A Manning's equation open channel calculator for your team's phones

Your Manning's equation sheet works on your laptop: section type, depth, n, slope, and out comes the flow. The crew in the field needs the same numbers on a phone, without Excel, without the file, and without someone's Goal Seek settings. This guide gives the formulas with their sources, the geometry of each section, a solver that replaces Goal Seek, a test table to check the result against your sheet, and the prompt to give your AI.

The FHWA, USBR, and Microsoft pages quoted here were read on November 6, 2026. Every number in the examples and the test table was computed with our own script and checked against a published worked example where one exists.

Manning's equation, with its units

Manning's equation gives the mean velocity, and the flow, of steady uniform flow in an open channel. FHWA's Introduction to Highway Hydraulics (HDS-4, 2008, section 4.3.1, equations 4.1 and 4.5) writes it as:

V = (k / n) · R^(2/3) · S^(1/2)
Q = V · A = (k / n) · A · R^(2/3) · S^(1/2)
R = A / P

Why 1.486. n is the same number in both systems, so the constant carries the unit change: 1.486 is the cube root of 3.2808, the feet in a metre. HDS-4 rounds it to 1.49 ("Ku = Units conversion factor equal to 1 (1.49 in English units)"), and the USBR Water Measurement Manual (chapter 13, section 6) writes Q = (1.486/n)·A·R_h^(2/3)·S^(1/2) with "Q = discharge (ft3/s)". The two constants differ by 0.27%, so a sheet built on 1.49 and a page built on 1.486 don't match in the fourth digit. Write down which one your sheet uses.

Where it holds. FHWA's older Design Charts for Open-Channel Flow (HDS-3, 1961, section 2.3) says the equation "will give a reliable estimate of velocity only if the discharge, channel cross section, roughness, and slope are constant over a sufficient distance to establish uniform flow conditions." Near a culvert, a drop, a bend, or a change of section, the depth is set by the control, not by Manning. Enter slope as a decimal: 0.3% is 0.003.

Area, wetted perimeter, and top width of each section

The calculator needs A and P for R, and the top width T for the Froude number and critical depth. HDS-4 gives the trapezoid as equations 4.8 to 4.10, with bottom width B, depth y, and side slope Z (horizontal to 1 vertical). A rectangle is the same section with Z = 0.

For the pipe, θ is the central angle the water surface cuts, in radians: θ = 2·acos(1 − 2y/D). The same formulas are in Ed Maurer's open textbook Hydraulics and Water Resources: Examples Using R (Santa Clara University, section 5.4, equations 5.13 to 5.16), which adds "Be sure to use theta in radians." At y = D/2, θ = π and the pipe is half full; at y = D, θ = 2π and T = 0.

A box culvert running part full is a rectangle whose top is not wetted. HDS-3 warns that its pipe charts "are not suitable for use in connection with most types of culvert flow, since culvert flow is seldom uniform" (section 5.1), and the same goes for a calculator built on Manning alone.

Manning's n: put your own table in the file

Use the n table your agency or client accepts, and write it into the file so the crew picks a lining from a list. FHWA's HDS-4 gives a short one in Appendix B, Table B.2 for channels and Table B.3 for closed conduits. Some of its values:

Table B.3 adds that "Concrete pipe with poor joints and deteriorated walls may have n values of 0.014 to 0.018." For natural channels and floodplains, HDS-4 points to the USGS guide by Arcement and Schneider (Water-Supply Paper 2339). The USBR manual says the selection of n is "at best, an estimate," so give the page a custom n field next to the list.

Normal depth: a solver in place of Goal Seek

The usual field question runs the other way: this flow, this channel, how deep? Depth is inside both A and R, so there is no formula to rearrange. HDS-3 says "the solution requires repeated trials" (section 2.3), and HDS-4's Example Problem 4.2 tries y = 0.70 m (Q = 4.03 m³/s, too much), then y = 0.60 m (2.96 m³/s), and stops there.

In Excel that's Goal Seek. Two things about it matter when the sheet moves to a phone. Microsoft's Goal Seek help lists only desktop versions under Applies to (Excel for Microsoft 365, 2024, 2021, 2019, and 2016, some for Mac as well), with no Excel for the web, iPad, iPhone, or Android, and says "Goal Seek works only with one variable input value." Its precision page says Goal Seek uses "iteration in a controlled way" without saying where it stops. A JavaScript page should state its own stopping rule instead.

Bisection does the job for every section here, because Q rises with depth (up to a pipe's peak, below). Start with a bracket that holds the answer, halve it, keep the half where Q crosses the target, and stop when the bracket is narrow enough:

// Normal depth by bisection. qAt(y) returns Manning's Q at depth y.
// lo..hi must bracket the answer: qAt(lo) <= target <= qAt(hi).
function normalDepth(qAt, target, lo, hi) {
  if (qAt(hi) < target) return { depth: NaN, note: "Q is more than this section carries" };
  for (let step = 1; step <= 100; step++) {
    const mid = (lo + hi) / 2;
    if (qAt(mid) > target) hi = mid; else lo = mid;
    if (hi - lo <= 1e-9 * hi) return { depth: (lo + hi) / 2, steps: step };
  }
  return { depth: (lo + hi) / 2, steps: 100, note: "stopped at the step limit" };
}

On HDS-4's Example 4.2 (B = 1.5 m, Z = 2, n = 0.013, S = 0.002, Q = 3 m³/s), this function with a bracket of 0 to 2 m stops after 32 steps at y = 0.6040 m, V = 1.834 m/s. HDS-4 stops its trials at 0.60 m and 1.85 m/s. A tolerance of 1e-6 of the depth takes 22 steps, and 1e-3 takes 12; the 100-step cap only guards against a bracket that doesn't hold the answer. Newton's method gets there in fewer steps but needs the derivative and can step outside the section, so it needs a bisection fallback anyway. For a channel with no top, start the bracket at 0 to 1 m (or 1 ft) and double its top until Q there passes the target, at most 60 times. A slope of zero or less never reaches the target, and an n of zero or less isn't a roughness: the page should say so and give no depth. If your sheet checks bank height, add it to the prompt as an input and have the page flag a depth above it.

Critical depth and the Froude number

Two extra outputs tell the crew whether a Manning depth is believable. HDS-4 defines the Froude number as Fr = V / √(g·y) (equation 4.16). For a section that isn't rectangular, use the hydraulic depth A/T for y, as HDS-4 does at critical flow (equations 4.25 and 4.26), and as the USBR manual describes the hydraulic mean depth: "For use in Froude number and energy relationships." Fr below 1 is subcritical, above 1 supercritical.

Critical depth for any section comes from HDS-4's equation 4.24, A³/T = Q²/g at the critical depth. For a rectangle it reduces to y_c = (q²/g)^(1/3) with q = Q/b (equation 4.21); for a trapezoid or a pipe, solve it with the same bisection. HDS-4 adds: "Uniform flow within about 10 percent of the critical depth is unstable and should be avoided in design." Have the page flag that band. Use g = 9.81 m/s² or 32.2 ft/s² (HDS-3 lists 32.2), and keep it the same as your sheet's.

A pipe flowing partly full carries the most at about 0.94 D

With a constant n, a circular pipe carries more just before it runs full than when full, because the last sliver of depth adds more wetted perimeter than area. HDS-3 (section 5.2) puts the maximum at "a depth of 0.94 diameter"; HDS-4 (section 7.1) says "the peak flow occurs at 93 percent of the height of the pipe, and the average velocity flowing one-half full is the same as gravity full flow." Our script, with a golden-section search over y/D:

Three things follow for the calculator:

• Two depths carry the same Q above 0.82 D. HDS-3 says the same of its charts: "If dn is greater than 0.82 diameter, two values of dn will be shown by the slope line hook for a particular value of Q." Search for normal depth from 0 to 0.938·D only, and report the lower depth.
• A flow above 1.076 × Q full has no normal depth. The pipe surcharges and flows under pressure. Have the page say so instead of printing a number.
• The peak assumes n doesn't change with depth. Maurer's Example 5.4 solves a 0.2 m pipe both ways: 0.158 m with a constant n and 0.174 m with an n that varies with depth. If your sheet uses a depth-varying n, give the AI that table too.

The examples, checked against FHWA's

Our script computed each case below; the published values are in brackets. The small differences are rounding in the published steps (HDS-4 rounds R to 0.35 m before the power) and the 1.49 constant.

• HDS-4 Example 4.1, SI. Trapezoid, B = 2 m, Z = 2, S = 0.003, y = 0.5 m, n = 0.02: A = 1.500 m², P = 4.236 m, R = 0.3541 m, V = 1.371 m/s [1.36], Q = 2.056 m³/s [2.0].
• HDS-4 Example 4.1, US customary. B = 6.5 ft, y = 1.6 ft, with 1.49: V = 4.444 ft/s [4.45], Q = 68.97 ft³/s [69.06]. With 1.486: 4.432 ft/s and 68.79 ft³/s.
• HDS-4 Example 4.2, normal depth. SI: y = 0.6040 m [0.60]. US customary, B = 5 ft, Q = 105 ft³/s, with 1.49: y = 1.959 ft [1.96], V = 6.010 ft/s [6.01].
• Maurer Examples 5.1 and 5.2. Trapezoid, b = 10 ft, side slope 2, n = 0.016, S = 0.0006, Q = 225 ft³/s: normal depth 3.406 ft with 1.486 (3.402 ft with 1.49) [3.406]; critical depth 2.154 ft [2.154].
• Maurer Example 5.10. Rectangle, b = 3 m, Q = 8 m³/s: critical depth 0.8983 m [0.898].
• HDS-4 Example 7.1. Concrete pipe, D = 0.6 m, n = 0.013, S = 0.005, full: Q = 0.434 m³/s [0.43]. At Q = 0.40 m³/s the depth is 0.454 m and V/V full = 1.135 [1.13, read from HDS-4's Figure 7.1].

An AI writes the code fast, and its arithmetic still needs this check. On November 6, 2026, we asked ChatGPT, in Spanish, how to make a Manning's open channel calculator for a team. Both of its answers printed a worked example that doesn't compute. One used HDS-4's Example 4.1 channel (b = 2 m, z = 2, y = 0.5 m, n = 0.020, S = 0.003) and printed A = 1.00 m² and Q = 0.356 m³/s, where FHWA gets 1.5 m² and 2.0 m³/s. The other printed Q = 1.42 m³/s for a trapezoid (side slope 1.5, n = 0.025) that computes to 1.529 m³/s.

The prompt for ChatGPT, Claude, or Gemini

Paste your sheet's formulas with each cell's name and unit (in Excel, Formulas > Show Formulas, or =FORMULATEXT(B13) next to each result), your n table, and the test table below. Then add this:

Write one self-contained HTML file: a Manning's equation calculator my field crew opens on a phone. Use the formulas and tables I pasted.

Rules
- Plain JavaScript functions only. No eval, no new Function, no formula-parsing library, no fetch or other network request. Write every table (our n values, pipe sizes) into the file as a JavaScript array.
- Use Math.pow or ** for powers, never ^ (in JavaScript ^ is bitwise XOR).
- A units switch: SI (k = 1, g = 9.81 m/s², metres) and US customary (k = 1.486, g = 32.2 ft/s², feet). Every input and output shows its unit. Slope is entered as a decimal, with the percent shown next to it.
- Sections: rectangle (b), trapezoid (b, side slope z as z horizontal to 1 vertical), circular pipe flowing partly full (D) with theta = 2*acos(1 - 2y/D) in radians, A = D^2 (theta - sin theta)/8, P = D theta/2, T = D sin(theta/2).
- Two modes. Depth given: compute Q and V. Q given: find the normal depth by bisection. For a pipe, bracket 0 to 0.938 D. For a rectangle or trapezoid, bracket 0 to 1 (metre or foot) and double the top until Q there is above the target, at most 60 times. With S <= 0, give no depth and say uniform flow needs a positive slope; with n <= 0, give no depth and say n must be above 0. Stop when the bracket is narrower than 1e-9 of the depth or after 100 steps, and show the step count. If Q is more than a pipe carries at 0.938 D, give no depth: say so, and show that largest Q.
- Always show A, P, R, T, V, the Froude number V/sqrt(g A/T), and the critical depth y_c from A^3/T = Q^2/g (same bisection, up to D for a pipe). Flag "supercritical" when Fr > 1, and "within 10% of critical depth" when |y - y_c| is 0.1 y_c or less.
- n: a list from our table, plus a field for a custom value.
- Phone layout: one column, inputs first, then a results card with the intermediate values. Put the calculator's version number and date in the footer.
- A "Run the test cases" button at the bottom of the page: it runs the cases from my test table and shows expected, computed, and the difference in percent for each.

The page computes in the phone's browser from what's in the file. On Convika, a script's fetch and XHR requests reach only the page's own address, plus, on Pro, the analytics tools Convika supports and the hosts you list for your own ad network, so a live link to your sheet, or an n table on another site, won't load. Scripts from a CDN do run; eval and new Function don't.

The test table: check the page against your sheet

Run each case through your spreadsheet and through the page, and compare to 4 significant figures. These expected values come from our script with k = 1.486, g = 9.81 m/s² and 32.2 ft/s²; add your own cases from jobs you have already checked.

Cases 3 and 5 show why the flag matters: HDS-4's own Example 4.2 sits 5.5% above its critical depth, and the half-full pipe in case 5 is almost exactly at it. Keep the table in a file next to the calculator and run it again after every change.

Where a sheet and a page drift apart

• ^ is not a power in JavaScript. Excel's R^(2/3) copied as is turns into bitwise XOR, and MDN's reference says the operator "converts both operands to 32-bit integers." 0.3541 ^ (2/3) gives 0, so Q comes out as 0 for any R below 1. The prompt rules it out; check the code for it anyway.
• Radians. Excel's ACOS returns "radians in the range 0 (zero) to pi" (Microsoft), and so does JavaScript's Math.acos, but a sheet that converts θ with DEGREES for display can leak degrees into the formula.
• A Goal Seek result is a number saved in the sheet. It holds the depth from the last run, at whatever point that run stopped. The page solves again on every input, so compare the two to the tolerance you need, not to the last digit.
• Blank cells. Excel reads an empty input cell as 0; a page reads an empty field as nothing. Have the page refuse to compute until every input has a number.
• Percent slope. A sheet formatted as % stores 0.003 and shows 0.3%. The page should take one form and show the other.
• Precision. Excel "stores and calculates with 15 significant digits of precision" (Microsoft), and JavaScript numbers are the same 64-bit doubles, so matching formulas match to well past the fourth digit. A bigger difference means a different formula, constant, or input.

Ready-made calculators, your sheet, and your own page

The AI answers we studied on November 6, 2026 pointed field staff to calculators like these. They are good for a quick check; your sheet is what carries your n table, your sections, and the method your reviewer signed off.

Anyone can open the three ready-made calculators. Your Excel file opens for whoever you send it to, with the formulas showing. Your page opens for whoever its Who can open this page? setting allows, below.

Publishing it for the crew

• Who can open it. When you publish, Who can open this page? offers Public ("Anyone with the link can open it. It may show up in search results and in Convika's gallery."), Private link ("Anyone with the link can open it, with no password or account. It stays out of search results and out of the gallery."), and Only people with the password. A private link keeps the calculator out of search; it doesn't stop someone the link is forwarded to. For a client's design data, choose the password, which Convika's server checks before it sends the page. Both are Pro, at $12 a month. A Public page is free, with the Convika bar.
• One link for every version. When n values or a section change, upload the corrected file as a new version of the same page and click Publish v2 at the same link (with your version number). The crew's link and any QR code keep working. Put the version number and date in the page's footer, so a screenshot from the field says which version made it. Updating a page at the same link covers versions and rollback.
• A second calculator. On Free, 1 page can be live at a time. Publishing a second Public calculator, such as a slurry or culvert one, opens a choice: Light, an add-on to Free, keeps up to 5 public pages live for $3 a month, or Swap it in for the live page (free) takes the first calculator back to a draft and stops its link. A second calculator on a private link or behind a password goes through Pro, which keeps up to 100 live.
• On the phone. One column, big inputs with the unit beside each, and the results card with A, P, R, V, Fr, and critical depth, because a field engineer checks those before trusting Q. Adding the link to the home screen gives it an icon (how). Convika doesn't make the page work offline: the phone opens the link over its connection.

You stay responsible for checking the page's results against your own design standard.

Put your Manning sheet on the crew's phones in 5 steps

1. Collect the formulas and the test cases

Show the sheet's formulas, list each input and output with its unit, and export the n table. Run the 8 cases above through the sheet and note what it gives.

2. Ask the AI for one HTML file

Give ChatGPT, Claude, or Gemini the formulas, the table, the test cases, and the prompt above. Ask for the whole file in one reply.

3. Run the test table before anyone sees it

Open the file in the free HTML viewer, which runs its scripts in your browser at phone width without uploading it. Press the page's "Run the test cases" button, or type the cases in by hand, and fix anything off by more than your tolerance.

4. Upload it and publish

Drop the file into the box below and click Get my preview link. That preview runs no scripts, so the calculator shows no results there; don't test it there. Click Keep this page — free, no card and sign up: the page moves to your dashboard, and its scripts run when you open it from there. Click Publish page, choose who can open it, and click Next. For a Public page, the first time, click Publish free. A Private link or Only people with the password opens an order for Pro: click Subscribe, free for 1 day and add your card, and the page goes live once the trial starts.

5. Send the link, and publish fixes at the same link

Post the address in the crew's chat. For each fix, click Upload a new version under Update from an HTML file, run the test cases in the new version's preview, and click Publish v2 at the same link.

Frequently asked questions

What is the Manning equation in SI and US units?

Q = (1/n)·A·R^(2/3)·S^(1/2) in SI, with Q in m³/s and A and R in metres, and Q = (1.486/n)·A·R^(2/3)·S^(1/2) in US customary units, with Q in ft³/s and A and R in feet. S is the energy slope in m/m or ft/ft, and n is the same number in both. FHWA's HDS-4 rounds 1.486 to 1.49.

How do I calculate flow in a trapezoidal channel with Manning's equation?

Compute A = y·(b + z·y) and P = b + 2y·√(1 + z²) for bottom width b, depth y, and side slope z:1, then R = A/P and Q = (k/n)·A·R^(2/3)·S^(1/2). For b = 2 m, z = 2, y = 0.5 m, n = 0.020, and S = 0.003, that gives 2.056 m³/s, which FHWA's HDS-4 Example 4.1 rounds to 2.0.

How do I use Manning's equation for a partially full pipe?

Find the central angle θ = 2·acos(1 − 2y/D) in radians, then A = D²(θ − sin θ)/8 and P = Dθ/2, and put them into Manning's equation. Half full, the velocity equals the full-pipe velocity and the flow is half the full-pipe flow.

At what depth does a circular pipe carry the most flow?

At about 0.94 of the diameter, where the flow is about 7.6% more than when the pipe runs full, with a constant n. FHWA's HDS-3 gives 0.94 D and HDS-4 gives 93 percent; our calculation puts it at 0.938 D. The velocity peaks lower, at about 0.81 D.

How do I solve for normal depth without Goal Seek?

Use bisection: bracket the depth between 0 and the top of the section, compute Q at the middle, keep the half where Q crosses the target, and repeat until the bracket is narrower than your tolerance. Each step halves the bracket, so about 30 to 40 steps reach a billionth of the depth. Goal Seek is listed for desktop Excel only, so a phone calculator needs its own solver.

Can I use Manning's equation for a box culvert?

Only for a culvert barrel long enough for uniform flow to develop, treated as a rectangle with its top not wetted. Most culvert flow is set by the inlet or the outlet: FHWA's HDS-3 says its charts "are not suitable for use in connection with most types of culvert flow, since culvert flow is seldom uniform."

Where do I get Manning's n values?

From the table your agency or client uses. FHWA's HDS-4 has a short one in Appendix B (Tables B.2 and B.3), for example 0.013 for ordinary concrete lining and 0.011 to 0.013 for concrete pipe, and it points to USGS Water-Supply Paper 2339 for natural channels. Put the table in the calculator file with a custom field beside it.

Can my crew use the calculator without Excel or an app?

Yes. Published as a page, it opens from a link in the phone's browser, with nothing to install and no sign-in for a Public page or a Private link. With Only people with the password, each phone asks for the password and then remembers it for 7 days.

Why doesn't my calculator match the spreadsheet?

Look for a different constant (1.49 against 1.486, or g), slope entered as a percent, a ^ used as a power in JavaScript, or a Goal Seek result saved at a looser tolerance. Compare the intermediate values first: A, P, and R show which formula differs.

Sources

• FHWA, HDS-4, Introduction to Highway Hydraulics (FHWA-NHI-08-090, June 2008): section 4.3.1 (equations 4.1 to 4.5), section 4.3.2 (4.8 to 4.10, Examples 4.1 and 4.2), section 4.3.5 (4.16), section 4.6.2 (4.21 to 4.26), section 7.1 and Example 7.1, Appendix B Tables B.2 and B.3. Read November 6, 2026.
• FHWA, HDS-3, Design Charts for Open-Channel Flow (August 1961, archived): sections 2.3, 5.1, and 5.2, and the list of symbols. Read November 6, 2026.
• USBR, Water Measurement Manual, chapter 13, section 6, Slope-Area Method, and chapter 2, section 11. Read November 6, 2026.
• Ed Maurer, Hydraulics and Water Resources: Examples Using R, chapter 5 (Santa Clara University): sections 5.1, 5.2, and 5.4, Examples 5.1, 5.2, 5.4, and 5.10. Read November 6, 2026.
• Microsoft, Use Goal Seek to find the result you want by adjusting an input value Change formula recalculation, iteration, or precision in Excel, and ACOS function. Read November 6, 2026.
• MDN, Bitwise XOR (^). Read November 6, 2026.
• Omni Calculator, Open Channel Flow, Auburn University, Open Channel Flow Calculator, and NWS Alaska-Pacific River Forecast Center, Normal Depth tool. Read November 6, 2026.