so when F is appreciably less than 2; but when
F > 2, an approximate theory predicts that the strength increases without bound
and the theory ceases to apply.
The criterion of F > 2, or from (21) its equivalent form S C2/g > 4, is satisfactory
since other considerations show that F = 2 will be critical. If -vo and vo+V(gho)
are taken as typical velocities for kinematic and dynamic waves, respectively,
F = 2 is the value at which these velocities become equal. If F = 2, the energy
carried by the kinematic waves goes along with the dynamic wave front; if F > 2,
kinematic waves cannot carry the energy (continuously), and the general descrip-
tion of flood waves given earlier would cease to apply. Again, Dressler (I949),
Dressler & Pohle (I953) found that SoC2/g>4 is a necessary condition for the
instability of steady flow and the formation of roll waves. This fits in admirably
with the above results.
The critical value of F for resistance laws other than the Chezy law may be
deduced most simply by equating the kinematic and dynamic wave velocities. Thus
the critical value is when /(gh) = hdv/dh where v = C /(Soh). When voc hn, for
example, this gives v/J/(gh) = l/n; for the Manning formula n = , leading to F = .
We postpone further discussion of the more extreme flows and return to the
question of the roles of kinematic and dynamic waves in flows with F < 1. This
question has now been elucidated to some extent by separate consideration of the
two types of wave; the discussion will be completed by an account of the linear
theory of small disturbances. The approximations of linearization are, in some
respects, severe, but the compensating advantage of the theory is that a complete
solution containing both kinematic and dynamic waves can be found. The main effects
of non-linearity may be sketched in afterwards. The linear equations are obtained
by substituting v = v + u, h = ho+ g in (19) and (20), and retaining only first-order
terms in u and q. Thus, using (21), the equations may be approximated by

Ut+VoUx+ 2-
x+YSgSho ) = 0, (26)

gt+ V o x+- h,= 0. (27)

A single equation for r is obtained by differentiating (26) with respect to x and

substituting for uz from (27), to give
(gho - v2) xx- 2o xt- t - 2Ah - 3AVo x = 0, (28)
 Kinematic waves. I 295
where A = gSo/vo. Making a rather surprising appearance in this subject, equation
(28) is a form of the 'telegraph equation' which occurs in the study of transmission
lines; methods for its solution are, therefore, well known.
The typical initial-value problem for flood waves is as follows. For t 0, the river
downstream of a certain observation point, x = 0 say, is undisturbed with h =hA ,
v = v0. Starting at t = 0, a flood wave passes the point x = 0, and one quantity, h
say, is observed as a function of t. Hence, the boundary conditions are that for
t = 0, r= = 0 in x> 0, and q is equal to a given functionf(t) at x = 0; the value
of y is required in x > 0 and t > 0. The problem is well set provided F < 1 and the
derivation of the solution is standard; both Riemann's method and the operational
method have been applied to its solution (Deymie 1938; Masse 1938). Here we

employ the Heaviside calculust in which the operation f dt is denoted by lIp.

Then, if Y(p, x) is the representation of 1, equation (28) becomes

(gh, - v) Yx - (2p + 3A)vo - (p2+ 2Ap) Y = 0.
The general solution is
Y = Al(p) ePl(P)x/V/(ho)+ A2(p) eP2(P)x/v(gho), (29)
where P1 and P2 are the roots of the quadratic

(l-F2)P2-(2p + 3A) FP-(p2+2Ap) = 0,

and A1, A2 are arbitrary functions of p; solving the quadratic, we have

(p + 3A)F (p2 + p(22) + A2F2)

1' A 1/\F2 (30)

For large p, the behaviour of the roots is given by

PI-p/(l-F), P2 -p/(l + F);

hence, the first term in (29) represents waves travelling upstream and is zero for
x < -t{V(gho)- vo, while the second term represents waves travelling downstream
and is zero for x > t{V(gho)+ vo}. For our problem, therefore, only the second term
can appear, and we drop the suffix 2, taking

Y(p, x) = A(p) eP(P)x/(gho), (31)

where P is (30) with the negative sign for the square root. Since y = f(t) for x = 0,
it follows immediately that A(p) must be the operational representation of f(t).
The interpretation of (31 ) as an integral involving a Bessel function may be obtained
by application of the usual methods of Heaviside calculus, but the expressions are
lengthy and the details are relegated to the Appendix; the results required here
can be deduced directly from (31).
We are interested in two questions. First, the behaviour of the solution near the
dynamic wave-front, and, secondly, the location of the main disturbance. The
t Readers more familiar with the Laplace transform may note that the Heaviside repre-
sentation of a function differs from the Laplace transform by an additional factor p.
296 M. J. Lighthill and G. B. Whitham
values of q near the wave-front correspond to the values of Y(p,x) for large p.
For large p, (- A _
Y =A(p)exp x-~F
(1)}
+
Y=(p)expvoV(gho) (gho)1+ F
and the interpretation of this is

11= expf{2 k fQt- +o(i (t-+$h- ) (32)

This
exession describes the dynai ves and their exponentialcon-(o)
decay is
This expression describes the dynamic waves and their exponential decay is con-
firmed. In fact, the linear approximation of (25) is h1 oc exp { -A(2 - F) t} which is
in exact agreement with (32) on the wave-front.
Although the solution (31) requires F < 1 (otherwise the probem is not correctly
set since the other family of characteristics t +x/{/i(gh)- vo = constant also
carries disturbances downstream), V is given near the wave-front by (32) in all cases;
hence, the prediction that for F > 2 the disturbance increases is also valid. Since
we are using a linear theory, the propagation speeds, v + (gh), of the individual
wavelets are approximated by the same value 0+ V(gho) and the possibility of
bore formation due to later wavelets overtaking the wave-front is excluded. This
failing may be corrected, however, by methods already used in analogous problems
of gas dynamics to predict and determine shocks from an improved linear theory
(Whitham 1952). It is then found that a bore will ultimately be formed provided
thatf'(t) exceeds the value K (equation (24)) for some range of t. (This is a valuable
check with the earlier result.) The strength of the bore can be evaluated, and when
F< 2, it is found that after an initial formation period, the strength decays ex-
ponentially at the rate shown by (32). When F > 2, the whole disturbance, including
the strength of the bore, is predicted to become large and this theory breaks down.
It must be remarked that the result for F < 2 has been deduced under the assump-
tion that the disturbance is small. More generally, as will be seen below, the value
of F which must be exceeded if a bore of constant strength is to be maintained,
depends on the strength; this value of F is always less than 2 and tends to 2 as the
strength approaches zero. The required conditions for bore formation and the
prediction of bore height are of more interest for the bores which are formed in
certain rivers by the rising tide. This problem has been investigated by M. Abbott,
and his results will be published in due course. In that case the bore decays under all
conditions, at least in a uniform channel, a result which we associate with the fact
that kinematic waves do not propagate upstream; a permanent increase in the
height at the mouth of the river is accommodated through a steady flow in the river
(the profile of its surface being a so-called 'backwater curve'), and the bulk of the
disturbance does not propagate upstream at all.
In order to show the main features of the solution away from the wave-front and,
in particular, to locate the main disturbance, an approximate form of the solution
for large values of t is next obtained. To be precise the solution for large t is found
in the range t < xl{v+ /(gho)} < ( - ) t, where 6 is any small positive number.
The reason for this condition will become apparent later; it excludes only the
initial observation point and the wave-front where the flow is already known.
 Kinematic waves. I 297
We consider two problems: (a) the original disturbance is a 'hump' with f(t)
returning to zero after a sufficient time, and (b) the disturbance is a 'smoothed
step' so thatf(t) tends to a constant positive value at t -> oo. Problem (b) follows from
the solution to (a), since rt satisfies exactly the same conditions in (b) as does r in (a).
Therefore, we consider first the case whenf(t) - 0 as t -> o, and we shall require that
0o

f(t) dt is convergent.
fo
The solution y may be expressed in terms of Y(p, x) by the contour integral
1 f+iioo Y(p, x) evt
(x,t) = 22iJ dp,

where I is so large that all the singularities of Y(p, x) lie to the left of the path of
integration. Introducing the expression for Y(p,x), and for convenience setting
m = x/t(gho), we have 1 +iA(
1 t+i
A(p)et(mP+P)dp

The behaviour of y for large t is now found by estimating the integral by the method
of steepest descents. The asymptotic expansion (of which we shall find the first
term) will be valid for values of t which are large compared to some quantity having
the dimensions of time and, in fact, the precise conditiont is t> 1/A. The contour
is chosen to pass through the saddle-point of the function mP(p) +p which is where
l+mP'(p)= 0; (33)
the main contribution to the integral then comes from the neighbourhood of the
saddle-point. For large t, we have, according to the standard formula for this
method, 1 A(pi) e{Po+mP(po)}t
metho, 1 A(P0) (34)
Po (34)
P((pO) tm}
V{27TIr
where p = po(m) is the solution of (33). (This formula does not apply at x = 0, or
at the wave-front where m = 1+ F, since P"(po) vanishes for this value of m.)
For fixed t, the exponential term (which dominates the expression) is maximum
for the value of m which is given by
d
-(Po + mP(po)) =0.
dm
But in view of (33) this reduces to P(po) = 0. The zero of P(p) is p = 0; hence g
attains its maximum when m =- 1/P'(O). From (30) we deduce that m = -F,
i.e. x = 3vot. Hence the position of maximum depth ultimately travels downstream with
velocity 3vo,showing that the main disturbance is carried downstream as a kinematic
wave. To find the value of the maximum depth we substitute po = 0 in (34), noting
co
that A(po)/po is now replaced by lim A(p)/p = f(t) dt. Thus we find that
p-o o

/A\g 3F fC
(max. )' -2(1 F) f(t)dt (35)

t The argument t(mP +p) may be written in dimensionless form as At{mQ(q) +q}, where
q =p/A and Q = P/A.

20 Vol. 229. A.
298 M. J. Lighthill and G. B. Whitham

For problem (b) it is the position of maximum slope which moves downstream
with speed 3v0, and the magnitude of the maximum slope falls off as 1/i/t.
In interpreting the results, it is essential to remember that the linear theory in
eludes the 'diffusion' effects but does not include the equally important non-linear
features. The latter would introduce modifications in the same way as for dynamic
waves. That is, the kinematic wavelets instead of being lumped together with the
same propagation speed should have individual speeds 3v, taking into account
the variation in v; in regions which have higher values of v the wavelets travel with
higher velocity. In particular, in problem (b) the diffusion which acts to smooth
out the step (the slope decreasing as l//t) is counteracted by the non-linear steep-
ening due to the higher values of v in the rear. The two opposing effects eventually
achieve a balance and the wave is translated down the river without change in
shape. This 'steady profile wave' is nothing but the kinematic shock separating
constant flow regimes.
In (a), the propagation speed will be greatest at the peak, producing tendencies
to steepen near the front and smooth out at the rear. Near the front, equilibrium
between diffusion and non-linear steepening will be attained as in (b), and a shock
appears at the head. The flow will be as represented in figures 2 and 3. The detailed
solution of this problem is worked out in ? 4, and it is found that the strength of the
shock decreases like IlVt for large t. Thus, for this case non-linearity distorts the
profile and concentrates the disturbance near the head; the strength remains
proportional to l//t as in the linear theory but the constant of proportionality is
different.
A necessary condition for the approximation (35) for large t was that t> 1/A.
Therefore 1/A provides an estimate of the time-scale which is required if the theory
of kinematic waves is to be applied. This may also be noted directly from (28);
for, if t > 1/A, one is led to approximate the equation as
(gho- ) x - 2A(qt+ Vox) = 0. (36)
Without the diffusion term, x, the equation has solution y =- q(x - vot), repre-
senting kinematic waves. Therefore, the full equation (36) represents waves travel-
ling with speed Iv0, but with amplitude decreasing like Il/lt (as is typical in diffusion
problems). Equation (36) also indicates that when appreciable changes in x occur
only over distances x which are large compared to gho/Avo,the diffusion may be
neglected, and the solution y = g(x-3vot) taken. Since gho/Avo= ho/So, this con-
firms the earlier remarks that appreciable diffusion is limited to relatively thin
shock waves whose thickness (as we shall also see below), is of order ho/So, and the
problem may be treated accordingly.
The steady profile solution or 'monoclinal flood wave', which the foregoing
arguments indicate as the ultimate wave-form in problem (b), is of great importance
in the subject. It may be determined exactly by assuming in (19) and (20) that
h and v are functions of a single variable cr = x- Ut; this is equivalent to describing
the wave relative to axes moving with the velocity U of the wave, in which the flow
is steady. The equation of continuity integrates to the obvious form for steady flow
relative to the moving axes
h(U-v) = Q, (37)
 Kinematic waves. I 299
where Q is a constant. The values of U and Q are determined if, for example, the
limiting values of depth at large distances ahead of and behind the wave are
specified. If these values are ho and h,, respectively, the correspondingvalues of
v are vo = CV(Soho),v- = CV(Soh1),since at large distances the flow tends to be
uniform. Then it follows from (37) that
v1hl-v0h0 -
i -hi-
0 (38)
h1 - ho ? =
VU_lhl--?
hS- 1-hho
-
hi,-
- o= -
and -
hoh hh Co
1Q hOh hi (39)
- '.
ho
The expression for U is the simplified form (appropriatefor the simplification
of this section) of (7). It reduces, for small values of (h1- ho)/ho,to 3voin accordance
with linear theory. Moreaccurately (and this will be referredto later), U is given by
1(tvl + -ov) + 0(V1- Vo)2, (40)
i.e. the mean value of the propagation speeds of the kinematic wavelets on each
side of the wave.
The shape of the profileis given by the momentum equation (20) as the solution of
dv dh v2
o
(v-U) =(= dg( - 2'

which, from (37), may be written as

dh (Q- Uh)C2 - Soh3
do'- - *'
h3-Q1g

Substituting for the values of Q and U in terms of hoand hl, it becomes

dh _ (h-ho)
(h-l-}h) (h-H)
drr
do-S0 h3 h3
hah- (42)

where hohl
H = (h+h_) <ho, h3 = Q2/g; (43)

here hi is the critical depth for the flow relative to the moving system of reference.
With h1> h > h0 the sign of dh/dcrdepends on the sign of h-hc; for flood waves,
however, we may take h > h, and dh/drois negative, giving the monoclinal wave.
To integrate (42) we write
S0do- 1 aao A
hodh ho h-ho hl-h h-H'
where
_ Ah- h-h3-A _ h-H3
ao - h
ao ho(h,- ho)(ho- H)' ho(h,- o - H)' o(o - H) (h - H)'
(45)
whence eSomho =
(hll -h)mae
) ehlho. (46)
(h - ho)ao(h - H)A
There is, of course, no definite shock thickness, but it is assumed (as usual in such
cases) that for practical purposes, only the variation of h from h0+e(hl -ho) to
20-2
300 M. J. Lighthill and G. B. Whitham

h, - e(h, - ho)is significant (and can be measured), where e is a suitably small positive
number. The distance over which this increase in depth takes place is taken as the
measure of shock thickness. This distance is easily calculated from (46); assuming
that e is very small, it may be approximated as
ho 1
(ao +a )ln 1. (47)

Fortunately, this definition of the thickness is not sensitive to changes in e, and

e = 0-05 is adequate. As pointed out earlier the thickness is of the order ho/So. It
may be noted that if the depth is well in excess of the critical value, he may be
neglected in (45), and a0 + a becomes a function of the strength (h - ho)/ho alone;
a graph of this function is given in figure 4.
The steady profile solution for the more general case in which C and R are func-
tions of h may be obtained similarly, but numerical integration is required. Equation
(37) expressing v in terms of h is unchanged but
= C(hl){SoR(h1))} and v0-C(h o)SoR(ho)=)
lead to modifications in (39); the profile is given by
dh _ S (hU-Q)2 h(C2RSo)- 1
do- 0 h3 - 2/48)

ao+al

10-

8-

6-

4 1I I I I1 I
40 2 3 4 h-h
ho
FIGURE4

Keeping h0and hAfixed, we consider the form of (46) as F increases to and exceeds
2. Equally well, in view of the relation F2 = S C2/g, we may interpret the variation
in F as an increase in the value of the slope S0, or a decrease in the friction coefficientf.
In fact, it is convenient to study (42) and (46) through their dependence on he which
is proportional to Fl. When he < ho, the expected monoclinal wave is obtained. But
when h. = ho, corresponding to F = F1say, the profile reaches the form shown in
figure 5, with a finite slope at h = ho. In a sense the wave is on the point of breaking;
when F = F1 is exceeded, (46) describes a curve of the form sketched in figure 6
and physical reality is lost. The critical value F1is given from (39) and (43) by

hc= h0hhlh- hg 0

that is, F -ho( ) (49)

I il,
 Kinematic waves. I 301
We observe that Fl< 2 (since h > ho), and F1= 2 is reached only for hI = ho. Two
results have great significancein view of previous remarks. First, when F = E1,the
velocity of the wave is, from (38),
hM-ho

I
hih'W1 (gho);
+h)h}i
={(h V
(ith
hence, substituting from (49), we have

-F
t= +1 h)
()= vo+(gho). (50)

Secondly, the maximum value of the slope, which is attained at h = o is

(dh\ so(h i-^ho)(3 . ,i 1(h
.i.S'(h 1
h,h1, 3hg -S?o h +
After some manipulation this reduces, using (49), to

and the correspondingvalue of ah/atis

-vUdh _ So(1 ) J(gho).

do- 3 F- FI) (2- (51)

This is identical with the K of (24). Thus the steady profile wave (and this is the
kinematic shock) has exactly the same speed as the dynamic wave-front (equation
(50)), and its slope is exactly equal to the criticalvalue which governs bore formation
by dynamic waves. We may say then that at this value of F the two types of wave
coalesce to move downstreamtogether, and bore formation is imminent.
A remedy for the solution when F exceeds FTis now clear. A bore must be fitted at
the front of the wave increasing the depth discontinuously from h0 to some value
h* on the profile (figure 6), to give a physically acceptable solution as shown in
figure 7.
The equations expressing conservation of mass and momentum across the dis-
continuity are (Lamb 1932, p. 280)
Q2 g(ho + h*) hoh*, (52)
Q = (U-vo)ho =(U-v*)h*, (53)
where v* is the particle velocity immediately behind the discontinuity. Equation
(53) is alreadysatisfied since h = h* is a point on the steady profile,and (52)remains
to determine h*. Since Q2/g= h3,the positive root of (53) is
h*_ (V{ + 8(hIho)3}-(-4) -*
h 2 h,- ~~~~2 (54)
o
Finally, it must be verifiedthat this value for h* exceeds the value h = h, the depth
at which dh/dc becomes infinite in the solution representedby figure 6. But this is
a fundamental property of bores; the flow ahead is subcritical and the flow behind
302 M. J. Lighthill and G. B. Whitham
is supercritical. Hence h* > h, > h0. (The result may also be verified directly from
(54).)
We see then that, as one expects, a bore will be formed whatever the Froude
number of the initial flow, if the increase of discharge is great enough. As h, increases
further, the jump in height of the bore in this 'combined kinematic-dynamic shock'
increases until at some value h* becomes equal to h, and the bore separates regimes
of uniform flow. Before this value is reached, however, F exceeds 2, the flow
becomes unstable and breaks down into a series of roll waves. Solutions repre-
senting roll waves have been obtained by Dressler (I 949) as a series of bores separated
by steady profile waves which satisfy (41); the solution is periodic and the whole
configuration moves downstream at a steady speed. For the details, which do not
directly concern us in the present work, the reader is referred to Dressler's paper.

h=hh

h=hl

= h=h h-ho

FIGURE 5 FIGURE 6

h=hh

h=h* .
l h=ho

FIGURE7

4. THE KINEMATICWAVE THEORYOF FLOODMOVEMENT

In this section we set out the basic theory of floods treated as kinematic waves,
with shock formation and prediction included as an essential part of the theory.
Afterwards, improvements such as some allowance for diffusion effects in the kine-
matic waves may be added (see ? 5). But the description of flood movement given
in this section is the appropriate first approximation.
In general, the q-k relation, from which kinematic waves are deduced, will
depend upon the position x. The theory for the completely general relation will be
described in this section, but first as an introductory example we consider a special
form of the relation which has sound practical value and leads to mathematical
simplification. This arises when the dependence of k on q and x is separable; that is,
k = a(x) f(q), (55)
say. An example covered by this relation is the uniform river for which k is a func-
tion of q alone, and thus a(x) = 1; the inclusion of the additional factor a(x) does not
 Kinematic waves. I 303
introduce any essential complication into the analysis, yet it greatly increases the
practical value of the example. Thus, Seddon in his thorough investigation of flow
conditions in the Missouri found that the variations of stage with discharge at
different stations were related linearly. That is, if a station located at x0 is taken as
reference, the stage at position x is given by
h(x, q) = a(x) h(x0, q). (56)
(It is unnecessary to add a further function of x to the right-hand side of (56), since
by definition zero stage corresponds to the same value of q at x and x0.) Hence the
stage-discharge relation is separable. If, further, the breadth B is a function of x
multiplied by a power of h (as, for example, in the cases of rectangular and triangular
cross-sections), the concentration-discharge relation will be separable. Hence,
observational results indicate some special interest in (55). From a theoretical point
of view, it is a consequence of the simple Chezy or Manning formulae, when the
slope S is a function of x alone and R is replaced by h. Then v = C{hS(x))}, so that
with q = vhB, we have BS() = q2
Clh3B2S(x) = q2. (57)
Thus, if C is a function of x alone (as in the Chezy formula) or the product of a func-
tion x with a power of h (as in the Manning formula) the relation between h and q is
separable, provided B is again of the form b(x) hn.
The equation of the characteristics, which represent in the (x, t) plane the paths
of wavelets, assumes a simple form when the q-k relation is (55). The velocity of
propagation is (aqlak)xconstant= {a(x) f(q))-l; hence the characteristics satisfy

d) = a(x)f'(q). (58)
q constant

It is convenient to label a characteristic by the value of t when the wavelet passes

a fixed observation point. If we choose this observation point as x = 0, and use T
to denote the times at which the wavelets pass it, (58) integrates to
t = A(x)f'(q) + T, (59)
where A (x) = a(x) dx. Since q remains constant on a wavelet, q retains the value
taken at x = 0 at the time T; therefore, we may set q = q(T). If we suppose that this
function q(T) is known from observations made at x = 0, the flow is now deter-
mined-at least until shocks appear, with their attendant modifications of the flow.
For, at any position x at the time t, the corresponding value of T with the appro-
priate value of q is determined so that (59) is satisfied. In the special case of the
uniform river discussed in ? 3, a(x) may be taken as unity and A(x) = x; hence the
characteristics are straight lines. More generally the characteristics are straight
in the (t, A(x)) plane. The (t, A(x)) plane for a typical problem will be as in figures 2
and 3 with x replaced by A(x).
If the disturbance observed at x = 0 were a decrease in discharge, then, since
f'(q) increases as q decreases,t successive wavelets leaving x = 0 would, according
to (58), have smaller velocities and the flow would spread out smoothly with no

t Greater discharge always gives greater propagation speed (cf. ? 2).

304 M. J. Lighthill and G. B. Whitham

tendency to shock formation. In the (x, t) plane the characteristics diverge forming
an' expansion wave'. In this case, the above solution is adequate by itself to describe
the flow. However, whenever q(T) increases, the velocities of successive wavelets
increase so that the earlier ones are ultimately overtaken by later ones, resulting in
shock formation. This is represented by the convergence and eventual overlapping
of characteristics in the (x, t) plane. When this occurs, the solution given by (59)
does not give a unique value for q (since T is not unique when there is more than one
characteristic through a given point (x, t)) and modifications must be introduced;
a shock wave, changing the values of q and k discontinuously, must be fitted in. The
shock is in fact a relatively narrow region in which, due to the relatively rapid
change of q, the assumed q - k relation becomes invalid. But, in the first instance,
it may be treated as a discontinuous wave producing the appropriate abrupt changes
in q and k; the more detailed behaviour of q and k in the shock region is represented
by the steady profile solution of ? 3 and can be included afterwards.
At each point of the shock, characteristics of the flow ahead and behind intersect
as shown in figure 3. All the characteristics and the values of q on them are known;
it only remains to determine where they are cut off and separated by the shock.
It is obvious (graphically) that this will be achieved, if we can find how the pairs
of characteristics which meet at the shock are related; in particular, a determination
of the relation between the labels T1 and T2 of two such characteristics will suffice.
This is now obtained.
rT2
With T2> Tl, the total flow across x = 0 between the times T1and T2is q(T) dT.
JT,
If t is the time at which the wavelets are at the same point x on the shock, then this
quantity of fluid must flow out of the region between the wavelets by time t. Fluid
passes a wavelet travelling with speed c, at a rate of q - kc; hence the total amount
t
passing a wavelet is (q -kc) dt. Since cdt = dx, we may write this in the alter-
rx T
native form (q/c-k)dx; in view of (58) it is evaluated as {qf'(q)-f(q)}A(x).
Thus the required expression of continuity becomes

Tq(T) dT =-[qf'(q)-f(q)]q A(x), (60)

- JT
where q1 = q(T1)and q2 = q(T2). The characteristic equations for the wavelets give
t = A(x)f'(ql)+ T1, (61)
t = A(x)f'(q2)+T2; (62)

hence A(x) _' -T l (63)

Equations (60) and (63) provide the required relation between T1and T2. It is often
convenient to introduce the excess of the discharge over its undisturbed value q0,
rT2
and write the left-hand side of (60) as (q - qo)dT + qo(T2- T); from (63) this gives
n~~TT2
= -[(q- q) f'(q)-f (q)]2 A(x), (64)
(q-qo)dT
in (60).
placeof (60).
in place
 Kinematic waves. I 305
These shock equations are most easily dealt with when the flow on one side is
the uniform flow q = qo. This will ultimately be the case, for example, at the head
of a 'hump' (as shown in figure 3), since even if the shock forms in the interior it
eats its way through to head the flood. Then, q1 takes the constant value q0, and if
Tf is the time of the first arrival of the disturbance at x = 0 (i.e. the label of the first
characteristic of the disturbed flow), (64) becomes

f (q-qo) dT = -(q2-qo) f'(q2) -f(q2)+f(qo)} A(x). (65)

J Tf
This relation between q(T2)and x, coupled with equation (62) for t, gives the equation
of the shock path with T2 as parameter. Equation (63) which would determine the
corresponding value of T1is no longer required.
For the problem of a 'hump', wavelets are continually fed into the shock as time
goes on; if Tzis the value of T for the last characteristic of the 'hump', T2-> TJand
q2->q0as t-* cc. We may approximate (62) and (65) to give this asymptotic behaviour.
For, when q2- q0 is small,

{(q2-qo) f'(q2) - (f(q2) -f(q))} = tf (qo)(q2- qo)2+ O(q2-qo).

Hence (65) is approximately

(q-qo) dT =-f"(qo) (q2-qo)2 A(x) (66)

and we have the valuable result: the increase in discharge at the shock is proportional
to A-i(x). Equation (62) becomes
t = A(x) f'(qo) + (q2-qo) f"(qo) A(x) + O(q2-qO)2 A(x) + T1;
therefore the shock path is given by

t = A(x)f'(q)-A(x) -2f(qo) q(T) dT + 0(1).

In the special case of the uniform river for which A (x) = x, this is a parabola in the
(x, t) plane; in general it is a parabola in the (A (x), t) plane. We note that at the shock
A (x) c t to a first approximation, so that q2- q0 oc t- and the width of the disturbed
region increases proportional to ti. As remarked in ? 3, the rate of decay like t-i is
also obtained by linear theory.
When a disturbed flow on each side of the shock must be considered, the implicit
relation between T1 and T2cannot be avoided. Eliminating A(x) from (63) and (64)
it may be written T2

J! = [(q - o)f'(q)-f(q)]q q
\Ylt. (67)
^T-T^ -
f'(q2)-f'(ql)
In general, determination of T2as a function of T1from this relation may be rather
laborious. There is, however, an approximate form which offers great advantage in
following the progress of the shock. The approximation applies rigorously when
(q2--q)/ql is small, i.e. when the shock is weak. This will be true in the earlier stages
after the shock is formed, and again for the ultimate decay of the shock; some
306 M. J. Lighthill and G. B. Whitham
correction may be needed between these two extremes (although the approximate
results would still be of qualitative value). On the other hand, for such problems as
the 'hump' (which is a typical one for floods), the shock soon moves to the head of
the flood. Thus, for most of the motion, the (exact) description already given can
be used; to supplement this, details of the shock near its formation may be sufficient.
In order to deduce the approximate form, we first remark that the right-hand side
of (67) may be written
(q- qo)df (q)
(68)
f'(q2)-f(q) (68)
(The relation has a nice symmetry now, since it states that the mean values of q- q
with respect to T and with respect to f'(q) are equal; this fact does not appear to
help in the practical solution, however!) Then, if (q2-q1)/q1 is small, (68) is
{((q -qo) + (q2- qo)} correct to the first order in (q2- ql)/ql, and we have

JT,

On a graph of the function q(T) - q, the left-hand side of (69) is the area under the
curve between the ordinates T = T1 and T = T2; the right-hand side is the area of
the trapezium under the segment joining the points (Ti, q,) and (T1,q2)of the curve.
Hence, the areas of the lobes (between the segment and the curve) on either side of the

q-qo
A'

E D C T

FIGURE 8

segment must be equal. Moreover, it should be noticed that in (63), f'(q2) -f'(ql)
will be approximately proportional to q2- q1 so that the slope of a segment is pro-
portional to 1/A(x). Thus, the slope of the segment approximately determines the
corresponding position of the shock. But it must be emphasized that this second
approximation is only used as a rough guide in recognizing immediately the change
in position of the shock from the change in slope of the segments; in any actual
calculation, the value of x would be determined accurately from (63).
The progress of a shock after its formation can now be interpreted graphically by
means of the segments which cut off lobes of equal area. We first describe the
simple (yet most important) problem of a humped disturbance (figure 8). The
 Kinematic waves. I 307

position at which the shock first appears will correspond to the limiting case when
the segment becomes the tangent, AA', at the point of inflexion of the curve. The
values of q1and q2becomes equal, and since the slope of the segment is a maximum,
the value of A (x), and hence x, will be a minimum. As x increases, the corresponding
segment decreases in slope. At first, the jump in q (the difference in the values at
the end points of the segment) increases (BB', CC'), but it ultimately decreases
(DD', EE') as x--oo, the segments tending to the axis. After CC' is reached, one
end of the segment is on the axis, i.e. q1 = q0, and the flow on the upstream side of
the shock is uniform. Then the exact determination of the shock (63) and (65) can
be used instead of the present approximate method.
The point of shock formation is represented by the tangent AA'; hence, the
shock forms on the characteristic specified by T = Ti, where T is the solution of
q"(Ti) = 0 (with q'(Ti) > 0). The distance of the shock formation from the observation
point x = 0 is found from (63) to be given by

A(x) =-{ff"(q(T)) q(T,)}-.

The subsequent motion and increase of strength of the shock very near its point of
formation may be found analytically by approximating the relation between T1
and T2for T - T1and T2- T small. But eventually the full solution of (69) must be
found. The above graphical interpretation is then valuable. After a little practice,
the positions of the segments can be guessed fairly accurately and the details of
the shock determined quite easily. In any case, such a guess provides a first approxi-
mation which can be checked by numerical evaluation of the expressions in (69).
If tables of the functions q(T)-qo and (q-qo) dT have been made, this initial
estimate can be improved with little labour. When (69) is considered to be an in-
sufficient approximation to (67), we suggest that it is simplest to make a first
estimate by the 'segment method', and then adjust the values to satisfy (67) more
accurately.
All graphs of the function q(T) - q may be treated by the 'equal-area segments'
in the way described above. There is one occurrence, however, which needs special
mention, and which we illustrate by an example. Suppose there are two humps,
corresponding to successive surges of the flood, as shown in figure 9. Shocks will be
formed corresponding to each of the points of inflexion where the tangents have
positive slopes; the tangents are shown as AA', a'a. The propagation of each shock
will proceed, represented by segments such as BB' and b'b, respectively. But there
may eventually be segments CC' and c'c which have the same slope. If this occurs,
the equality of slopes indicates that the corresponding x is the same for each shock,
and since the characteristic corresponding to the common point c', C' meets both
shocks, the time is also the same. Thus the two shocks coalesce. When this state of
affairs is reached, all the points on the q(T) - curve between the two points of
inflexion have been used for segments of one of the shocks. This means that all the
wavelets represented by these points have been fed into one or other of the shocks;
the remaining wavelets are represented by points on the curve either to the right of
c or to the left of C. After the shocks combine a single shock continues, and it is
308 M. J. Lighthill and G. B. Whitham
represented by segments such as Dd; this part of the motion is described exactly
as for the single hump.
Finally, in connexion with the segment method, it may be remarked that the
approximate form (69) corresponds to the assumption that the shock velocity is
the mean of the characteristic velocities of the flow on each side of it. (This result
was proved in ? 3 and is displayed as equation (40).) In the (x, t) plane this means
that the shock line bisects the angle between characteristics which meet on it. This
property is useful in picturing the (x, t) plane correctly and may even be used for
a rapid (but rough) determination of the shock.

q-qo
A' a

~D C T
FIGURE 9

General k-q relation

For the completely general relation k =k(q,x), steps analogous to the above
may be written down, but the practical solution of the various equations is more
difficult. As before, q may be prescribed as a function of T alone, since it remains
constant on a characteristic. The propagation speed is c =(aq/lk)x and the
characteristics are
t= n d x+T,
J(
0T x constant

or
aV(q,x) + T,
t= (70)
aq
x
where V(q, x) = k(q, x) dx is the volume of water in the river between 0 and x.
Evaluation of V, as a function of x for a range of values of q, may be lengthy, but
at least it may be computed once and for all for any particular river; only the
function q(T) depends upon the particular flood. When q(T) has been observed at
x = 0, (70) provides the solution until shocks appear.
The determination of a shock again depends upon deducing the connexion
between characteristics T1 and T2 which meet on it. Now, the total amount of
fluid which passes a wavelet between 0 and x is

( k)dx =q- q-V.

J c /
 Kinematic waves. I 309

Therefore, corresponding to (64) we have

q(T) dT = - [q . (71)

Further relations between t, x, T1and T2are provided by the characteristic equations,

t= V-(ql,x)+TI, (72)
t-Vq(q2,x)+T2; (73)
in particular, x) - Vq(q2x).
T2- T1 = Vq(qq, (74)
The latter equation may be used to write (71) in either of the alternative forms:
JT

(q-qo) dT --[(q-qo) Vq- V]q2 (75)

T2 Tq2

J(q-o) dT J(q- qo)dV(q, x)

(76)
T2 - T, Vq(q2,x)- Vq(ql, x)

In general, then, the shock determination depends upon the solution of two implicit
equations (74) and (75) (say) for three quantities x, T1, T2. Without simplification
their practical solution is not feasible. However, we may again consider the
important case of uniform flow, q = q0, ahead of the shock. Then Tf replaces T,,
and qo replaces q1 in (75) to give the equation corresponding to (65). This provides
an implicit relation between T2 and x which may be solved numerically. Then the
discharge behind the shock q(T2)can be found as a function of x, and the value t can
be obtained from (73).
Apart from this case, we must again approximate (76) by (69), and describe the
shock propagation by the segments cutting off lobes of equal area. This proceeds
exactly as in the earlier discussion of the separable k-q relation and further
comment is unnecessary.
When the shock line has been obtained together with the values of q on each side
of it, the final step is to replace the discontinuous shock by the appropriate mono-
clinal flood wave. Of course, since the latter has a steady profile, it applies strictly
only to uniform flow conditions. Hence, mean values of the slope and the cross-
section of the river must be taken in the neighbourhood of the shock, and these
values used in applying the steady profile solution described in ? 3. From the known
values of q1 and q2, the corresponding values hi and h2 of the depth may be found;
then, for example, if the Chezy law is assumed, the required solution is given by (46).
For other friction laws and general dependence of the hydraulic radius on depth,
(48) must be integrated. This monoclinal flood wave is then centred on the shock
line to provide details of the transition.

Tributaries and run-off

An important modification of the flood will occur when the waves pass a junction
with a tributary. The discharge from the tributary will influence the flow in the main
river upstream of the junction and produce modifications in the kinematic waves.
310 M. J. Lighthill and G. B. Whitham
But we suggest that a good approximation to the effect on the flood will be obtained
if the upstream influence is ignored, and as each kinematic wave passes the junction
the discharge of the tributary is added to the value of q carried by the wave; the
increase in q will produce a corresponding jump in the propagation speed. Although
this method will give rise to a discontinuous increase in the stage, it is expected to
be satisfactory provided that the tributary is appreciably smaller than the main
stream. The discontinuity in stage would in fact be smoothed out by an adjustment
in the flow upstream of the junction, and it is expected that the variation of stage
would be rather like that given by the 'backwater curves' of steady flow. For a
junction of streams of comparable size it may be necessary to take some explicit
account of this.
The effect of run-off from the surrounding terrain will be similar except that the
increase of the discharge will take place continuously. If the run-off has a volume
flux /(x, t) per unit length of the river, the equation of continuity must include ut
as a source term, and we have k aq
3k (x, t).

Introducing k = k(q, x), the kinematic waves will now be described by

l q aq
c at +~9x = ,(X, t).
The rate of change of q along a characteristic is then It per unit distance, and in
general the solution would be found by numerical integration along the charac-
teristics. However, it is unlikely that the functional form of / is known precisely,
and the main interest is in deducing the effect of an estimated constant run-off.
When It is constant or a function of x alone, the equation for q may be solved
explicitly (assuming that c is already known). For it may be written

at- )+) ca= 0;I

c J
hence, q- j/dx remains constant on each characteristic. If q = q(T) at x = 0,

then q = q(T) + /dx on the characteristic labelled by T. The characteristic itself

is then given by T dx
t-= + .---
Jo c(q,x)'
The case of a tributary is obtained when the run-off is concentrated at a point so
that f/ dx is a step function, and the curvature of the characteristics (introduced

by the run-off) is concentrated at a point.

5. DIFFUSION OF KINEMATIC WAVES

In the theory described in ?4, diffusion is confined entirely to the interior of

shock waves, where its effect is crucial in arresting the steepening of the kinematic
wave profile. Outside these shock regions, diffusion is certainly small, but it may
 Kinematic waves. I 311
be valuable to include its effect as a second approximation. Mathematically,
diffusion corresponds to the inclusion in (4) of an additional term proportional to
a second derivative of q. This will arise if the flow-concentration relation involves,
in addition to q, k and x, some dependence on a derivative of q or k. The existence of
such a dependence is demonstrated by the well-known observational result that the
stage-discharge rating curves for increasing and decreasing stage differ slightly.
Thus, for example, we may assume that for each x, q is a function of k and akl/t;
alternatively, since aklat = - aq/ax, k would then be a function of q and aq/3x. If
we take the latter form and substitute for k in the equation of continuity, we have
akaq ak a2q 3q 0.
at +t+SqD8xSt+x=
axat ax (77)
aq aq
If, further, the coefficients of the derivatives of q in (77) are approximated as
functions of q and x alone (by setting qx = 0 therein, for instance), the equation
may be written aq aq a2q
at^+cax+^-=
xt 0, (78)
(78)
wherec = aqlakis the kinematic wave velocity as before,and v = c aklqx. Equation
(78) is typical of the equations representing the diffusion of kinematic waves. Other
derivations may lead to one of the other second derivatives of q in place of a2q/axat
(in fact, the interchange can be carried out directly using the first approximation
q/lat- -c(aqlax), but the consequences will be the same.

q /
/
A/B

/ /
! /

FIGTRE 10

In practice the most satisfactory course may be to estimate a suitable form for
the coefficient v in (78) from the observational data for previous floods. For a given
stage, the discharge is greater when the stage is rising than when it is falling, and
the graph of q against k as a hump passes the observation point, is of the form shown
in figure 10. To determine v we must estimate ak/aqx, which is the same as
- (ak/kt)qconstant. If the two points on the curve corresponding to one value of q
are denoted by A and B (see figure 10) then

(3ko\ kB-kA
a constant (t)B - (kt)a'
312 M. J. Lighthill and G. B. Whitham
where subscripts A and B denote values at the points A and B. From the graph of
k against time, the values of kt can be found and therefore the value of v deduced.
The value of v could be obtained for the different values of q to give v as a function
of q. The dependence of v on q obtained in this way may not be too significant how-
ever, and it is probably sufficient to take a suitable average value for all q.
As in the case of the velocity c, values for v may also be predicted on the basis of
simple theories. Numerous approximate formulae have been suggested (see the
article by Gilcrest in Rouse 1950) in attempts to described the deviations of the
rating curves from the curve corresponding to steady flow conditions. The simplest
of these is obtained by including the 'wedge storage term' in (11), i.e. S is taken to
be So - h rather than So, where So is the surface slope for uniform stage (the slope
of the bottom for a uniform channel).
Then we may write q = q*(h, x)}(1-hx/So), (79)
where q*(h, x) is the discharge for steady-flow conditions. This formula is of special
importance since it is the starting point of the Forchheimer method for predicting
the subsidence of a flood wave. Forchheimer considers the simplified problem of ? 3,
in which the variations of So, B and q* with respect to x are neglected. Then, sub-
stituting (79) in the equation of continuity which may be written
ah aq
B -+ = o,
at ax

we
have
we have ah
B1 dq*
-+t+ T3 ( - hXISO)h
dh (1
* 2h == 0. (80)
(80)
a9x 2BSo(1-hxSo)aX2
At the crest of the flood ahlax = 0; therefore, the rate at which the height of the
crest decreases is given by dh q* a2h
dt 2BSoax (81)
From observations of the flood profile at one time, a2h/ax2can be determined, and
the change in the height of the crest at a time At later is predicted as
* a2h
2BS At;x2
in this time, the crest will have reached a distance cat downstream. In practice,
it is more convenient to obtain the values of a2q/aX2from observational data and
-(aq*lah) 2h/ax2= Bca2h/ax2to deduce a2h/ax2.
use a2q/ax2
If hx/So is neglected in the coefficients of equation (80), we have
a q* a2h
r.--(82) ^a^
a^- 2BSoax2
a +cax
at (82)
as an equationt representing the diffusion of kinematic waves, with q*/2BSO as
a coefficient of diffusivity. Alternatively, an equation of the form (78) may be
deduced if (79) is modified by replacing hx by -htlc. Then

q = q*/(1 + ht/cSo); (83)

t It may be noted that the linearized form of (82) is the same as (69) if gh- v2 is approxi-
mated by gho in the latter.
 Kinematic waves. I 313
this relation is known as the 'Jones formula'. Rouse (I950) states that (83) is a
better approximation to reality than (79). Since ht = kt/B, (83) gives q as a function
of x, k and kt. We may then write it in the alternative form (using qx = -kt):

k= (1-qx/cBS0) ), (84)
where k = f(q,x) is the k-q relation for steady flow conditions. Hence, finally,
the coefficient v in (78) is q/2BSoc.
We now assume that (78) can be formulated either from observational data or
from the above theoretical discussions, and turn to a consideration of its con-
sequences in the theory of ? 4. Previously, q was constant on each characteristic,
but with the diffusion term we have

dq aq aq _ 2q
att+V
dt dtC= =^z
Ca~x a= - t'
xt(at) -(85)

so that q varies slightly, the rate of change depending on the values of q on neigh-
bouring characteristics. For example (as in the Forchheimer method), where q is
a local maximum so that 9q/lx2 < 0, q will decrease along the characteristic, since
a2qlaxat- - a(caqlax)/3x> 0. Conversely, when q is a local minimum it will have a
tendency to increase. In this way diffusion smooths out the values of q, and this
effect will be superimposed on the solution of ? 4.
The introduction of a variation in q along the characteristics will also change the
position of the characteristics in the (x, t) plane, since the propagation speed c
depends on q. The changes in the values of c will be small; nevertheless, the total
displacement of a characteristic may become large if the solution is continued far
along it. However, in a first estimate for the correction due to diffusion, the charac-
teristics could be left unchanged and determined as in ? 4, but the variation of the
values of q on them would be determined according to (85). For this purpose, it
is convenient to take (85) in the form

dq 13q aq a23q
dx ct x+ ct2at (86)
()
At x = 0, q is known as a function of t, and the integration of (86) along the charac-
teristics can be carried out by the usual numerical methods. The simplest method is
to evaluate the right-hand side of (86) from the data at x = 0, and deduce the
increments in q at a distance Ax downstream along the characteristics from the
formula
Aq = a x. (87)

With the new values of q, the procedure can be repeated to furnish values of q at
a further distance Ax downstream; this process is continued to extend the solution
to points downstream. More refined schemes would include higher-order differences
in the integration. Near a shock, the changes in q will become relatively large, and
the integration is only continued until the values have been joined smoothly on to
the steady profile solution which replaces the shock.

21 Vol. 229. A.
314 M. J. Lighthill and G. B. Whitham

In some cases this adjustment of q on the existing characteristic network may be

sufficient; it may even indicate that the unmodified solution of ? 4 is adequate for
the purpose in hand. But, in other cases, it may show the desirability of modifying
the characteristics progressively with the values of q. The step-by-step procedure
is easily adjusted to incorporate this, but the labour is increased. As before, a2q/at2
and c can be determined from the data at x = 0. Segments can be drawn, for the
intervals Ax, in the characteristic directions (given by At = Ax/c); the increment
of q along each of these segments is then given by (87). In this way q is obtained as
a function of t at a distance Ax downstream. Repetition of this process continues
the solution downstream. The shocks require special consideration, however. The
above scheme would, in fact, include shocks automatically as regions of relatively
large increases in q, since (78) is capable of providing steady profile shock solutions.
But unless a special choice of v is made, the shocks would not be described accurately;
equation (78), particularly when obtained from theoretical considerations, has been
introduced to estimate only small corrections to continuous kinematic waves. For
example, qx has been approximated by zero in the coefficients of (77) and this will
not be valid inside the shocks. One method of describing the shocks accurately is
to include them explicitly in the calculation. That is, when the presence of a shock
is recognized (by the relatively large changes in q), a shock line should be drawn at
each stage in the direction determined by the shock velocity (q2-ql)/(k2-kl)
appropriate to the flows which it separates. The final step would then be to centre
on the shock line the steady profile solution which produces the required changes
in q. In some respects, however, this method of determination is not as convenient
as one in which the shocks are included automatically. We, therefore, investigate
whether (78) may be artificially modified in order to give the shocks accurately.
As mentioned above, (78) cannot really describe the flow inside a shock when c
and v are taken as functions of q and x alone. However, the dependence of v on q
may be so chosen that the shock will have the correct velocity; this is the main
requirement. The shocks which will be described by (78) may be investigated by
finding the steady profile solution. Taking q to be a function only of - = x- Ut,
we have
(c- U)ddo- vU dcrdo-2

f (C 1\, - dq
or on integration, J U- d
U ) dor

Now, if q1and q2are the values of q on the two sides of the shock, dq/do' must vanish
for both q = q1 and q = q2; therefore

f (c l) dq= 0

and thus U is given by U= -dq -dq.

We require that this be the accurate shock velocity (q2-q)/(k2-kl). Since

c = (ak/lq)-1, it will be so if v is proportional to c. Of course, v oc c is not in agreement
 Kinematic waves. I 315
with the variation of v outside the shock as given by (84), for example. But the
variation of v with q outside the shock would probably be ignored in any case and
some constant value assigned to v. Certainly, it is expected that observational data
will only provide an approximate estimate for v without showing its dependence
on q. Thus, provided the constant of proportionality is adjusted to agree approxi-
mately with the value outside the shocks, we may take v oc c. Then the shocks will
be deduced by the step-by-step solution in their correct positions, without explicit
consideration at each step.

APPENDIX
In this appendix, the details are given of the interpretation of (31) as a real
integral involving the Bessel function Il(z). First, (31) is written as
y = e(p+JA)F {e-gV{(p+a)22-2 - e-(P+a)} A (p) + e-(1-F)[p+A(1-F) A (p), (88)
x
where = (1_-2) a= A(1+
( F2), ,8= A(1-F2)J(1-J F2). (89)
(gho)

The second term in the expression for Y can be interpreted immediately as

e 1+-(gh) (90)
(xp gh) vo +
since ift f(t) = A(p), then
f(t -/) = e-P A(p). (91)

The interpretation of the first term can be deduced by the usual rules of Heaviside
calculus from the known result (Doetsch 1947, p. 105) that

p(e-V(2-2)- e-6P) = 1(t I{(t2 _ 2)} H(t- 6). (92)

For, if G(p) = g(t) then pG(p + a)/(p + ca) = g(t) e-at; hence, the first term in (88) is

e6(P+ A)FG1(p) A(p)

p

where Gl(P) = gl(t) = g(t) e-t and g(t) is given by the right-hand side of (92). Now

p-',(p) A(p) = lgl(t- t')f(t') dt';

hence, again using the 'shift rule' (91), the first term of (88) is
rt+F
e-AF g(t + F - t') e-a(t+F'-t)f(t') dt'.
Jo

t As is customary, the equality sign is used rather loosely between functions and their
operational representations.
21-2
 316 M. J. Lighthill and G. B. Whitham
On substituting for g(t) from (95), and for 6 from (89), this becomes
o or
i (x
vo -(gho)IJ o
XeAFx/V(gho) +l1[ {(t -'- + <(o)) (t-t + V
o)h0 dt,'
e_1(t-t,)f(t)
(1 -F2)V(gh0) t( V tIt;+ ((93)
o Vo + V(g o) V(gho)- v (93)
Finally, y is the sum of (90) and (93).

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