❲❡✐rs ❛♥❞ ❋❧✉♠❡s ❘❡❝❛♣ ❙t❡❛❞② ♥♦♥✲✉♥✐❢♦r♠ ✌♦✇ ❇r♦❛❞ ❝r❡st❡❞ ❲❡✐rs ❛♥❞ ❋❧✉♠❡s ✇❡✐rs ❚❤✐♥ ✇❡✐rs ❙❖❊✷✶✺✻✿ ❋❧✉✐❞s ▲❡❝t✉r❡ ✼ ❘❡❝t❛♥❣✉❧❛r ♥♦t❝❤ ✇❡✐r ❱❡❡ ♥♦t❝❤ ✇❡✐r ❋❧✉♠❡s
❲❡✐rs ❛♥❞ ❋❧✉♠❡s ❘❡❝❛♣ ❘❡❝❛♣ ❙t❡❛❞② ▲❛st ❧❡❝t✉r❡ ④ ❛ss✉♠❡❞ t❤❡ ❝❤❛♥♥❡❧ ✇❛s ✉♥✐❢♦r♠ ✭❝♦♥st❛♥t ♥♦♥✲✉♥✐❢♦r♠ ✌♦✇ ❞❡♣t❤✱ s❤❛♣❡✱ s❧♦♣❡ ❡t❝✳✮ ④ st❡❛❞② ✉♥✐❢♦r♠ ✌♦✇ ❇r♦❛❞ ❝r❡st❡❞ ❋♦✉♥❞ t❤❛t ✿ ✇❡✐rs ❚❤✐♥ ✇❡✐rs ✎ ❧♦❝❛t✐♦♥ ♦❢ ❢r❡❡ s✉r❢❛❝❡ t♦ ❜❡ ❞❡t❡r♠✐♥❡❞ ❘❡❝t❛♥❣✉❧❛r ♥♦t❝❤ ✇❡✐r ✎ ✷ ♣♦ss✐❜❧❡ s♦❧✉t✐♦♥s ④ s✉❜❝r✐t✐❝❛❧ ❛♥❞ s✉♣❡r❝r✐t✐❝❛❧✳ ❱❡❡ ♥♦t❝❤ ✇❡✐r ❙♦❧✉t✐♦♥ ❝❛♥ ❥✉♠♣ ❜❡t✇❡❡♥ t❤❡s❡ ✭❤②❞r❛✉❧✐❝ ❥✉♠♣✮ ❋❧✉♠❡s ✎ ❋r♦✉❞❡ ♥✉♠❜❡r ✐♠♣♦rt❛♥t ✎ ✌♦✇ ❝❛♥ ❛❧s♦ ❜❡ ❝r✐t✐❝❛❧ ④ ❋ r ❂ ✶ ✎ ❙✐♥❣❧❡ s♦❧✉t✐♦♥ ❢♦r t❤✐s ❝❛s❡ ✿ ♠❛①✐♠✉♠ ✌♦✇ ❢♦r ❛ ❣✐✈❡♥ s♣❡❝✐☞❝ ❡♥❡r❣② ❲❤❛t ✐❢ t❤❡ ❝❤❛♥♥❡❧ ✐s ♥♦♥✲✉♥✐❢♦r♠❄ ④ st❡❛❞② ♥♦♥✲✉♥✐❢♦r♠ ✌♦✇ ❄
❲❡✐rs ❛♥❞ ❋❧✉♠❡s ❙t❡❛❞② ♥♦♥✲✉♥✐❢♦r♠ ✌♦✇ ❋r♦✉❞❡ ♥✉♠❜❡r st✐❧❧ ✐♥❞✐❝❛t❡s ✇❤❡t❤❡r t❤❡ ✌♦✇ ✐s s✉❜✲ ♦r ❘❡❝❛♣ s✉♣❡r✲❝r✐t✐❝❛❧✳ ❆s t❤❡ ❝r♦ss✲s❡❝t✐♦♥ ❝❤❛♥❣❡s✱ t❤❡ ✌♦✇ ❝♦♥❞✐t✐♦♥s ❙t❡❛❞② ♥♦♥✲✉♥✐❢♦r♠ ✇✐❧❧ t♦♦✳ ✌♦✇ ❇r♦❛❞ ❝r❡st❡❞ ❚❤❡ ❢r❡❡ s✉r❢❛❝❡ ♠❛② ♣❛ss t❤r♦✉❣❤ t❤❡ ❝r✐t✐❝❛❧ ❞❡♣t❤✳ P♦✐♥ts ✇❡✐rs ✇❤❡r❡ t❤✐s ❤❛♣♣❡♥s ❛r❡ ❛ ❧✐♠✐t✐♥❣ ❢❛❝t♦r ✐♥ t❤❡ ❞❡s✐❣♥ ♦❢ t❤❡ ❚❤✐♥ ✇❡✐rs ❝❤❛♥♥❡❧✱ ❛♥❞ ❛r❡ ❦♥♦✇♥ ❛s ❝♦♥tr♦❧ s❡❝t✐♦♥s ✳ ❘❡❝t❛♥❣✉❧❛r ♥♦t❝❤ ✇❡✐r ❱❡❡ ♥♦t❝❤ ✇❡✐r d1 dc Hydraulic ❋❧✉♠❡s jump s<sc d2 Critical depth s>sc s<sc ❊①❛♠♣❧❡s ♦❢ t❤✐s ❜❡❤❛✈✐♦✉r ✐♥❝❧✉❞❡ ✇❡✐rs ❛♥❞ ✌✉♠❡s✱ ✉s❡❞ ✭♣r✐♥❝✐♣❛❧❧②✮ t♦ ♠❡❛s✉r❡ ✌♦✇ r❛t❡s ♦❢ ✇❛t❡r✳ ❲❡✐rs ❛r❡ ✈❡rt✐❝❛❧ ♦❜str✉❝t✐♦♥s ④ ❝❛♥ ❜❡ t❤✐♥ ♦r ❜r♦❛❞✲❝r❡st❡❞✳ ❋❧✉♠❡s ❛r❡ ♥❛rr♦✇✐♥❣s ✐♥ t❤❡ ❝❤❛♥♥❡❧✳
❲❡✐rs ❛♥❞ ❋❧✉♠❡s ❇r♦❛❞ ❝r❡st❡❞ ✇❡✐rs ❘❡❝❛♣ ❚②♣✐❝❛❧ ❛r❛♥❣❡♠❡♥t ✿ ❙t❡❛❞② ♥♦♥✲✉♥✐❢♦r♠ Total energy ✌♦✇ ❇r♦❛❞ ❝r❡st❡❞ E ✇❡✐rs h dc ❚❤✐♥ ✇❡✐rs ❘❡❝t❛♥❣✉❧❛r ♥♦t❝❤ ✇❡✐r ❱❡❡ ♥♦t❝❤ ✇❡✐r ❋❧✉♠❡s ✎ ❙✉✍❝✐❡♥t❧② ❜r♦❛❞ t❤❛t t❤❡ ❢r❡❡ s✉r❢❛❝❡ ✐s ♣❛r❛❧❧❡❧ t♦ t❤❡ ❝r❡st ✎ ❋❧♦✇ ✉♣str❡❛♠ tr❛♥q✉✐❧ ④ ❋ r ❁ ✶ ✎ ❋❧♦✇ ❞♦✇♥str❡❛♠ ✉♥✐♠♣❡❞❡❞ ④ ❢r❡❡ ❥❡t ④ ❞✐s❝❤❛r❣❡ ♦✈❡r t❤❡ ✇❡✐r ✇✐❧❧ ❜❡ t❤❡ ♠❛①✐♠✉♠ ♣♦ss✐❜❧❡ ✎ ❚❤✉s ✌♦✇ ♦✈❡r ✇❡✐r ❛t ❝r✐t✐❝❛❧ ❞❡♣t❤
❲❡✐rs ❛♥❞ ❋❧✉♠❡s ❋❧♦✇ r❛t❡ ♦✈❡r ✇❡✐r ❣✐✈❡♥ ❜② ◗ ❂ ❈❜❊ ✸ ❂ ✷ ❘❡❝❛♣ ❙t❡❛❞② ♥♦♥✲✉♥✐❢♦r♠ ✐♥ t❡r♠s ♦❢ t❤❡ ✉♣str❡❛♠ s♣❡❝✐☞❝ ❡♥❡r❣② ❊ ✳ ❯s✉❛❧❧② ❛ss✉♠❡ ✌♦✇ ❱ ❂ ✵ ✉♣str❡❛♠✱ s♦ ❊ ✬ ❍ ✱ ❛♥❞ t❤✉s ❇r♦❛❞ ❝r❡st❡❞ ✇❡✐rs ❚❤✐♥ ✇❡✐rs ◗ ❂ ❈❜❍ ✸ ❂ ✷ ❘❡❝t❛♥❣✉❧❛r ♥♦t❝❤ ✇❡✐r ④ ❣❡♥❡r❛❧ ✇❡✐r ❡q✉❛t✐♦♥ ✳ ❆ s✐♥❣❧❡ ♠❡❛s✉r❡♠❡♥t ♦❢ t❤✐s ✈❛❧✉❡ ❱❡❡ ♥♦t❝❤ ✇❡✐r ❣✐✈❡s t❤❡ ❞✐s❝❤❛r❣❡✳ ❋r♦♠ t❤❡♦r❡t✐❝❛❧ ❛♥❛❧②s✐s ❋❧✉♠❡s ❈ ❂ ✶ ✿ ✼✵✺ ❜✉t t❤✐s ✐s ✉s✉❛❧❧② ❞❡t❡r♠✐♥❡❞ ❢r♦♠ ❡①♣❡r✐♠❡♥t✳ ◆✳❇✳ ❚❤❡ ❞❡♣t❤ ♦✈❡r t❤❡ ❝r❡st ♦❢ t❤❡ ✇❡✐r ✐s ☞①❡❞✳ ❘❛✐s✐♥❣ t❤❡ ✇❡✐r ❝r❡st ✇✐❧❧ ♥♦t ❛❧t❡r t❤✐s✱ ❜✉t ✇✐❧❧ ❛❧t❡r t❤❡ ♦✈❡r❛❧❧ ❞❡♣t❤ ✉♣str❡❛♠✳
❲❡✐rs ❛♥❞ ❋❧✉♠❡s ❚❤✐♥ ❲❡✐rs ❘❡❝❛♣ ❙t❡❛❞② ♥♦♥✲✉♥✐❢♦r♠ ✌♦✇ ❇r♦❛❞ ❝r❡st❡❞ ❆❧t❡r♥❛t✐✈❡ ❛r❛♥❣❡♠❡♥t ④ ✇❡✐r ✐s t❤✐♥ ✭✌❛t ♣❧❛t❡✮ ✇✐t❤ s♣❡❝✐☞❡❞ ✇❡✐rs ♦♣❡♥✐♥❣✳ ❚❤✐♥ ✇❡✐rs ❘❡❝t❛♥❣✉❧❛r ♥♦t❝❤ ✇❡✐r ❆ss✉♠♣t✐♦♥s ✿ ❱❡❡ ♥♦t❝❤ ✇❡✐r ✎ ❋❧♦✇ ❜❡❤✐♥❞ ✇❡✐r ❱ ❂ ✵ ❋❧✉♠❡s ✎ Pr❡ss✉r❡ ✐♥ ♥❛♣♣❡ ❛t♠♦s♣❤❡r✐❝ ✎ ◆♦ ❡♥❡r❣② ❧♦ss❡s ✎ ❙tr❡❛♠❧✐♥❡s ❤♦r✐③♦♥t❛❧ ♦✈❡r ❝r❡st
❲❡✐rs ❛♥❞ ❋❧✉♠❡s ❘❡❝t❛♥❣✉❧❛r ♥♦t❝❤ ✇❡✐r ❘❡❝❛♣ ❙t❡❛❞② ♥♦♥✲✉♥✐❢♦r♠ ✌♦✇ ∆ ❇r♦❛❞ ❝r❡st❡❞ H1 H ✇❡✐rs δ h ❚❤✐♥ ✇❡✐rs ❘❡❝t❛♥❣✉❧❛r ♥♦t❝❤ ✇❡✐r ❱❡❡ ♥♦t❝❤ ✇❡✐r ❙t❛rt ❢r♦♠ ❋❧✉♠❡s ◗ ❂ ❆❱ ❈♦♥s✐❞❡r t❤❡ t❤✐♥ str✐♣ ♠❛r❦❡❞ ❆ ❂ ❜ ✍ ❤
❩ ❍ ✷ ❣ ✷ ♣ ♣ ✸ ❍ ✸ ❂ ✷ ❂ ❜ ✷ ❣❤ ❞❤ ❂ ❜ ✵ ❲❡✐rs ❛♥❞ ❋❧✉♠❡s ❘❡❝❛♣ ❆♣♣❧②✐♥❣ ❇❡r♥♦✉❧❧✐ ❣✐✈❡s ❙t❡❛❞② ♥♦♥✲✉♥✐❢♦r♠ ✌♦✇ ♣ ❱ ❂ ✷ ❣❤ ❇r♦❛❞ ❝r❡st❡❞ ✇❡✐rs ❚❤✐♥ ✇❡✐rs s♦ ❘❡❝t❛♥❣✉❧❛r ♣ ✍ ◗ ❂ ❜ ✍ ❤ ✷ ❣❤ ♥♦t❝❤ ✇❡✐r ❱❡❡ ♥♦t❝❤ ✇❡✐r ❛♥❞ t❤✉s ❋❧✉♠❡s ❩ ❍ ◗ ❂ ❞◗ ✵
✷ ❣ ✷ ♣ ✸ ❍ ✸ ❂ ✷ ❂ ❜ ❲❡✐rs ❛♥❞ ❋❧✉♠❡s ❘❡❝❛♣ ❆♣♣❧②✐♥❣ ❇❡r♥♦✉❧❧✐ ❣✐✈❡s ❙t❡❛❞② ♥♦♥✲✉♥✐❢♦r♠ ✌♦✇ ♣ ❱ ❂ ✷ ❣❤ ❇r♦❛❞ ❝r❡st❡❞ ✇❡✐rs ❚❤✐♥ ✇❡✐rs s♦ ❘❡❝t❛♥❣✉❧❛r ♣ ✍ ◗ ❂ ❜ ✍ ❤ ✷ ❣❤ ♥♦t❝❤ ✇❡✐r ❱❡❡ ♥♦t❝❤ ✇❡✐r ❛♥❞ t❤✉s ❋❧✉♠❡s ❩ ❍ ❩ ❍ ♣ ◗ ❂ ❞◗ ❂ ❜ ✷ ❣❤ ❞❤ ✵ ✵
❲❡✐rs ❛♥❞ ❋❧✉♠❡s ❘❡❝❛♣ ❆♣♣❧②✐♥❣ ❇❡r♥♦✉❧❧✐ ❣✐✈❡s ❙t❡❛❞② ♥♦♥✲✉♥✐❢♦r♠ ✌♦✇ ♣ ❱ ❂ ✷ ❣❤ ❇r♦❛❞ ❝r❡st❡❞ ✇❡✐rs ❚❤✐♥ ✇❡✐rs s♦ ❘❡❝t❛♥❣✉❧❛r ♣ ✍ ◗ ❂ ❜ ✍ ❤ ✷ ❣❤ ♥♦t❝❤ ✇❡✐r ❱❡❡ ♥♦t❝❤ ✇❡✐r ❛♥❞ t❤✉s ❋❧✉♠❡s ❩ ❍ ❩ ❍ ✷ ❣ ✷ ♣ ♣ ✸ ❍ ✸ ❂ ✷ ◗ ❂ ❞◗ ❂ ❜ ✷ ❣❤ ❞❤ ❂ ❜ ✵ ✵
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