f(u) f(u) We see that the fur function in this case is neither conver nor concave. In fact, it has RAREFACTION u(x,0) Physically we see that as the water moves in, it displaces a certain fraction of the oil immediately. Behind the shock there is a mixture of oil and water with less and less oil as times goes on We note that for r707-c07we fur functions it is possible to have a riemann solution whi ch involves both a shock and a rarefaction wave. If the fut function had more inferion points, the solution might involve more shocks and rarefaction Note 13 raphic determinat ion of the shook location The solution to the Riemann problem can be determined from the graph of f(u) n a simple manner (u°)0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ♦❚♣✭q✢♣✎♣sr✆t❇✉♠r✁r✆t❇♣❱✈①✇❤②①③✎✇❳④☞⑤✢r⑦⑥❞⑧♠④✩⑥✆④⑨r⑩tP⑥❥qs⑤✎✉■q✢♣s⑥❥q❶④☞♣✢⑥✆r⑩t❝♣✢❷s⑤✎⑧♠④❇❸♠♣✽②❢④☞⑧♠❷✞⑤✎⑧♥④☞⑤✎✉♠❸♠♣✑❹❀❺✒④s③✒✉❤⑤✢r✛❻ ⑥✆r✁t❇✉■q❼✉♥④❽⑥✆④✑✈✭♣✽②❾⑥❞⑧♠④➀❿❇⑧♠⑥✆④✤r➁❹ ➂☞➃✖➄✆➅✫➆❽➇✤➈ RAREFACTION SHOCK ➉✔➊➌➋ ➍➎t❳➏■q✎⑥❞⑤✎✉♥➐✆➐➑➏❚➒▲♣❢q✢♣✎♣➓r✆t❇✉♠r✞✉♠q❼r⑩t❝♣➓➒▲✉♥r➁♣✢❷❼➔➀⑧♠❸♠♣✒q❼⑥✆④❝❻✭⑥✆r✞→♥⑥❥q✛❿❪➐➣✉❤⑤✎♣✒q❧✉❚⑤✽♣✢❷↔r↕✉♠⑥✆④✄③✎❷✍✉❾⑤✒r⑦⑥❞⑧♠④❘⑧❩③ r⑩t❝♣➀⑧♠⑥✆➐✱⑥✆➔❢➔➀♣✽→♥⑥❞✉♠r↕♣✒➐✠➏❤❹❢➙s♣✽t❳⑥✆④☞→✙r⑩t❝♣➓q❩t❇⑧✑⑤✎➛✩r⑩t❝♣✢❷✍♣❢⑥❥q➓✉✩➔❢⑥✠②❾r✛✇❳❷✽♣❢⑧❩③❢⑧♥⑥✆➐▲✉♠④❪→✩➒▲✉♠r↕♣✒❷❼➒✱⑥✆r✆t ➐➣♣✒q✎q✔✉♥④☞→✄➐➣♣↔q✎q❏⑧♠⑥✆➐✁✉■q✷r⑦⑥✆➔➓♣✒q✞➜❾⑧➌♣↔q❏⑧♥④☞❹ ♦❚♣✙④☞⑧♥r➁♣✙r⑩t❝✉♥r▲③✒⑧♠❷✙④❪⑧♠④❇➝❩⑤✎⑧♥④❇❸♠♣✍②❧✈①✇❤②✄③✎✇❳④☞⑤✢r✛⑥❞⑧♥④❝q⑨⑥✆r❏⑥❥q✔❿❇⑧■q✎q↔⑥❞➞✒➐➣♣❽r↕⑧➟t❝✉♠❸♠♣✺✉❚➠✞⑥❞♣✢➔➀✉♠④❇④ q✢⑧♠➐➑✇❳r⑦⑥❞⑧♠④✩➒➡tP⑥❞⑤✎t✩⑥✆④✤❸■⑧♥➐➑❸■♣✒q➢➞✎⑧♠r⑩t✩✉❏q❩t❇⑧✑⑤✎➛➓✉♠④❪→➓✉❏❷✽✉♠❷✍♣✛③✒✉❤⑤✢r✛⑥❞⑧♥④❚➒▲✉♠❸♠♣✑❹✴❺⑩③✞r✆t❇♣✁✈✴✇❤②①③✎✇❳④❪⑤✒r⑦⑥❞⑧♠④ t❝✉❾→s➔➓⑧♥❷✍♣✴⑥✆④✑✈✭♣✽②❤⑥❞⑧♥④❶❿❇⑧♠⑥✆④✤r❞q✎❻✲r⑩t❝♣▲q✢⑧♠➐➑✇❳r⑦⑥❞⑧♠④➓➔❢⑥✠➜➌t❳r✲⑥✆④❇❸♠⑧♠➐➑❸♠♣✴➔➓⑧♥❷✍♣①q✍t❝⑧➌⑤✽➛♠q✴✉♥④☞→s❷✍✉♠❷✽♣❞③✒✉❾⑤✒r⑦⑥❞⑧♠④ ➒▲✉♠❸♠♣✒q✒❹ ➤✺➥✲➦↔➧❽➨❇➩ ➫➓➭♠➯❾➲➎➳➡➵❊➸✩➺❱➧❝➦↔➧♥➭■➻❴➵❞➼✁➯✲➦✍➵↕➥❳➼➽➥❝➾✷➦❩➳▲➧⑨➚➌➳➎➸❾➥✖➪➹➶↕➥◆➸♠➯✲➦❩➵❊➥P➼ ➘❀➴❝➷✭➬✍➮❤➱✚✃❳❐✍❒✚➮❤❮❼❐✍➮➢❐✍➴❇➷❶❰✭❒➣➷✑Ï➀Ð♥❮❝❮❢Ñ❝Ò✽➮❤Ó❝➱✚➷✢ÏÕÔ✑Ð♥❮❼Ó✤➷❶Ö❳➷✢❐✍➷✑Ò✍Ï➓❒✚❮❝➷✑Ö❏×⑩Ò✍➮❾ÏØ❐✍➴❇➷✭Ù❤Ò✎Ð♥Ñ❝➴❼➮♥×❪Ú✁Û❞Ü❪Ý ❒✚❮✙Ð➓➬✍❒➣Ï➓Ñ❝➱✚➷✷Ï➀Ð♥❮❝❮❇➷✢Ò➌Þ ß ➊