---
title: "Prove q = k times focal length"
subject: "Soil Mechanics"
chapter: 7
key: "semester_4/soil_mechanics:discharge-equals-k-focal-length"
source: https://www.bctengineer.com/q/semester_4/soil_mechanics/discharge-equals-k-focal-length
site: BCT Engineering
---

# Prove q = k times focal length

Soil Mechanics · Chapter 7: Seepage through Soils

- Asked in: 2078 Baisakh, 5 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/soil_mechanics/discharge-equals-k-focal-length
- Whole chapter: https://www.bctengineer.com/md/semester_4/soil_mechanics/chapter-7

## Question

Prove that the discharge per unit width of an earthen dam with a horizontal filter at its toe is equal to the coefficient of permeability times the focal length.

## Short answer

Starting from the parabola property y(dy/dx)=y0, Dupuit’s assumption gives the discharge per unit width q = k i A = k (dy/dx) y = k y0. The focal length y0 is the distance from focus to directrix, so q equals k times that focal length.

## Answer

**Meaning of focal length:** the Casagrande base parabola has its focus F at the beginning of the horizontal filter. Let $y_0$ be the distance from the focus to the directrix (the "focal length" in this context, the parabola's parameter). The distance from focus to the vertex is $y_0/2$.

```text
   y
   |   .  .           directrix
   |        .   .  <--- parabola
   |   y0 --->   .
   +------F-----------------> x   (filter at the toe)
```

#### Proof

Take the origin at F, $x$ positive towards the upstream side and $y$ vertical. Any point P(x, y) on the parabola is equidistant from the focus and from the directrix ($x = -y_0$):

$$
\sqrt{x^2 + y^2} = x + y_0
$$

$$
x^2 + y^2 = x^2 + 2xy_0 + y_0^2 \quad\Rightarrow\quad y^2 = 2y_0\,x + y_0^2
$$

Differentiate with respect to $x$:

$$
2y\frac{dy}{dx} = 2y_0 \quad\Rightarrow\quad y\frac{dy}{dx} = y_0
$$

By Dupuit's assumption, the hydraulic gradient on a vertical section at height $y$ is $i = dy/dx$ (slope of the free surface), and the flow area per unit width is $y \times 1$:

$$
q = k\,i\,A = k\,\frac{dy}{dx}\,y = k\,y_0
$$

Hence the discharge per unit width is equal to the coefficient of permeability times the focal length $y_0$. $\blacksquare$

At the focus ($x = 0$) the ordinate is $y = y_0$ and the slope is 1, so $q = k \times 1 \times y_0$, which agrees.

(If the focal length is taken as the focus-to-vertex distance $a = y_0/2$, then $q = 2ka$.)

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What property of the parabola starts the proof?
   A. A point is equidistant from the focus and the directrix
   B. A point is at constant height
   C. The vertex is at the origin
   D. The tangent is horizontal
   Answer: A. sqrt(x^2 + y^2) = x + y0 expresses equal distance from focus and directrix. (Key point: Parabola property)

2. What equation results?
   A. y^2 = 2 y0 x + y0^2
   B. y = y0 x
   C. y^2 = y0 x
   D. x = y0 y
   Answer: A. Squaring gives y^2 = 2 y0 x + y0^2. (Key point: Parabola equation)

3. What gradient and area does Dupuit's assumption use?
   A. i = dy/dx and area y x 1
   B. i = y/x and area x
   C. i = 1 and area y0
   D. i = dy/dx and area y0
   Answer: A. Gradient equals free surface slope, area is depth y per unit width. (Key point: Dupuit assumption)

4. Since y dy/dx = y0, what is q?
   A. k y0
   B. k y0^2
   C. k/y0
   D. 2 k y0^2
   Answer: A. q = k (dy/dx) y = k y0. (Key point: Result)

## Related in Soil Mechanics

- [Earth dam with filter, seepage quantity](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Aearth-dam-horizontal-filter-seepage) — Soil Mechanics, chapter 7
- [Top flow line, dam with horizontal filter](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Atop-flow-line-horizontal-filter) — Soil Mechanics, chapter 7
- [Earth dam flow net and discharge](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Aearth-dam-flow-net-compare-discharge) — Soil Mechanics, chapter 7
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- [Why a filter on dam downstream](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Afilter-downstream-earth-dam) — Soil Mechanics, chapter 7
- [Prove q = k h Nf/Nd for earth mass](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Aflow-net-discharge-formula-proof) — Soil Mechanics, chapter 7

## Related in other subjects

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- [Seepage and phreatic line: horizontal filter at toe](https://www.bctengineer.com/question.md?key=semester_7%2Fhydropower_engineering%3Aseepage-horizontal-filter-derivation) — Hydropower Engineering, chapter 4, asked 3×
- [Gravity dam equilibrium and overtopping discharge coefficient](https://www.bctengineer.com/question.md?key=semester_7%2Fhydropower_engineering%3Agravity-dam-reservoir-capacity-curve-hrwl) — Hydropower Engineering, chapter 4
- [Seepage discharge through dam body (k = 1×10⁻⁶ m/s)](https://www.bctengineer.com/question.md?key=semester_7%2Fhydropower_engineering%3Aearthen-dam-seepage-gradient-figure-2072k) — Hydropower Engineering, chapter 4