---
title: "MOC numerical, characteristics not on diagonal (25 cm)"
subject: "Computational Techniques in Civil Engineering"
chapter: 6
key: "semester_8/computational_techniques_in_civil_engineering:moc-numerical-off-diagonal-j"
source: https://www.bctengineer.com/q/semester_8/computational_techniques_in_civil_engineering/moc-numerical-off-diagonal-j
site: BCT Engineering
---

# MOC numerical, characteristics not on diagonal (25 cm)

Computational Techniques in Civil Engineering · Chapter 6: Method of Characteristics

- Asked in: 2079 Jestha, 4 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/computational_techniques_in_civil_engineering/moc-numerical-off-diagonal-j
- Whole chapter: https://www.bctengineer.com/md/semester_8/computational_techniques_in_civil_engineering/chapter-6

## Question

If the MOC is applied for $t_1 = 1$ s and $t_2 = 2$ s, time levels for a pipe with diameter 25 cm carrying water. If $Q_A = 0.6$ m$^3$/s, $Q_B = 0.65$ m$^3$/s, $Q_C = 0.64$ m$^3$/s, $H_A = 20$ m, $H_B = 20.6$ m, and $H_C = 20.4$ m are the values at grid points. Find the values of $Q$ and $H$ at $t_1 = 1$ s that will be required for finding $Q$ and $H$ at P when characteristics do not lie on diagonal. Here, $\Delta x = 1000$ m, $\Delta t = 1$ s, $f = 0.02$ and $c = 850$ m/s.

## Short answer

Since aΔt < Δx, characteristics intersect the t₁ line between grid points. With θ = cΔt/Δx = 0.85, linear interpolation gives for C⁺: Q_R = Q_C − θ(Q_C−Q_A) = 0.606 m³/s, H_R = 20.06 m; for C⁻: Q_S = Q_C − θ(Q_C−Q_B) = 0.6485 m³/s, H_S = 20.57 m. These replace A and B in the characteristic equations.

## Answer

#### Idea
When the characteristics do not lie on the diagonal of the grid, $a\Delta t<\Delta x$: the $C^+$ and $C^-$ lines through P (at $t_2$) meet the line $t_1$ at points R and S that lie **between** the grid points, not at A and B. $Q$ and $H$ at R and S are found by linear interpolation along the time line $t_1$.

```text
 t2 :                P
                    /|\
               C+  / | \   C-
                  /  |  \
 t1 :   A--R---------C---------S--B
           |<-a dt ->|<-a dt ->|
        |<--- dx --->|<--- dx --->|
```
Here R lies on the $C^+$ line between A and C, and S on the $C^-$ line between C and B, each at a distance $a\Delta t$ from C.

#### Interpolation factor
$$
\theta=\frac{a\,\Delta t}{\Delta x}=\frac{850\times1}{1000}=0.85
$$
**Point R (between A and C, $C^+$ foot):**
$$
Q_R=Q_C-\theta\,(Q_C-Q_A),\qquad H_R=H_C-\theta\,(H_C-H_A)
$$
$$
Q_R=0.64-0.85(0.64-0.6)=0.606\ \text{m}^3/\text{s},\qquad
H_R=20.4-0.85(20.4-20)=20.06\ \text{m}
$$
**Point S (between C and B, $C^-$ foot):**
$$
Q_S=Q_C-\theta\,(Q_C-Q_B),\qquad H_S=H_C-\theta\,(H_C-H_B)
$$
$$
Q_S=0.64-0.85(0.64-0.65)=0.6485\ \text{m}^3/\text{s},\qquad
H_S=20.4-0.85(20.4-20.6)=20.57\ \text{m}
$$

**Answer:** at $t_1=1$ s, $Q_R=0.606$ m$^3$/s, $H_R=20.06$ m on the $C^+$ line, and $Q_S=0.6485$ m$^3$/s, $H_S=20.57$ m on the $C^-$ line. These replace $Q_A,H_A$ and $Q_B,H_B$ in the $C^+$ and $C^-$ equations
$$
H_P=H_R-B_p(Q_P-Q_R)-R_pQ_R|Q_R|,\qquad H_P=H_S+B_p(Q_P-Q_S)+R_pQ_S|Q_S|
$$
with $B_p=a/(gA_p)$ and $R_p=f\,a\Delta t/(2gDA_p^2)$ (the friction length is $a\Delta t=850$ m).

## Check yourself

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

1. When a dt < dx, Q and H at feet of characteristics are found by:
   A. linear interpolation between grid points at t1
   B. ignoring them
   C. using Newton-Raphson
   D. using values at P
   Answer: A. The feet R and S lie between grid points. (Key point: Why interpolate)

2. The interpolation factor theta is:
   A. 0.85
   B. 1.18
   C. 0.15
   D. 850
   Answer: A. a dt/dx = 850 x 1/1000 = 0.85. (Key point: Factor)

3. Q_R at the C+ foot is:
   A. 0.606 m^3/s
   B. 0.634 m^3/s
   C. 0.640 m^3/s
   D. 0.6485 m^3/s
   Answer: A. 0.64 - 0.85(0.04) = 0.606. (Key point: Q_R)

## Close variants set in other exams

- [MOC numerical, characteristics not on diagonal (30 cm)](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Amoc-numerical-off-diagonal-a2) — Computational Techniques in Civil Engineering, chapter 6, asked 2×

## Related in Computational Techniques in Civil Engineering

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