)S x=-((uR)(Ur) &=uu u Note 8 The Equal Area Rule One technique that is sometimes useful for determining the discontinuous solu tion is to start with the solution constructed using characteristics. This solution may be multi-valued if characteristics cross. The multi-valued parts can be elim inated by inserting shocks at appropriate locations. The shock location can be determined by the equal area rule which is depicted in the figure below. The shock is placed so that the shaded areas cut off on either side have equal areas 3.3.1 Example 1+t 0 1+t 3.3.2 Variable States SLIDE 24 If u is not const ant at both sides of the shock, the jump condition still applies locally→ shock path isã♣äå✚æ❩ç➎è✎é ã è✓ê✲ë å✚ì❬í ã ç❥î➲ç➎è✎é➲ë ã î➲ç✌è✓ê✓ë✦ë ïPð✷ñ❩ò❯ó➀ô✦õ✎ö➄ö❭÷ ø í å✚ì å✚æ í î➲ç✌è✓é➲ë ã î➲ç➎è✎ê✲ë èé ã èê ùûúî➢ü úè➢ü ý➀þ✘ÿ✁￾✄✂❥ÿ✆☎✞✝✠✟☛✡✌☞✎✍➢ÿ✌✂✏✍✒✑✔✓✖✕✘✗õ✞ò➄ñ✚✙✘÷✜✛✣✢✠✛✤ñ✞✙ ✤✦✥ ✧✩★✫✪✭✬☛✮ ✯✱✰✌☎✳✲✵✴✫✡⑦þ✜✶✸✷✺✹✻☎Pþ♠ý✺✡✌✶✏☎ ✼✽✙✷ö✽✢✦ö➦ò❯ð✜✙✜✛✿✾✕ö✽✢❚ð✘❀❁✢❂✛➓ô✡ôÒñ✗ö❃✢❄✛✗ö❭ô ✕ô✦ö❃❅✕✜❆ ❅✌ñ✞❇✡÷❩ö❃✢❚ö❈❇✗✛❉✙✜✛❉✙✜❊❋✢✦ð✷ö❁÷●✛➓ô❚ò❘ñ✚✙❍✢✠✛■✙✕ñ✕ô❤ôÒñ❆■✕●❏ ✢❄✛❲ñ✚✙❋✛➓ô❑✢✦ñ❁ô▲✢✠❀▼❇❄✢❖◆❂✛✣✢❚ð❋✢✦ð✷ö✡ô✦ñ❆❉✕✢❄✛❲ñ✚✙↔ò❘ñ✞✙✘ô✖✢✠❇✕ò✭✢✦ö➦÷ ✕ô▲✛❉✙✜❊❁ò❯ð✘❀▼❇P❀✼ò❃✢✦ö◗❇❄✛➓ô✖✢✠✛❲ò❭ô❈❘✆❙❤ð✜✛❲ô⑤ôÒñ❆■✕✢✠✛✤ñ✞✙ ✗❀✻❚❱❯➢ö ✗❲✕✜❆✢❄✛ ❏❨❳❀❆■✕ö❭÷❩✛■❅➢ò❯ð✘❀✚❇✠❀◗ò✭✢✦ö◗❇❄✛➓ô▲✢❄✛➓ò➄ôsò❈❇✦ñ❶ô✦ô◗❘✁❙❤ð✷ö ✗❲✕✜❆✢❄✛ ❏❨❳❀❆■✕ö❭÷Óõ✒❀▼❇❄✢❚ô➒ò❈❀▼✙❲❯➢ö❝ö❆ ✛✗❋❏ ✛❉✙✘❀❁✢❚ö❭÷❬❯❭❚✺✛❉✙✘ôÒö◗❇▲✢✠✛■✙✘❊✄ôÒð✷ñ❩ò❯ó❩ô❂❀❁✢✦❀✼õ✷õ✜❇❚ñ✼õ✜❇✠✛✿❀❁✢✦ö ❆ñ❩ò❈❀❁✢✠✛✤ñ✞✙✘ô◗❘❪❙❤ð✷ö✹ô✦ð✷ñ❩ò❯ó ❆ñ❩ò◗❀❁✢❄✛❲ñ✚✙✑ò◗❀▼✙❫❯➢ö ÷❩ö❈✢✦ö❈❇✗✛■✙✘ö❭÷✩❯❭❚✳✢✦ð✘ö❵❴✠❛❈❜✘❝❁❞✸❝❁❡❄❴P❝✳❡✭❜●❞■❴❱◆✡ð✜✛➓ò❯ð❢✛❲ôÓ÷✷ö➄õ✜✛➓ò✭✢❚ö❭÷❢✛❉✙❣✢❚ð✷ö✐❤✘❊✕❇❚ö❋❯✎ö❆ñ❁◆❩❘✺❙❤ð✷ö ô✦ð✷ñ❩ò❯ó❋✛➓ô❤õ❆❀✼ò➄ö❭÷➀ô✦ñ✱✢❚ð✘❀❁✢✸✢❚ð✷ö❁ôÒð✘❀◗÷❩ö❭÷❥❀▼❇❚ö◗❀◗ô➞ò✕✢❝ñ✚❦✞ñ✚✙❱ö❈✛■✢✦ð✷ö◗❇❤ô❄✛➓÷❩ö■ð✘❀❳ö❄ö◗✾✕❀❆ ❀✚❇✦ö❧❀✼ô◗❘ ♠✁♥❉♠✁♥✏♦ ✲✽♣þ✜qsr❑✶✏☎ t✎✉❍✈①✇✁②✩③✜④ ⑤ñ✞❇✡ñ✕❇✡ö❈⑥✜❀✗õ❆ö✚⑦ ø í úî➢ü úè✘ü í ⑨⑧ ã❶⑩ ❷ ⑧ ã ä í ä ❸ ì✎❹✡í ä❻❺ æ ❸ SHOCK ♠✁♥❉♠✁♥✿❼ ❽þ●✹❁✂➎þ✜❾✌✶❿☎✩➀✄✑❭þ●✑❧☎❍➁ t✎✉❍✈①✇✁②✩③❭➂ ➃❅ è ✛➓ô➄✙✷ñ✚✢☛ò❘ñ✞✙✘ô✖✢P❀▼✙❍✢✽❀▼✢✦❯➢ñ✚✢✦ð✞ô❄✛❲÷✷ö❭ô✻ñ✚❅✆✢❚ð✷ö✾ô✦ð✷ñ❩ò❯ó➅⑦●✢✦ð✘ö ✓✖✕✜✗õ✞ò❘ñ✞✙✘÷●✛■✢❄✛❲ñ✚✙✞ô▲✢❄✛ ❆■❆ ❀✵õ✷õ❆ ✛❲ö❭ô ✶✏✍✁➆✼þ✜✶❿✶➈➇➊➉ ô✦ð✷ñ❩ò❯ó✄õ✘❀▼✢✦ð❫✛❲ô ➆✞✡✁✹✻➋✎☎❍➌ ❘ ä✼ä