Problem 58324. Compute the critical depth of a channel

Created by ChrisR

Solve Later

{"parts":[{"partUri":"/matlab/document.xml","contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?><w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Problem statement</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Write a function to compute the critical depth of a channel with discharge </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"Q\"/></w:customXmlPr><w:r><w:t>Q</w:t></w:r></w:customXml><w:r><w:t>. The unit system will be specified in units as either ‘SI’ or ‘USCS’ (U.S. customary system); take the acceleration of gravity </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"g\"/></w:customXmlPr><w:r><w:t>g</w:t></w:r></w:customXml><w:r><w:t> to be either 9.81 </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"m/s2\"/></w:customXmlPr><w:r><w:t>\\rm m/s^2</w:t></w:r></w:customXml><w:r><w:t> or 32.2 </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"ft/s2\"/></w:customXmlPr><w:r><w:t>\\rm ft/s^2</w:t></w:r></w:customXml><w:r><w:t>. The geometry of the channel’s cross section will be specified by a structure channelStruct as in </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/58314-compute-the-normal-depth-of-a-channel\"><w:r><w:rPr><w:u/></w:rPr><w:t>Cody Problem 58314</w:t></w:r></w:hyperlink><w:r><w:t>.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Background</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Specific energy for a flow is </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"E = y + V^2/2g\"/></w:customXmlPr><w:r><w:t>E = y + V^2/2g</w:t></w:r></w:customXml><w:r><w:t>, where </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"y\"/></w:customXmlPr><w:r><w:t>y</w:t></w:r></w:customXml><w:r><w:t> is the water depth and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"V\"/></w:customXmlPr><w:r><w:t>V</w:t></w:r></w:customXml><w:r><w:t> is the velocity averaged over the cross section. The critical depth is the depth of minimum specific energy. Using the definition of the average velocity </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"V = Q/A\"/></w:customXmlPr><w:r><w:t>V = Q/A</w:t></w:r></w:customXml><w:r><w:t> and differentiating with respect to depth gives</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"dE/dy = 0 = 1+(-2) (Q^2/2gA^3)dA/dy = 1-(Q^2/gA^3)dA/dy\"/></w:customXmlPr><w:r><w:t>\\frac{dE}{dy} = 0 = 1 + (-2)\\frac{Q^2}{2gA^3}\\frac{dA}{dy} = 1-\\frac{Q^2}{gA^3}\\frac{dA}{dy}</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Then using the definition of the top width </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"T = dA/dy\"/></w:customXmlPr><w:r><w:t>T = dA/dy</w:t></w:r></w:customXml><w:r><w:t> gives the condition for the critical depth in terms of a dimensionless parameter called the Froude number </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"Fr\"/></w:customXmlPr><w:r><w:t>Fr</w:t></w:r></w:customXml><w:r><w:t>:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"Fr^2 = Q^2T/gA^3 = 1\"/></w:customXmlPr><w:r><w:t>Fr^2 = \\frac{Q^2T}{gA^3} = 1</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Flows with depths smaller than critical are supercritical—fast and shallow (</w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"Fr &gt; 1\"/></w:customXmlPr><w:r><w:t>Fr &gt; 1</w:t></w:r></w:customXml><w:r><w:t>), and flows with depths greater than critical are subcritical—deep and slow (</w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"Fr &lt; 1\"/></w:customXmlPr><w:r><w:t>Fr &lt; 1</w:t></w:r></w:customXml><w:r><w:t>). A flow changes from supercritical to subcritical with a hydraulic jump, which can be observed in the </w:t></w:r><w:hyperlink w:docLocation=\"https://www.youtube.com/watch?v=qsC9FUYpHqU\"><w:r><w:t>field</w:t></w:r></w:hyperlink><w:r><w:t>, in the </w:t></w:r><w:hyperlink w:docLocation=\"https://youtu.be/5etwhZ0d2GU?t=373\"><w:r><w:rPr><w:u/></w:rPr><w:t>laboratory</w:t></w:r></w:hyperlink><w:r><w:t>, and even in a </w:t></w:r><w:hyperlink w:docLocation=\"https://youtu.be/OoA1ASjMfag?t=24\"><w:r><w:rPr><w:u/></w:rPr><w:t>kitchen sink</w:t></w:r></w:hyperlink><w:r><w:t>. </w:t></w:r></w:p></w:body></w:document>","relationship":null}],"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","target":"/matlab/document.xml","relationshipId":"rId1"}]}

<div style = "text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; "><div style="block-size: 409.2px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 204.6px; transform-origin: 407px 204.6px; vertical-align: baseline; "><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; ">Problem statement</div><div style="block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 226.642px 8px; transform-origin: 226.642px 8px; unicode-bidi: normal; ">Write a function to compute the critical depth of a channel with discharge Q<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 139.633px 8px; transform-origin: 139.633px 8px; unicode-bidi: normal; ">. The unit system will be specified in units as either ‘SI’ or ‘USCS’ (U.S. customary system); take the acceleration of gravity g<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 54.45px 8px; transform-origin: 54.45px 8px; unicode-bidi: normal; "> to be either 9.81 <img src="data:image/png;base64,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" alt="m/s2" style="width: 33px; height: 19.5px;" width="33" height="19.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 25.6667px 8px; transform-origin: 25.6667px 8px; unicode-bidi: normal; "> or 32.2 <img src="data:image/png;base64,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" alt="ft/s2" style="width: 30.5px; height: 19.5px;" width="30.5" height="19.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17.8833px 8px; transform-origin: 17.8833px 8px; unicode-bidi: normal; ">. The geometry of the channel’s cross section will be specified by a structure channelStruct as in <a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/58314-compute-the-normal-depth-of-a-channel">Cody Problem 58314</a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; ">.</div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.8333px 8px; transform-origin: 40.8333px 8px; unicode-bidi: normal; ">Background</div><div style="block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 87.125px 8px; transform-origin: 87.125px 8px; unicode-bidi: normal; ">Specific energy for a flow is <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAL4AAAAnCAYAAACxHfIWAAAIKElEQVR4Xu2d28uvUxDH9/4L5HAlSQ4XckHOiQsuEBJRiAtFTrmhHLJJytmFcuEUd4Rwo4gUhZTDBUWUQ8KlQ/IHMJ/8RrPXXmvNWut51u/9ve+znpr2731/6zjznVkzs+Z59+5d4xkcWCAHdi9wz2PLgwO7BvAHCBbJgQH8RYp9bHoAf2Agx4Fj5ctThfYT+lno5Z3CrgH8nSLJ+ffxpAx5udCnQueshv9c/r1W6MuG6a6XPr8IvdXQd/YuA/izs3RHDHi77OL4FfDZ0IFCbwudKPSO0LkNu/xD+ty0KafGAH6DBBfQ5TPZ42VCP5q94vZ8sfr5IPn39wo+nCdtXxA6oKJP16Ye8NH81gfL0HIkts635H64JIcmGPCX/P6ZyHcp2T4vbU8Rirkk/zQCH7eJB4ufemw8QZuPhT7qJVQP+MzLgl4XOsIs4in5/FOwqMPk57NNO9pbi9FrD2Pc/ziAVX3TMOOHlTxyMkBhXlr1wVBdKZSz5AAfP/+kCqbjJv0mdL5QTJkOl9+jGBpH2KGnxBTZJZYAnwEeELprNVJu4+oLAvqNOdYqhLTdm+Ki4IfzPCi0x9mQui9/rvrllOR0afOh0A1CsRMkNRXKdb/QkZEG4OU7of2FwBVKRztrZFHgS4Rm9R5KgU9goxrpMZQj9CyhlgBouwOvdP2AiOeblbBL+3ntsJw3rhp5cqKZGrQSMDM2lr7G2jMHKVDAG1NCvsNLODMANvx5XEiV+BX5jALN9pQCX307Jj5DKOd7AXzyvp61mW0T23Ag5edxgcCnboWU4dOrQTyXRK0toPTAzMnwfgSg3npxYxg/ts/cd4yr3/OZE+kooZqAOru2EuDrEVe6ABZMQDXbIj3ubsPvewHfygrAEaSm5KCng6d8KMiLQrgrtcEmikjeP6ZYJQaSE4HsEo+3zioYlADf+vepHC4bfG2AvZj3vYCvgaQuJJV2VN++xB0CfFj7Gr9e5yfmeC7RVy/HcnEFyvHIVgHfBkwxX5ANXCQ0qw9WDKF5GnJKHSx0iBBpwUcjw6LcuHCk+6aeZr2Az7K5KCJY5Em5pcRsJwt57gOnAkFlCHoySJ84fFDlmpLdU+Dj6pQkSzjxTlvtHa/jXaGoYnkWP7Qg4XHD9zDgsQhzaiGZy0XXjBUDba6/znudNNJsQph6sy7EHIFWT+DbREQshahpzytkv7naG0AP2EKDBi9ujvw+5DGeAvycYhDV1fFOJozSQ0IoPVj8WuhuIRIyeCk8KMD/9wge8G2el853mN1h/fC/2NwUrdYhrcBqgG7bllqG2Pg2tRZz6dQnbr2yt3P2BL51TUPAqKH6XhaTy7rZIDnGK09p6MMcgG9KYZueXjl8qVzCANgaK+6dPrBr8YBv02NkCV41XKCWA+ATRMVytLXgncPip24pS9diQRNjNsr5nlDtqRLO3xP4FrQI3N6W4jrcKRSmD+36UI5rHIZ5+9cSBc+Vyk2jRhdjm5rPgjtURvvdPiefB3y0Vo//mJbj/6MMHiNKgbfV7Ww9Smy/8KPkMsVTYg3YsMgoa+pp4asVuE1pEsfwM8rQO9VcUqKQk7U9mXK3ySlrz9jWAFQB34KAgWIZAoB/i1BtmmurAZ6bX5U99OX1UsXLeTP2VrptNv9tU5p6WVQSJE6Rj7qMV8kgrSXIAJqLrVw61vI5dmeh8UH0DiBn8UsuQ1jgPUJTsxxTGD1335QVAcxPFAqz1OLHap50P1PcNnvhiME6Wohyg1S9zJw8zJUolMxDf2TAra1X66WyCoFvlT8aGOeAby8Pcn5WyWZK2nhgKRmDNi3ugR3bBvSaxaqx9iXr7OnjhycOYL9PCOs/JcNSsi/a5EoUvDFqQM9Y6tbZWiO9cCOjk8zA5YBfU6agGyJ4ai1H3kr3wArEWgu9t8Clu7fQ2nvC5fvewLdJCeRBzr7EgpasPdfGK0PI9a0FvY5FIM1JDPiJN0lL8zwrlDSCKeDXlikw0dRIfg6LP8U9sEJRP59jkndNLxaas+iuN/DtjSf7WseJzTy4x7cJ1Wb5wNsbjnKiVChwmB4lE0dhHtmrX4X+FnIrOVPAt4wryVtrEROLyr1sYMG1yZ/VYuIe8JRkcmr20xv4Yc1OLRBr9mLb5koUUmMqdnLBsJa7h6eu4rTk3YO95k8B35YphLlgOwBaeKmQpue8ys1Whq67nw3se1jL3sC37tq6ZKJz1lxmKuiRLy+1px4sPZdZoQLbOBRXh5+x9m7dWAh8NAsga2krC2FAwG/zzdzakm7SemnazXWRtW6Qx+ZTi9lrT72Bz56YI2e05uYzLscJQqUuoYJe64q89cQMkLrX9LXjgNmHhYp9fI6O1qfrO5Kti2rspz7nhdK/xx3FOoBPzESR1rpSzbUlCrUxnS0OxEDfKkTlgOb6UYILhPRFHESfrPHxbm4bcbPtu00pxS3ZfK83sErm7tFGT8jav77QuhZccVyqWEkELhdpTLyR5AssA/j7sp5Tz/5NmVbhLKnf1BKFGl7pPUuuYtPGOFFlHMDfm+VTbx1rBLiT2hJ4TilRqOGFZnJywNdy+mRGcunA1+wVQSDZAHLQXn1IjZCW0FYvnnrXACkvtYYsl8LkBGJdyRht6cC3byuV/F2ZJQC5do9TShRq59L2qmzIjxtaEis8vEF3tRAWP/xLcHvNtXTg6z3ETspItYKptR8ZqllfBK9YCEH1MUKk13m+EvpWyCtuG/8xRAWTR9M4B3A93BKBTWPe0i3+psljrGdNHBjAXxOjxzSbxYEB/M2Sx1jNmjjwL4e77DeFMTLgAAAAAElFTkSuQmCC" alt="E = y + V^2/2g" style="width: 95px; height: 19.5px;" width="95" height="19.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24.8917px 8px; transform-origin: 24.8917px 8px; unicode-bidi: normal; ">, where y<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 72.7333px 8px; transform-origin: 72.7333px 8px; unicode-bidi: normal; "> is the water depth and V<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 122.908px 8px; transform-origin: 122.908px 8px; unicode-bidi: normal; "> is the velocity averaged over the cross section. The critical depth is the depth of minimum specific energy. Using the definition of the average velocity <img src="data:image/png;base64,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" alt="V = Q/A" style="width: 61.5px; height: 18.5px;" width="61.5" height="18.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; "> and differentiating with respect to depth gives</div><div style="block-size: 36.6px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.3px; text-align: left; transform-origin: 384px 18.3px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><img src="data:image/png;base64,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" alt="dE/dy = 0 = 1+(-2) (Q^2/2gA^3)dA/dy = 1-(Q^2/gA^3)dA/dy" style="width: 257.5px; height: 36.5px;" width="257.5" height="36.5"></div><div style="block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 127.583px 8px; transform-origin: 127.583px 8px; unicode-bidi: normal; ">Then using the definition of the top width <img src="data:image/png;base64,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" alt="T = dA/dy" style="width: 70px; height: 18.5px;" width="70" height="18.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 211.6px 8px; transform-origin: 211.6px 8px; unicode-bidi: normal; "> gives the condition for the critical depth in terms of a dimensionless parameter called the Froude number <img src="data:image/png;base64,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" alt="Fr" style="width: 18px; height: 18px;" width="18" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; ">:</div><div style="block-size: 36.6px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.3px; text-align: left; transform-origin: 384px 18.3px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><img src="data:image/png;base64,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" alt="Fr^2 = Q^2T/gA^3 = 1" style="width: 95.5px; height: 36.5px;" width="95.5" height="36.5"></div><div style="block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 230.283px 8px; transform-origin: 230.283px 8px; unicode-bidi: normal; ">Flows with depths smaller than critical are supercritical—fast and shallow (<img src="data:image/png;base64,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" alt="Fr > 1" style="width: 44px; height: 18px;" width="44" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 114.742px 8px; transform-origin: 114.742px 8px; unicode-bidi: normal; ">), and flows with depths greater than critical are subcritical—deep and slow (<img src="data:image/png;base64,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" alt="Fr < 1" style="width: 44px; height: 18px;" width="44" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 240.533px 8px; transform-origin: 240.533px 8px; unicode-bidi: normal; ">). A flow changes from supercritical to subcritical with a hydraulic jump, which can be observed in the <a target='_blank' href = "https://www.youtube.com/watch?v=qsC9FUYpHqU">field</a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 22.9417px 8px; transform-origin: 22.9417px 8px; unicode-bidi: normal; ">, in the <a target='_blank' href = "https://youtu.be/5etwhZ0d2GU?t=373">laboratory</a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 47.8417px 8px; transform-origin: 47.8417px 8px; unicode-bidi: normal; ">, and even in a <a target='_blank' href = "https://youtu.be/OoA1ASjMfag?t=24">kitchen sink</a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; ">. </div></div></div>

Problem statement
Write a function to compute the critical depth of a channel with discharge . The unit system will be specified in units as either ‘SI’ or ‘USCS’ (U.S. customary system); take the acceleration of gravity  to be either 9.81  or 32.2 . The geometry of the channel’s cross section will be specified by a structure channelStruct as in Cody Problem 58314.
Background
Specific energy for a flow is , where  is the water depth and  is the velocity averaged over the cross section. The critical depth is the depth of minimum specific energy. Using the definition of the average velocity  and differentiating with respect to depth gives

Then using the definition of the top width  gives the condition for the critical depth in terms of a dimensionless parameter called the Froude number :

Flows with depths smaller than critical are supercritical—fast and shallow (), and flows with depths greater than critical are subcritical—deep and slow (). A flow changes from supercritical to subcritical with a hydraulic jump, which can be observed in the field, in the laboratory, and even in a kitchen sink.

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Problem 58324. Compute the critical depth of a channel

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