---
title: "Pressure-time diagram at 2/3 L from valve"
subject: "Hydraulics"
chapter: 4
key: "semester_4/hydraulics:pressure-time-diagram-two-thirds-length"
source: https://www.bctengineer.com/q/semester_4/hydraulics/pressure-time-diagram-two-thirds-length
site: BCT Engineering
---

# Pressure-time diagram at 2/3 L from valve

Hydraulics · Chapter 4: Unsteady flow in pipes

- Asked in: 2072 Magh, 2 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/hydraulics/pressure-time-diagram-two-thirds-length
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-4

## Question

In the figure below, water flowing through a pipe from the reservoir is suddenly stopped by closing a valve at point B. Draw pressure-time diagram at the 2/3 L from valve of the pipe for one cycle of wave motion. [Figure: reservoir connected to a pipe of length L ending in a valve at B]

## Short answer

At 2/3 L from the valve, pressure remains static H0 for 0–2L/3c, then rises to H0+ΔH until 4L/3c. It returns to H0 until 8L/3c, drops to H0−ΔH from 8L/3c to 10L/3c, then H0 again, repeating every 4L/c.

## Answer

**Case.** Reservoir at the upstream end, valve at B (downstream end), pipe length $L$, wave speed $c$, valve closed suddenly at $t = 0$. The point is $\tfrac{2}{3}L$ from the valve (so $\tfrac{1}{3}L$ from the reservoir). Let $\Delta H = cV_0/g$ and the static head be $H_0$.

#### Wave timing at the point ($d = 2L/3$ from the valve)
- $t_1 = \dfrac{d}{c} = \dfrac{2L}{3c}$: the positive wave from the valve arrives; pressure rises to $H_0 + \Delta H$
- $t_2 = \dfrac{2L - d}{c} = \dfrac{4L}{3c}$: the negative wave reflected from the reservoir arrives; pressure returns to $H_0$
- $t_3 = \dfrac{2L + d}{c} = \dfrac{8L}{3c}$: the negative wave reflected from the valve arrives; pressure falls to $H_0 - \Delta H$
- $t_4 = \dfrac{4L - d}{c} = \dfrac{10L}{3c}$: the positive wave reflected from the reservoir arrives; pressure returns to $H_0$
- At $t = \dfrac{4L}{c}$ the cycle is complete and repeats.

| Interval | Pressure at the point |
|---|---|
| $0$ to $\dfrac{2L}{3c}$ | $H_0$ |
| $\dfrac{2L}{3c}$ to $\dfrac{4L}{3c}$ | $H_0 + \Delta H$ |
| $\dfrac{4L}{3c}$ to $\dfrac{8L}{3c}$ | $H_0$ |
| $\dfrac{8L}{3c}$ to $\dfrac{10L}{3c}$ | $H_0 - \Delta H$ |
| $\dfrac{10L}{3c}$ to $\dfrac{4L}{c}$ | $H_0$ |

#### Pressure-time diagram (one cycle, $t$ from 0 to $4L/c$)
```text
        ********


********--------****************--------*********


                                ********
(top = 1.0, bottom = -1.0; '-' is zero line)
```
(+1 = $H_0 + \Delta H$, 0 = $H_0$, $-1$ = $H_0 - \Delta H$.) The positive pulse lasts $2L/3c$ and the negative pulse lasts $2L/3c$, separated by a period of $4L/3c$ at the static pressure.

## Check yourself

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

1. The point is 2L/3 from the valve. When does the first pressure rise reach it after instantaneous closure?
   A. L/c
   B. 2L/(3c)
   C. 2L/c
   D. 4L/(3c)
   Answer: B. The positive wave from the valve travels distance d = 2L/3 at speed c, so t = 2L/(3c). (Key point: First wave arrival)

2. What is the pressure at the point just after the first wave arrives?
   A. H0 - dH
   B. H0 + dH
   C. H0
   D. 2 H0
   Answer: B. Sudden closure sends a positive wave of magnitude dH = cV0/g upstream, raising pressure to H0 + dH. (Key point: Pressure at first arrival)

3. What does the wave reflected from the constant-level reservoir do to the pressure?
   A. It returns the pressure to H0 as a negative wave
   B. It makes the pressure H0 - dH immediately
   C. It leaves the pressure unchanged
   D. It doubles the rise
   Answer: A. At the reservoir the head stays fixed, so the wave reflects with opposite sign and cancels the rise. (Key point: Reflection at reservoir)

4. What is the period of one complete cycle of pressure variation?
   A. 4L/c
   B. L/c
   C. 2L/(3c)
   D. 2L/c
   Answer: A. The wave travels to the reservoir, back, to the valve and back again: 4L/c. (Key point: Cycle period)

## Related in Hydraulics

- [One cycle of water hammer pressure wave](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Awater-hammer-one-cycle-wave-sketch) — Hydraulics, chapter 4, asked 2×
- [Pressure-time diagram 1200 m from reservoir](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Apressure-time-diagram-1200m-from-reservoir) — Hydraulics, chapter 4
- [Valve suddenly opened: time to 95% flow](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Asudden-valve-opening-time-to-95-percent-flow) — Hydraulics, chapter 4
- [Water hammer in 20 m steel pipeline with valve](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Awater-hammer-20m-pipeline-pressure-time-curves) — Hydraulics, chapter 4
- [Water hammer pressure and stress, 2300 m pipeline](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Awater-hammer-2300m-pressure-stress-1725m) — Hydraulics, chapter 4
- [Penstock pressure rise and pressure-time sketch](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Asurge-tank-penstock-pressure-rise) — Hydraulics, chapter 4

## Related in other subjects

- [Water hammer in elastic cast-iron pipe](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Awater-hammer-cast-iron-pipe) — Hydro Power, chapter 1
- [Water pipe thickness from pressure head](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Awater-pipe-thickness-pressure-head) — Strength of Materials, chapter 5
- [MOC single reach, reservoir pipe (2 s)](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Amoc-reservoir-pipe-sin-2) — Computational Techniques in Civil Engineering, chapter 6
- [MOC single reach, reservoir pipe (3 s)](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Amoc-reservoir-pipe-sin-3) — Computational Techniques in Civil Engineering, chapter 6