= H(W)z x, hu hu 2 + gh 2 /2., H(W) = gh

Transcription

1 THE EXACT RIEMANN SOUTIONS TO SHAOW WATER EQUATIONS EE HAN AND GERAD WARNECKE Abstract. We determine completely the eact Riemann solutions for the shallow water equations with a bottom step including the dry bed problem. The nonstrict hyperbolicity of this first order system of partial differential equations leads to resonant waves and non unique solutions. To address these difficulties we construct the M and R M curves in the state space. For the bottom step elevated from left to right, we classify the M curve into five different cases and the R M curve into two different cases based on the subcritical and supercritical Froude number of the Riemann initial data as well as the jump of the bottom step. The behaviors of all basic cases of the M and R M curves are fully analyzed. We observe that the non uniqueness of the Riemann solutions is due to bifurcations on the M or R M curves. The possible Riemann solutions include classical waves and resonant waves as well as dry bed solutions that are solved in a uniform framework for any given initial data. Key words. shock waves, rarefaction waves, velocity function, stationary waves, resonant waves, Froude number, M curve, R M curve, nonuniqueness solutions. AMS subject classifications. 65M6, 76M, 56. Introduction. In this paper we are concerned with the shallow water system of hyperbolic equations, which can be written in the form (.) where (.) W = z h hu W t + F(W), F(W) = = H(W)z, hu hu + gh /, H(W) = gh, see e.g. Stoker []. The independent variables z, h and u denote, respectively, the bottom topography, the water height and the water velocity, while g is the gravity constant. Usually the bottom topography z is assumed to be given a priori. The shallow water equations (.) model incompressible flows on a bottom bed under the assumption that the depth of the fluid is much smaller than the wave length of the disturbances considered. It has wide applications in fluid dynamics, for eample tidal flows in an estuary, hydraulic jumps, river beds and channels, tsunamis, etc. The system has also been studied from a mathematical point view. A particular feature of the system (.) is the presence of the bottom topography z(). This geometric variable is independent of time and leads to a stationary source and a nonconservative term. We only reference a few publications. efloch [8] complemented related non conservative system with an additional trivial equation z t =. This additional equation z t = introduces a linear degenerate field with a speed eigenvalue. As a result the system (.) becomes a nonstrictly hyperbolic system. Due to the coincidence Department of Mathematics, Magdeburg University, Universitaetsplatz 96 Magdeburg, Germany (han.ee@st.ovgu.de)., Department of Mathematics, Magdeburg University, Universitaetsplatz 96 Magdeburg, Germany (gerald.warnecke@ovgu.de).

2 of eigenvectors the system becomes degenerate at sonic states, see e.g. Alcrudo and Benkhaldoun [5]. Bernettia et al. [7] studied an enlarged system and used the energy to rule out solutions that are physically inadmissible. Andrianov [] proposed an eample which has two solutions for one set of initial data to show that different numerical schemes may approach different eact solutions. i and Chen [] studied the generalized Riemann problem for the current system. efloch and Thanh in [9, ] investigated the eact Riemann solutions. They obtained most of the possible solutions for given initial data. However they omitted one possible type of solution which is denoted as the wave configuration E in this work. Moreover they did not give complete proofs for the eistence and uniqueness of the solutions. Especially for the conjecture in [, Remark 6 p. 7646]. Besides the papers we just mentioned, considerable work has been devoted to the topic of the shallow water equations, see e.g. [,,, 4] and the references therein. In this work we propose a uniform framework to solve the system (.) with the following Riemann problem (.) W(, ) = { W, <, W R, >, where W q with q = or R are constant. There are three wave curves for the system (.). The first and second wave curves are given by the physically relevant parts of the rarefaction and shock curves. The third wave curve, denoted as the stationary wave curve, is due to the variation of the bottom step. Since the governing system is non strictly hyperbolic, the mutual positions of the stationary wave curve with respect to the rest of the two elementary waves cannot be determined a priori. To address this difficulty we introduce the M and R M curves in the state plane for the construction of solutions to Riemann problems. The idea is motivated by Marchesin and Paes-eme [] as well as our previous work for the eact Riemann solutions to Euler equations for duct flows in [9]. During this work we always assume without loss of generality that z < z R. The opposite case can be treated as the mirror image problem by reflecting the Riemann initial data in terms of =. We take into account the stationary wave curves by deriving a velocity function. Owing to this function, the M and R M curves with z < z R can be, respectively, classified into five and two different cases by the subcritical or supercritical Froude number of the Riemann initial data as well as the jump of the bottom step. This new classification is very helpful for a systematic consideration of solutions. It is given for the M curves at the begining of Section 4. and for the R M curves in Section 4.. Note that each of these curves leads to more than one wave configurations, depending on the Riemann initial data. We obtain the 7 wave configurations denoted as A, B, C, D, E, F, G that do not have a dry bed state and 6 that do have a dry bed state in the solution. The dry bed states are like vacuum states in gas dynamics and therefore we inde the corresponding wave configurations with subscript letter v. We find that the water can always spread across a lowered bottom step. But the water can go across an elevated bottom step if and only if a critical step size z ma is larger than the actual jump height of the bottom step. The critical step size z ma is determined by the height and Froude number of the inflow state. We carefully study the monotonicity and smoothness properties of the M and R M curves in each case. Note that the introduction of these curves and the use of the velocity function make our approach to the solution of the Riemann problem different

3 from the previous work. We feel that this makes the solution procedure clearer and simpler. Observe that a bifurcation occurs for certain cases. This bifurcation introduces nonunique solutions and validates the conjecture in [, Remark 6, p. 7646]. Especially we solve the dry bed problem of the solution in this framework. Here the dry bed problem refers two subcases. One is for the water propagating to a dry bed, see Toro []. The other one is for the dry bed state emerging due to the motion of the flow. The organization of the paper is as follows. We briefly review the fundamental concepts and notions for the governing system in Section. In Section we discuss the stationary wave curves. Our main focus is in Section 4, which contains the definition of the M and R M curves and the complete analysis of their structures. All the possible wave configurations are illustrated in this section. The algorithm for determining the eact solutions is eplained in Section 5. Finally we make some conclusions in Section 6.. The shallow water system. We now derive the quasi linear form of the system (.). Set V = (z, h, u) T, then (.) V t + A(V)V =, where the Jacobian matri A(V) is in the form A(V) = u h. g g u The eigenvalues of A(V) are (.) λ =, λ = u c, λ = u + c, where c = gh is the sound speed. They eigenvalues are referred to as the characteristic speeds. The system (.) is not strictly hyperbolic as a result of the fact that λ can coincide with any of the two other eigenvalues. The corresponding right eigenvectors are (.) R = c u c u h One can easily show that, R = c h, R = c h. (.4) R R k as λ k for k =,. Consequently the system (.) is degenerate for the states at which the eigenvalues λ or λ coincide with λ. Specifically, this state is the sonic state at which u = ±c. We use the terminology k-waves, k =,,, to denote the waves associated to the k characteristic fields when the eigenvalues are distinct from each other. Here the and waves are shocks, hydraulic jumps or rarefactions. Traditionally the wave is named, the stationary wave [5] due to the jump of the bottom step. Note that a speed shock or a transonic rarefaction wave will coincide with the stationary wave. In such kind a case these elementary wave will involved in the stationary wave [5]. We name these combined waves the resonant waves. They will be studied in details later, see also Han et al. [9].

4 4 We define the shock and rarefaction curves. et U q = (h q, h q u q ) T be any state in the state space. The shock speed σ k, where k =, represents the number of the wave family, and the velocity u can be epressed as follows (.5) (.6) ( g σ k = u q ± h h + ). h q ( g u = u q ± (h h q ) h + h q where h > h q. The shock speed σ takes the sign and σ takes the + sign in (.5), analogously in (.6). The detailed derivation can be found in Francisco and Benkhaldoun [5]. The admissible shock curves S k (U q ), k =, denote the states which are connected to the state U q by an admissible shock or shock respectively. Set ), (.7) S k (U q ) = {U F(U) F(U q ) = σ k (U U q ) with h > h q }, k =,. Generally the shock curve S k (U q ) contains three components, namely, (.8) S ± k (U q) = { U U S k (U q ) and σ k (U q,u) }, S k(u q ) = { U U S k (U q ) and σ k (U q,u) = }. We study the state set Sk (U q). Note that we have the shock speed ( g σ k = u q ± h h + ) (.9) = with h > h q. h q Therefore, introducing the Froude number F q := uq c q = uq, we obtain ghq (.) ( h There are two solutions to (.) which are (.) h q ) + h h q F q =. h = + +8F q h q, h = +8F q h q, Note that h > h q and h < < h q, so h is the physically relevant solution to (.9). Hence the set S k (U q) contains only one state. Hereafter we use Ûq = S k (U q) to denote it, then we have (.) ĥ q = + +8F q h q. Since ĥqû q = h q u q, we get û q = hquq. Direct calculation yields ĥ q (.) û q = + +8F q 4F q For the rarefaction curves, similarly, we use R k (U q ) to denote the states U which can be connected to U q by a k Rarefaction wave, i. e. (.4) u q. R k (U q ) = {U u = u q ± (c c q ) with h h q }, k =,.

5 .. The wave and wave curves. Generally the k wave curves T k (U q ), k =, are defined as the sets of states which can be connected to the initial state U q by admissible waves. That is to say we have 5 (.5) T (U ) = R (U ) S (U ), T (U R ) = R (U R ) S (U R ). Obviously T (U ) and T (U R ) are the admissible wave curves associated to the characteristic field with λ and λ in the state space respectively. For simplicity we define the following function (.6) f q (h; h q ) := ( gh c q ), if h h q, ( g (h h q ) h + h q ), if h > h q. We will consider f q (h;u q ) as a function of h for given parameter U q. Therefore the k wave curve T k (U q ), k =, can be rewritten as (.7) T (U q ) = {U u = u q f (h; h q ), h }, T (U q ) = {U u = u q + f R (h; h q ), h }. emma.. The function f q (h; h q ) is continuously differentiable, strictly increasing and concave. Proof. The function f q (h; h q ) is twice continuous due to lim f q(h; h q ) = lim f q(h; h q ) = h h q h h q+. The derivative of the function f q (h;q q ) is (.8) Therefore we have g h, if h h q, f q (h; h q) := h hq + hq h if h > h q. g q h + hq (.9) f q (h; h q) > and lim f q(h; h q ) = g. To see the conveity of the function, we need to consider hq h h q the second derivative of the function f q (h; h q ). Actually we have gh, if h h q, (.) f q (h; h q) := g 4 5 h + hq h 4 h + hq, if h > h q. Hence f q (h; h q) <. Moreover we have lim f q (h; h q) = h h q gh q. This is enough to confirm the lemma. emma. reveals that the wave curve T (U ) is a strictly decreasing concave curve, while the wave curve T (U R ) is a strictly increasing conve curve in the (u, h) state plane. Therefore these two curves have at most one intersection point. To find whether the intersection point eists or not, we need to consider the state with h =, which corresponds to the dry bed of the water, see Toro []. For the wave curve T (U ) and the wave curve T (U R ) we take h = in (.7) and (.6). We obtain two velocities (.) u = u + c,

6 6 and (.) u R = u R c R. These are the velocities of the water covering or uncovering a dry state h =. The two curves T (U ) and T (U R ) will interact if u u R, i.e. (.) u R u < (c + c R ). In thia case the intersection point of T (U ) and T (U R ) uniquely eists, In the other case u < u R there is no intersection point. Then we obtain a dry bed intermediate state. Now we want to study two specific dry bed problems. Both of them concern the water receding from the jump of the dry bed. The first problem has the Riemann initial data { (h, u (h, u)(, ) = ), <, (.4) (, ), >, with the restriction that u <. In such kind of case the wave of the solution is missing while the wave is a rarefaction wave on the left side. The corresponding solution is given as (.5) (h, u)(, t) = (h (, u ), ) t u c, (u+c t ) 9g, u+c+ t, u c < t < u, (, ), t > u. The other problem is has the Riemann initial data (.6) (h, u)(, ) = { (, ), <, (h R, u R ), >, with u R >. Similarly the wave of the solution is missing and the wave is a rarefaction wave on the right side. The eact solution of this case is shown in the following: (.7) (h, u)(, t) = ((h R, u R ), t u R + c R, (ur c R t) 9g, ur cr+ t ), u R + c R > t u R, (, ), t < u R. The jump of bottom step does not affect the solution in these two eamples. However for the Riemann problem (.), (.4) or (.6) but with u > or u R > respectively, the jump of the bottom step induces an additional wave. The motion of the flow becomes more complicated. Not to mention the general Riemann problem of (.) and (.) with h > and h R >. There the jump of the bottom step greatly affects the motion of the flow. So in the net section we study the stationary wave due to the jump of the bottom step.. The stationary wave curve. The stationary wave curve for the system (.) is defined by the ODE system (.) F(U) = H(U)z.

7 Motivated by Alcrudo and Benkhaldoun [5] and references cited therein, we have the following emma. emma.. For the smooth bottom topography the sonic state can only appear when the bottom function reaches a maimum. Proof. The ODE system (.) asserts the following equations 7 (.) Therefore we have (.) hu =, u u + g =. ( u c ) h u u = z. The relation (.) shows that for smooth lowered bottom topography, i.e. z <, the velocity of the water decreases when u < c and vice versa. Similarly for smooth elevated bottom topography, i.e. z >, the velocity increases when u < c and vice versa. So we can conclude that the quantity z as a function of has a maimum at the sonic state u = c. In this work we regard the stationary wave as a transition layer located at = with width. In this approach the discontinuous variation of the bottom step is viewed as the limiting case of locally monotonic bottom slope going to infinity. This idea has been used by Alcrudo and Benkhaldoun [5], efloch and Thanh [9, ], Toro [] etc. for the shallow water systems. Han et. al. [9] also adopted it to solve the Riemann problem for duct flows... The stationary wave. In this section we use the subscript i to represent the inflow variables while o represents the outflow variables. Assume that the piecewise constant bottom topography has the values z i and z o, while the upstream flow state is (h i, u i ) which is known and the downstream flow state is (h, u). Here z i will be z if u >, while it is z R if u <. With the analogous consideration z o will be determined. et assume that h i, h >. One can easily derive the following relations from the system (.) (.4) (.5) hu = h i u i, u + g(h + z o) = u i + g(h i + z i ). The formula (.4) implies the following conditions. u i and u have the same sign,. u i = u =. Our aim is to calculate the downstream state (h, u). Specifically if u i = and h i + z i z o > we have u = and h = h i +z i z o, otherwise if u i = and h i +z i z o <, we have u = and h =. In the following analysis we always assume that u i. For simplicity we can use the notation U = J(z o ;U i, z i ) to represent the eplicit solution U := (h, u) T implicitly given by (.4) and (.4). Our aim is to calculate the downstream state (h, u) of the flow for the known upstream flow (h i, u i ). A velocity function is derived from (.4) and (.5) to be (.6) Ψ(u;U i, z o ) := u + c i u i u u i gh i + g(z o z i ).

8 8 The behavior of the velocity function is analyzed in the following lemma. emma.. Consider (.7) u = ( u i c ) i, then the velocity function Ψ(u;U i, z o ) has the following properties. Ψ(u;U i, z o ) decreases if u < u ;. Ψ(u;U i, z o ) increases if u > u ;. Ψ(u;U i, z o ) has the minimum value at u = u and there u = c with the sound speed c = gh = g uihi u. Proof. The velocity function Ψ(u;U i, z o ) is smooth since if u i > the eistence region for u is u >, otherwise if u i < the eistence region for u is u <. Therefore the derivative of Ψ(u;U i, z o ) is (.8) Consequently we get (.9) Ψ(u;U i, z o ) u Ψ(u;U i, z o ) u = u u ic i u. <, if u < u, =, if u = u, >, if u > u. It follows that the velocity function Ψ(u;U i, z o ) is decreasing when u < u increasing when u > u and has the minimum value at u = u. Since and (.) we get the formula (.) c = gh = gh iu i u, u Ψ u (u;u i, z o ) = u gu h = u c. u From Ψ(u ;U i,z o) u = we obtain u = c. Corollary.. emma. shows that the equation Ψ(u;U i, z o ) = may have two, one or no solutions. Further discussions are as follows, ). If the minimum value Ψ(u ;U i, z o ) <, the equation Ψ(u;U i, z o ) = has two roots. Assume that the root closer to is u l and the other one is u r, c l and c r are the corresponding sound speeds. Then according to (.), u l c l < and u r c r >. It is well known that the transition from subcritical to supercritical channel flow can only occur at points of maimum of the bottom function [5]. So physically we can take the one which satisfies (.) sign(u q c q ) = sign(u i c i ) where q = l or r. However one special case is that if the inflow state U i is a sonic state, i.e. u i = c i, then (.) no longer holds. There are two possible solutions u l and u r, which one is to be chosen depends on the requirement of the specifical problem. The details will be given later.

9 ). If Ψ(u ;U i, z o ) =, the equation Ψ(u;U i, z o ) = has eactly one solution which is the sonic state, i.e. u = u. ). If Ψ(u ;U i, z o ) >, the equation Ψ(u;U i, z o ) = has no solution. The procedure for calculating the outflow state U = J(z o ;U i, z i ) is summarized in Algorithm. However it is necessary to analyze the eistence region for U = J(z o ;U i, z i ) to determine it a priori. Algorithm Algorithm for solving U = J(z o ;U i, z i ) Require: flag, z i, z o and (h i, u i ) : if z i = z o then : return (h i, u i ) : else if u i = then 4: if h i + z i < z o then 5: return (, ) 6: else 7: return (h i + z i z o, ) 8: end if 9: else : Ψ min Ψ(u ;U i, z o ) : if Ψ min < then : Solve Ψ(u;U i, z o ) = by the iteration method to obtain u l and u r : c l g hiui u l, c r g hiui u r 4: if sign(u l c l ) = sign(v i c i ) ( flag = vi = i) c then 5: return ( hiui u l, u l ) 6: else if sign(u l c l ) = sign(v i c i ) ( flag = vi = i) c then 7: return ( hiui u r, u r ) 8: end if 9: else if Ψ min = then : return ( hiui u, u ) : else if Ψ min > then : print No Solution. : end if 4: end if 9.. Eistence of the stationary wave. Corollary. reveals that the velocity function may have no solutions. To be more precise we now consider the eistence conditions for the outflow state (h, u) of the stationary wave introduced above. According to emma., it is equivalent to evaluate the minimum value of the velocity function Ψ(u;U i, z o ) being not larger than, i.e. Ψ(v ;U i, z o ) = ( ui c i) c i u i (.) + g(z o z i ). We introduce the Froude number F i := ui c i, then we have ( h i (F i) F i ) (.4) + z o z i. Therefore we obtain (.5) ( F z o z i h i i ) F i +.

10 We know that (.6) F i F i +. It reaches if and only if F i =. Hence, the above computation motivates the following theorem. Theorem.4. The eistence of the solution to the velocity function.. If z o < z i, Ψ(u;U i, z o ) always has solutions.. Otherwise if z o > z i, Ψ(u;U i, z o ) has a solution if and only if ( F z o z i h i i ) F (.7) i +. Theorem.4 indicates that on one hand the water can always spread across the lowered jump of the bottom step; on the other hand the water can overflow the elevated jump of the bottom step if and only if the bottom step is not too high. Specifically it should less than a critical value which is determined by the height and the Froude number of the inflow. Corollary.5. For the fied inflow state U i and two outflow bottom steps z o < z o, if J(z o;u i, z i ) eists then J(z o;u i, z i ) also eists. Since we regard the discontinuous bottom step as the limiting case of monotonic bottom step, it make sense to assume that the solution inside this transition layer is also continuous if there is no resonant wave. 4. M and R M wave curves. In this work we always assume without loss of generality that (4.) z < z R. According to emma. the sonic state can only be located on the side z = z R of the stationary wave. The opposite case z > z R can be treated as the mirror image problem by reversing the Riemann initial data and setting the velocity in the opposite direction. Here we study the general Riemann solution which contains a stationary wave. The sufficient condition for this requirement is that u > or u R <, where u and u R were defined in (.) and (.). Otherwise if u < and u R > the dry bed appears around the initial discontinuity point =. Specifically the solution has the wave configuration A v, see Figure 4.. Hereafter the symbols k r, k =, denote the k rarefactions. An eample of this case can be found in Figure 4.5. We can see that the jump of the bottom does not affect the motion of the flow. Therefore there is no stationary wave. The general eact Riemann solution for the system (.) with (.) with u > or u R < consists of a stationary wave which is located at = as well as a sequence of and shocks or rarefactions. Alcrudo and Benkhaldoun in [5] presented more than different solution patterns. Indeed the solution patterns without the dry bed under the condition (4.) can be classified into different wave configurations. We show them in Figures 4., 4.6, 4.8, 4., 4., 4.9, 4.4, 4.6, 4.7, and 4.8. In all the wave configurations the and wave represent a shock or a rarefaction. The dashed right arrow indicates that the velocity across the the bottom jump is positive, while the dashed left arrow indicates that the velocity across the bottom jump is negative. Note that the wave configuration E has been omitted by efloch and Thanh in [].

11 t dry bed t wave U U M u r (, ) (, ) u R r wave wave U U R U U R Fig. 4.. A v t Fig. 4.. A t wave (h M, ) dry bed (, ) r wave (h M, ) dry bed U U R U Fig. 4.. H Fig H The wave configurations A T, C T and D T, in some sense, can be viewed as the image reflection of the wave configurations A, C and D in terms of = respectively. Moreover the wave configurations B and G contain a resonant wave due to the coincidence of the stationary wave with a and rarefaction wave respectively. The wave configurations C and C T result from the coincidence of a statioanry wave with a speed and shock wave respectively. While the wave configurations E and F are the combination of a transonic rarefaction, a stationary wave and a speed shock. We point out that analogous resonant waves to these mentioned here for other systems can be found in Goatin and efloch [5], Rochette and Clain [7], Han et al. [9] etc. The solution patterns with a dry bed consist of the wave configurations A v, H and H, see Figures 4., 4., and 4.4 respectively. Also the wave configuration B v, see Figure 4.7, belongs to this category. Note that the wave configuration B v originated from the wave configuration B. But B v contains a dry bed intermediate state (, ) and the wave is a rarefaction wave. Here we should keep in mind that rarefaction wave will totally disappear if h R =. This is analogously to the wave configurations D v and E v, see Figures 4. and 4., which comes from the wave configurations D and E respectively. The wave configuration G v, see Figure 4.5, originated from the wave configuration G. Be advised that G v contains a dry bed state (, ) and a rarefaction if h >, or no wave if h =. The situation for the wave configuration D T v, see Figure 4.9, is similar. For one given set of initial data we cannot determine the wave configuration of the solution from the initial data in advance due to many possibilities of the mutual position between the stationary wave and shocks or rarefactions. This is the nature of non strictly hyperbolic system. Analogous to the Euler equations in a duct, see Han et al. [9], we here also introduce the M and R M curves to solve this problem. We merge the stationary wave curve into the wave curve T (U ) or the wave curve T (U R ). Here we also name them M and R M curves. These two curves can be regarded as an etension of the T (U ) and T (U R ) curves respectively. They will

12 Velocity Fig eft: The water free surface h + z at t =.5; Right: The velocity. The Riemann initial data are (z, h, u ) = (.,.5674, 6.) when <.5 and (z R, h R, u R ) = (.8,.558, 6.) when >. t r Ũc Uc wave t r Ũc Uc wave wave U M wave wave dry bed r (, ) U U R U U R wave t Fig B S (U ) wave Fig B v t wave S (U ) Uc U U + U M wave U M U+ U U z z z R U R U z z z R wave U R {z } = {z } = Fig C Fig F serve as a building block for the calculation of the Riemann solutions to the shallow water equation in a uniform way. There is precisely one stationary wave in a full wave curve from U to U R located either on the M curve or the R M curve. Due to the fact that the velocity does not change sign across the stationary wave, so the location of the stationary wave is determined by this rule: If u > the stationary wave is on the M curve; if u < the stationary wave is on the R M curve. Hence if u > the M curve always contains the segment (4.) P l (U ) = {U U T (U ) with u }, otherwise if u the M will be (4.) P l (U ) = {U U T (U ) with u u }.

13 t wave U Ū wave t wave U Ū dry bed U M wave (, ) r U R U R t wave Fig. 4.. D S (U ) Uc dry bed U + U r (, ) t wave Fig. 4.. D v S (U ) Uc dry bed U + U r (, ) z z z R z z z R U U R U U R {z } = {z } = Fig. 4.. E Fig. 4.. E v Similarly if u R < the R M curve always contains the segment (4.4) P r (U R ) = {U U T (U R ) with u }, otherwise if u R the R M curve will be (4.5) P r (U R) = {U U T (U R ) with u u R }. It is necessary to construct the remaining segments of M curves with u > and u >. Also for the R M curves with u R < and u <. Since z < z R, Theorem.4 implies that the stationary wave always eists if the fluid flows from z R to z. However the stationary wave equations (.4) and (.5) may not have solutions if the fluid flows from z to z R. Before constructing the M and R M curves, we need to consider the preliminaries for M and R M curves first. 4.. Preliminaries for M curves with u >. We now investigate the eistence of the state J(z R ;U, z ), where U T (U ) and connected to U by a negative speed wave. Theorem.4 suggests the study of the following function (4.6) ω(h ) := h ( F(h ) F(h ) + ) (z R z ), where the Froude number F(h ) := U(h ) and U(h ) = u f (h ; h ). Theorem gh.4 implies that if ω(h ) the state J(z R ;U, z ) eists and vice versa. So we need to study the behavior of ω(h ). emma 4.. The function ω(h ) is strictly increasing if < F(h ) <. Proof. The function ω(h ) is continuous and differentiable. The derivative of ω(h ) is ω (h ) = F(h ) ] (4.7) F(h ) + + h F(h ) [F(h ) 4 F (h ),

14 4 wave t r wave Ū c Uc wave U M dry bed t r wave Ū c Uc wave r (, ) U U R U U R Fig G t wave Fig G v S (U ) t wave wave U M U wave wave U + U z z z R U Fig A T wave Ū R t wave U R U {z } = Fig C T wave Ū R t wave U R wave U M r dry bed (, ) U U R U UR Fig D T Fig D T v where by (.9) and U(h ) >, we have F (h ) = f (h ; h ) gh U(h ) g h <. As we have mentioned in (.6), F(h ) F(h ) +. It takes the value if and only if F(h ) =. so we obtain that ω (h ) > if < F(h ) < and ω (h ) = if F(h ) =. Denote the minimum value of h as h min and the maimum as h ma. The curve T (U ) is strictly decreasing in the (u, h) state space. Also U T (U ) is connected to U by a negative speed wave. Hence if u c, h min is the height corresponding to the sonic state on the curve T (U ); while if u > c, h min is ĥ which is defined in (.). That is to say we have (4.8) h min = { (u+c ) 9g, if u c, ĥ, if u > c. Now we pay attention to h ma. It should satisfy (4.9) = u f (h ma ;U ).

15 5 If u, we have h ma Otherwise if u >, we have h ma equation (4.) < h which is the solution to the equation u ( gh c ) =. u (h h ) After a short calculation we have > h, Hence from (.6) it is the solution of the g ( h + ) =. h (4.) ( ) ( h h h h ) ( + F ) h h + =. Setting = h h >, (4.) becomes (4.) f() = ( + F ) +. Direct calculation yields the following facts. The function f() defined in (4.) reaches the maimum at l := + F < and the minimum at r := + + F >. When < l, f() increases from to the maimum value at = l ; When ] l, r [ it decreases from the maimum value to the minimum value at = r ; While when > r it increases from the minimum value to. Furthermore be advised that l < < r and f() = F <, so f( r) < f() <. Thus there is eactly one real solution to the cubic equation f() = when > r >. We denote this solution as u l which can be directly calculated by the method for the eact solution to cubic equations, see Nickalls [8]. Finally we have (4.) h ma = { (u+c ) 4g, if u, h u l, if u >. Thus the resonable region for considering ω(h ) is ]h min have the following lemma. emma 4.. Set, hma [. Moreover we (4.4) z ma := z + h ma. The stationary state U = J(z R ;U, z ) with < u c cannot eist if z ma < z R. Proof. Note that ω(h ma ) = h ma (z R z ) = z ma z R. So if z ma < z R, ω(h ma ) <. The function ω(h ) is increasing in terms of h ]h min, hma [. Hence ω(h min ) < ω(h ) ω(h ma ) < if z ma < z R. Theorem.4 implies that if ω(h ) < the stationary wave U = J(z R ;U, z ) cannot eist. emma 4.. Suppose that z R < z ma and u < c. There eists a state Ũc T (U ) which satisfies U c = J(z R ;Ũc, z ). Proof. Due to z R < z ma, we have ω(h ma ) = z ma z > and h min (u +c ) 9g =. A short calculation yields that ω(h min ) = z z R <. Since the function ω(h ) is continuous and increasing there is a unique solution to ω(h ) = by the

16 6 intermediate value theorem. Denote the solution to ω(h ) = as h c. Then the corresponding velocity ũ c can be calculated by (4.5) ũ c = u f ( h c ;U ). The velocity function of J(z R ;Ũc, z ) is Ψ(u;Ũc, z R ) := u + c cũc u ũ c g h c + g(z R z ). The minimum of this velocity function is Ψ(u ;Ũc, z R ) = gω( h c ) =. Hence Corollary. implies that the outflow state of stationary wave is a sonic state, i.e. U c = J(z R ;Ũc, z ). Corollary 4.4. emma 4. is totally consistent with emma.. Note that emma 4. states that in this case the flow coming from the left cannot spill over the obstacle caused by the jump in the bed height at =. Whereas in the case of emma 4. over spill occurs if the velocity is large enough leading to ω(h ) >. In case that z R < z ma and u > c, we have h min = ĥ. We define two critical bottom steps ( z S = z + ĥ ˆF ) ˆF (4.6) +, and (4.7) ( z T = z + h F ) F +, where ĥ and û were defined in (.) and (.) respectively. The Froude number (4.8) Since ĉ = (4.9) ˆF = û. ĉ gĥ, taking (.) and (.) into (4.8), we obtain ˆF = [ ] 8 F + + 8F. We invoke the eistence condition for resonant waves due to the coincidence of a speed shock and the stationary wave. emma 4.5. Suppose z < z R < z ma and u > c. We have the following facts.. The state U = J(z R ; S(U ), z ) eists if z R z S ; otherwise it fails to eist.. The state U = J(z R ;U, z ) eists if z R z T ; otherwise it fails to eist.. One always has z T > z S. Proof. From Theorem.4 the eistence condition for the state U = J(z R ; S (U ), z ) is that ( ( ) ( ) ) z R < z + ĥ (4.) ˆF ˆF + = z S.

17 Analogously we can prove the second statement. Now we investigate the relationship between z S and z T. From (.) and (4.9), we have [ ( z S = z + h + + 8F) ] + +8F + F (4.). = h F F By (4.6) and (4.), we have [ z T z S = h F F + F [ = h F = h 8F > F 4 + F 6 F F 6 ( + + 8F) + + 8F ( + 4F + + 8F ( + 8F ) 6, [ ] [ ] + + 8F + + 8F. ) + + 8F F 7 + F ] when F >. Assume that z < z R and u > c. We now consider resonant waves due to the coincidence of the speed shock with stationary waves. The speed shock splits the stationary wave into a supersonic part and a subsonic part. The corresponding wave curve is defined as follows. { (4.) U U = J(zR ;U +, z);u + = S(U );U = J(z;U, z ) }, where z ]z, z R [. We denote the Froude numbers for the states U ± in (4.) as F ± = u±. By using (.4) we have gh± ],, (4.) h + u + = h u = h u. Therefore we obtain the functions F ± in terms of h ± respectively: (4.4) F(h ± ) := F ± = u h. gh ± By (4.) the derivatives of the functions F(h ± ) are (4.5) df(h ± ) = F ±. dh ± h ± Similar to (4.9) we obtain the further relations for F and F + (4.6) F + = ( ) 8 F + + 8F. The resonant wave curve in (4.) is viewed as a function of z. Actually the variable h is more convenient to analysis the eistence of the wave curve in (4.). Specifically the following lemma holds.

18 8 emma 4.6. For the supersonic state U = J(z;U, z ) in (4.) with z z z R we have (4.7) h h h, where Ū = J(z R ;U, z ). Proof. Considering (.4) and (.5) for U = J(z;U, z ), we study the following equation (4.8) h u gh + h + z u g h z =. Taking z as a function of h, we obtain (4.9) z(h ) := h u gh h + u g + h + z. Using (4.), we have (4.) dz(h ) dh = F >. Note that h = h when z = z, while h = h when z = z R. Thus (4.) implies that h h h. To prove the eistence of the wave curve defined in (4.), we have to study the eistence of the supersonic state U = J(z;U, z ) and the subsonic state U = J(z R ;U +, z) with z z z R. We present the details in the following lemmas. emma 4.7. The region of z for the eistence of the subsonic state U = J(z R ;U +, z) defined in (4.) is as follows:. z ]z, z R [ if z S z R ;. z ]z c, z R [ if z S < z R where z c is defined in (4.4). Proof. Theorem.4 implies that U = J(z R ;U +, z) eists if ( z R z h + F + ) F (4.) + +. In addition by (.) and (.) we have (4.) h + = h ( ) + + 8F That is to say h + can be treated as a function of h. This suggests to consider the function. (4.) Θ(h ) := h + ( F(h +) F(h +) + ) + z z R. For simplicity we introduce the function A(h + ) := h + ( F(h +) F(h +) + ). Therefore Θ(h ) in (4.) can be rewritten as (4.4) Θ(h ) = A(h + (h )) + z(h ) z R.

19 9 By the chain rule we have (4.5) Using (4.5) we obtain Θ (h ) = da(h +) dh + dh + dh + dz(h ) dh. (4.6) dh + dh = + 4F. + 8F Besides by (4.5) and (4.6) we have da(h + ) = dh + F(h +) ( ) F(h +) + + h+ F(h + ) F(h + ) df(h+ ) dh + = F(h +) F(h ( ) +) + F(h +) F(h + ) F(h + ) = F(h + ) = ( ) 8 F + + 8F (4.7). By (4.), (4.6) as well as (4.7), we have Θ (h ) = (4.8) = ( >, 8 F ( ) ) + + 8F ( ) [ + + 8F ( F + F + 8F ) ( ) ] + 8F F F when F >. By emma 4.7, we have h h h. Thus due to (4.8) we have (4.9) Θ(h ) Θ(h ) Θ( h ). From (4.) the state J(z R ;U +, z) eists if Θ(h ). Remember that we denote ˆŪ = Sk (Ū). We have ( Θ( h ) = ˆ h ˆ F ) (4.4) ˆ F +. From (4.6) as well as (4.) we obtain that (4.4) Θ(h ) = z S z R. So on one hand if z S z R, we have Θ(h ) Θ(h ) Θ( h ). Thus the state J(z R ;U +, z) eists for any z z z R. On the other hand if z S < z R we have Θ(h ) < < Θ( h ). From the intermediate value theorem there is a unique solution, denoted as h cs, to the equation Θ(h ) = where h ]h, h [. The corresponding velocity can be calculated from (4.4) ũ cs = h u h cs,

20 and the related bottom step denoted as z c can be deduced from equation (4.8), i.e. (4.4) z c = h u g h c s + h cs u g h z. Hence Θ(h ) if z c z z R. emma 4.8. Assume that u > c, for z T given by (4.7) we have z T < z c if z T < z R. Proof. Denote U c,l = J(z R;U +, z T ). Taking z = z T in (4.8), we obtain that (4.44) α(h ) := h u h + gh ( u c ) =. The function α(h ) is continuous and differentiable. The derivative of this function is (4.45) α (h ) = h u h + g. Set h = h F. We have α (h ) < if h < h, while α (h ) > if h > h. It has the minimum value at h = h and α(h ) =. Therefore there is a unique solution to α(h ) =, i.e. h c,l = h = h F. Using (4.) we obtain that u c,l = u F = c F = gh c,l = c c,l. Thus the state U c,l is the sonic state. Hence we have h+ = h c,l and F + = in (4.) and (4.4) respectively. From (4.6) we have Θ(h c,l ) = z T z R < if z T < z R. Since Θ( h cs ) =, we have by (4.8) h c,l < h cs. Consequently we have z T < z c due to (4.9). Based on the previous emmas, we now study the eistence region for the wave curve defined in (4.). emma 4.9. Assume that z < z R and u > c, then we have. if z R z S < z T, the curve in (4.) eists.. if z S < z R z T, the curve in (4.) eists when z ]z c, z R [.. if z S < z T < z R, the curve in (4.) fails to eist. Proof. The wave curve defined in (4.) eists if the two states U = J(z;U, z ) and U = J(z R ;U +, z) eist. emma 4.5 implies that the state U = J(z;U, z ) eists if z z T. Thus in one case if z R < z T, the state U defined in (4.) with z ]z, z R [ always eists. emma 4.7 conveys that on one hand if z S z R the state U = J(z R ;U +, z) eists when z ]z, z R [. Thus the first statement is true due to z S < z T by emma 4.5. On the other hand if z S < z R the state U = J(z R ;U +, z) eists when z ]z c, z R [. This is the second statement. In the other case if z R > z T, the state U eists if z ]z, z T [. By emmas 4.8 and 4.7 we have ]z, z T [ ]z c, z R [=. This is enough for the third statement. Corollary 4.. Suppose that we have z < z R, u > c and z S < z R < z T. emma 4.9 reveals that there eists an h cs, such that Θ( h cs ) =. Moreover note that Θ( h cs ) is the minimum value of the velocity function to J(z R ;Ûc s, z c ), i.e. the outflow state of J(z R ;Ûc s, z c ) is the sonic state. We denote it as U c, i.e. U c = J(z R ;Ûc s, z c ).

21 that Corollary 4.. Suppose z S < z R < z T and u > c, i.e. h min = ĥ. Note (4.46) ω(ĥ) = z S z R <. Analogously to emma 4., there is a unique solution to ω(h ) =. Here we denote this as h c. The corresponding velocity ũ c can be calculated from (4.5) by setting h c = h c. Also we have U c = J(z R ;Ũc, z ), where U c is the sonic state. The subscript is used to distinguish the sonic state U c from the sonic state U c in Corollary Monotonicity. In this section we consider the monotonicity of two types curves as the preliminary step for the study of the M and R M curves. We define (4.47) P l (U R) = {U U = J(z R;U, z ) and U T (U ) } where h min < h < h ma and (4.48) P r (U R) = {U U = J(z ;U, z R) and U T (U R) } where < u + c < c. Note that P l (U ) and P r (U R ) are the composite of the or wave curve with a stationary wave. Before studying the behavior of P l (U ) and P r (U R ), we consider the following lemma first. emma 4.. For any state U T (U ) connected to U by a negative speed wave, we have (4.49) u h f (h ; h ) <, and u f (h ; h ) g <. Proof. We have u c < since the states U and U are connected by a negative speed wave. From (.8) we have u c, if h h, u h f (h ; h ) = u h g h + h + h (4.5) h q, if h h + > h. h If h h, obviously we have u h f (h ; h ) < ; Otherwise if h > h, we have g h + h + h h g = h + + h + h h + h h h h + h (4.5) So using (4.5) in (4.5) we obtain g + h h h > g h. (4.5) u h f (h ; h ) < u c. Hence we have u h f (h ; h ) < due to u c <.

22 Now we turn to u f (h ; h ) g. Note that (4.5) u f (h ; h ) g = u g g h (u c ) <, if h h, h + h + h h q h + h g, if h > h. So it is only necessary to consider the case that h > h. To ensure that the wave has a negative speed, we have h > h min where (4.54) { h min h, if u c, = ĥ, if u > c, where ĥ was defined in (.). Besides by (.) we have u f (h ; h ) h = (f (h ; h )) + u f (h ; h ) <. Therefore, when h > h we have (4.55) u f (h ; h ) < u min f (hmin ;U ), where u min (4.56) = u f (h min ;U ). Specifically by (4.54) we have u f (h ; h ) g < û g u g h g if u c, ĥ + h + h ĥ q ĥ + h g, if u > c. Note that when u c, u f (h ; h ) g < g h (u c ). Now we consider the case that u > c. By û = hu, we have ĥ û g + ĥ h + h ĥ = ĥ + h h u ĥ g Moreover from (.), we obtain that Set = h ĥ, then F = +. So we have g F h ĥ ( h ĥ + h + h ĥ ĥ + h = h = + + 8F ĥ 4F. g F h ĥ ( h ĥ ) + h ĥ + h ĥ + ) ĥ + h ĥ + = g + + < g by < <. h ĥ + 4 Hence by (4.56), we obtain that u f (h ; h ) g < when u > c. This completes the proof of the lemma.

23 Theorem 4.. The curve P l (U ) defined in (4.47) is strictly decreasing in the (u, h) state plane, while P r (U R ) defined in (4.48) is strictly increasing in the (u, h) state plane. Proof. It is enough to consider P l (U ). The other curve P r (U R ) can be dealt with in an analogous way. We need to prove that du dh <. Due to U = J(z R;U, z ), we have (4.57) (4.58) where (4.59) hu = h u, u + g(h + z R) = u + g(h + z ), u = u f (h ; h ), and f (h ; h ) is defined in (.6). By (4.57) and (4.58) we obtain the equations τ(h, h ) = and (u, h ) =, where (4.6) and τ(h, h ) = (h u ) h + g(h + z R ) u g(h + z ), (4.6) (u, h ) = u + g(h u + z R ) u u g(h + z ). With the implicit function theorem we obtain (4.6) and (4.6) So we have dh = dh du = dh τ h τ h h h = = τ h, u c h h u c u, (4.64) du du dh = dh dh = dh u h h τ. h emma 4.49 tells us that (4.65) τ h = h u h u (u h f (h ; h )) + u f (h ; h ) g <, and (4.66) h = g u (u h f (h ; h )) + u f (h ; h ) g <. Hence we have du dh lemma. < from (4.64) by h >, u >. This completes the proof of the

24 4 Now we define the wave curves (4.67) P k (U q ) = {U U = J(z o ;U +, z); U + = S k (U ); U = J(z;U q, z i ) }, where u q c q, z i z z o, as well as k = when u q > while k = when u q <. The state U = J(z;U q, z i ) is supersonic while U = J(z o ;U +, z) is subsonic. Note that this type of the resonant wave curves is the general case of the wave curve defined in (4.). Moreover we have the following monotonicity lemma for P k (U q ). emma 4.4. Assume that u q c dh du q, we have dz > ; while dz > when u q > as well as du dz < when u q < for the wave curves in (4.67). Proof. It is enough to consider the case that k =. The case for k = can be dealt with in a similar way. The curve P (U q ) defined in (4.67) is a function in terms of z. Note that dh dh dh dh dz (4.68). So we consider dh dh and dh dz in the following. Moreover we have h q u q = h u = h + u + = hu. From U = J(z;U q, z i ) and U = J(z o ;U +, z) we respectively have dz = (4.69) and (4.7) u q h q gh + h + z u q g h q z i =, u qh q gh + h + z o (h qu q ) gh h + z =, + where h + is defined in (4.). Similarly to (4.9) and (4.), we have (4.7) and (4.7) z(h ) := h qu q gh h + u q g + h q + z q. dz(h ) dh = F >. Taking (4.9) into (4.7), we introduce a equation ξ(h, h ) = where (4.7) ξ(h, h ) = u qh q gh + h + z o (h qu q ) gh h + z(h ). + So by the implicit function theorem we have (4.74) ξ dh dh = h = F ξ ξ h where F = u c. Using (4.6) and (4.7), we have (4.75) h ξ h = ξ h + dh + dh + ξ h, = (F + ) dh + dh F +, = Θ (h ) <.

25 So we obtain that ξ h < and dh dh 5 >. From (4.) and (4.75), we obtain that dh dz >. Since hu = h qu q, so du dz = u dh du h dz. Hence dz < if u > and vice versa. In the net section we study the M and R M curve case by case. The gravity constant g = 9.8 unless stated otherwise. 4.. M curves with u > and u >. There are respectively si different types of M curves. We list the classification for all cases in the following: CASE I : z ma < z R. CASE II : z ma z R, u < c F <. CASE III : z ma z R, u > c F >, z R < z S < z T. CASE IV : z ma z R, u > c F >, z S < z R < z T. CASE V : z ma z R, u > c F >, z S < z T < z R. ater we will construct the M curves for all cases. Before doing this we consider an eample given by Andrianov in [, (8)]. To match with the assumption z < z R, we reflect the Riemann initial data with respect to =.5. They become (4.76) (z, h, u) = { (.,.,.), <.5, (.5,.,.), >.5, with [, ]. Note that g = in this eample. For the given data c =., we have u c >. From (4.4), we obtain z ma =.79, z S =.8 and z T =.798. So the M curve of this eample belongs to CASE IV. Reducing z R from.5 to., we obtain the Riemann initial data for CASE III. ater these Riemann initial data will be used to give eamples for the eact solutions CASE I : z ma < z R. This is the case for which the jump of the bottom step is too high compared with the inflow state U of the stationary wave, which is connected to U by a negative wave. Mathematically we say that there is no solution to J(z R ;U, z ) for any U T (U ) with a negative speed wave. This was proved in emma 4.. Generally there are two different subcases for this case: h R =. u R >. We have the following two Riemann problems: (4.77) h t + (hu) =, (hu) t + (hu + gh ) =. (4.78) (h, u)(, ) = { (h, u ), <, (h, u ), >. (4.79) (h, u)(, ) = { (, ), <, (h R, u R ), >. We find that when h R = or u R >, the solution of the Riemann problem can be split into two parts: One part is the solution to the Riemann problem (4.77) and (4.78) in the region <. The other part is the solution to Riemann problem (4.77) and (4.79) in the region >. Note that if h R =, the solution to (4.77) and (4.79) is h = and u = for (, t) R R +. The wave configuration of u,r > in this

26 6 case can refer Figure 4.. The wave configuration of h R = in this case can refer Figure 4.4. Here we give two eamples to illustrate our construction. The first eample has the wave configuration H. The results are shown in Figure 4., where z ma =.5769 < z R = 4.7. The second eample has the wave configuration H. The results are shown in Figure 4., where z ma =.474 < z R = Velocity Fig. 4.. eft: The water free surface h + z at t =.75; Right: The corresponding velocity. The Riemann initial data are (z, h, u ) = (,, ) when < and (z R, h R, u R ) = (4.7,.,7.) when > Velocity Fig. 4.. eft: The water free surface h + z at t =.75; Right: The corresponding velocity. The Riemann initial data are (z, h, u ) = (,.,.) when < and (z R, h R, u R ) = (4.,,) when > CASE II : z ma z R, u < c. In this case the M curve consists of three segments which are defined as follows P l (U ) = {U U T (U ) with u < }, P l (U ) = {U U = J(z R ;U, z ) and U T (U ) with < u < ũ c, < u < u c }, P l (U ) = {U U T (U c ) with u > u c }, where U c = J(z R ;Ũ c, z ), Ũ c T (U ) which is defined in (4.5). The continuous of the three segments is obviously. According to Theorem 4., the segment P l(u ) is strictly decreasing in the (u, h + z) space. Also the segments P(U l ) and P(U l ) are strictly decreasing in the (u, ) space due to emma.. So the M curve Pk l(u ) is strictly decreasing in the (u, h + z) space. k=

27 7 We define (4.8) u = u c. If u > u R there is a unique intersection point between the M curve and the R M curve. If the intersection point lies on the segment P l (U ), the solution has the wave configurations A, see Figures 4.. Here we use an eample given by Alcrudo and Benkhaldoun in [5] to illustrate the corresponding M curve, the eact free surface of the fluids, as well as the Froude number in Figure 4.. If the intersection point lies on the segment P l (U ), the solution has the wave configuration B. An eample is shown in Figure 4.. We observe that the Froude number is larger than when the water go across the bottom jump P l (U ) h + z.5 P l (U ) u.5 P r (U R ) P l (U ) Froude number S Fig. 4.. Top: M curve Pk l(u ). Bottom left: The water free surface h + z at t =.; k= Bottom right: The corresponding Froude number. The Riemann initial data are (z, h, u ) = (., 4.,.) when < and (z R, h R, u R ) = (.,.,.) when >. Otherwise if u < u R and h R >, the eact Riemann solution contains a dry bed state and behaves in the manner of the wave configuration B v, see Figure 4.7. The eample for h R > is shown in Figure 4.4. The eample for h R = is shown in Figure 4.5.

28 P l (U ) h + z.5 P l (U ) u.5.5 P r (U R) P l (U ) Froude number S Fig. 4.. Top: M curve Pk l(u ). Bottom left: The water free surface h + z at t =.; k= Bottom right: The corresponding Froude number. The Riemann initial data are (z, h, u ) = (., 4.,.) when < and (z R, h R, u R ) = (.,.,.) when > CASE III : z ma z R, u > c, z R < z S < z T. In this case the M curve consists of the following four parts: P l (U ) = {U U T (U ) with u < }, P l (U ) = {U U = J(z R ;U, z ) and U S (U ) with < v < ˆv, < u < û }, P l (U ) = {U U = J(z R ;U +, z); U + = S (U ); U = J(z;U, z ), z z z R }, P l 4 (U ) = { U U T (Ū) with u > ˆū }, where Û = J(z R ;Û, z ) and Û = S (U ), while ˆŪ ) = S (Ū and Ū = J(z R ;U, z ). Due to z < z R, emma 4.4 tells us that h is increasing while u is decreasing when z varies monotonically from z to z R. So ĥ < ˆ h and û > ˆū. As shown in Figure 4.6, the M curve is folding in the (u, h + z) state space. We define (4.8) u = ū + c. Note that if u > u R, there are intersection points between the M curve and the R M curve. If the intersection point lies on the segment P l (U ) the solution is in

29 9.8.7 h + z P l (U ) u P r (U R )... P l (U ) P l (U ) Velocity S Fig Top: M curve Pk l(u ). Bottom left: The water free surface h + z at t =.; k= Bottom right: The corresponding velocity. The Riemann initial data are (z, h, u ) = (,.9,.) when < and (z R, h R, u R ) = (.,., 9.) when >. the pattern of the wave configuration A, This is analogous to the CASE II,. If the intersection point lies on the segment P l (U ), the solution has the wave configuration C. While if the intersection point lies on the segment P l 4 (U ), the solution has the wave configuration D. Due to the fact that the M curve is folding in the (u, h + z) state space, if the intersection point lies on the segment P l (U ), we can also find two other intermediate states lying on the segments P l (U ) and P l 4(U ) respectively. So for one given initial data there are three solutions with the wave configuration A, C and D respectively. An eample with g =. is shown in Figure 4.6. An eample for g = 9.8 is shown in Figure 4.7. Moreover if u < u R, the solution with the wave configuration D v occurs. An eample for h R > is shown in Figure 4.8. An eample for h R = is shown in Figure 4.9. Note that the computational region for these two eamples is [, ] and g =..