Pressure variation after sudden valve closure

Case. A long pipeline of length $L$ from a reservoir (upstream) to a valve (downstream) carries water at velocity $V_0$. The valve is closed instantaneously at $t = 0$. Neglect friction. Let $\Delta H = \dfrac{cV_0}{g}$ (pressure rise $\Delta p = \rho cV_0$) and let the static head be $H_0$ (shown as the zero line below).

At the valve end

The pressure wave starts at the valve at $t = 0$, so the pressure there rises at once by $\Delta H$ and stays at $H_0 + \Delta H$ until the negative wave reflected from the reservoir returns at $t = 2L/c$. The pressure then drops by $2\Delta H$ to $H_0 - \Delta H$ (the closed valve prevents backward flow) and stays there until $t = 4L/c$, when the positive wave from the reservoir restores it to $H_0 + \Delta H$. The variation is a square wave of period $4L/c$. Pressure at the valve over two cycles ($+\Delta H$ above and $-\Delta H$ below the zero line):

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(top = 1.0, bottom = -1.0; '-' is zero line)

At the mid-point of the pipe ($L/2$ from the valve)

• $0 < t < L/2c$: no change (the wave has not arrived yet); pressure $H_0$.
• $L/2c < t < 3L/2c$: the wave reaches the mid-point; pressure $H_0 + \Delta H$ (duration $L/c$).
• $3L/2c < t < 5L/2c$: the negative wave reflected from the reservoir cancels it; pressure $H_0$.
• $5L/2c < t < 7L/2c$: the negative wave reflected from the valve arrives; pressure $H_0 - \Delta H$.
• $7L/2c < t < 9L/2c$: the positive wave from the reservoir restores the pressure to $H_0$; the cycle then repeats from $t = 9L/2c$.

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(top = 1.0, bottom = -1.0; '-' is zero line)

The mid-point pressure has the same amplitude ($\pm\Delta H$) but is out of phase with the valve. It stays for $L/c$ at each extreme level, with $H_0$ in between, giving a "stepped" (rectangular pulse) shape.

Summary. At the valve the pressure alternates between $H_0 + \Delta H$ and $H_0 - \Delta H$ every $2L/c$; at the mid-point it follows the sequence $H_0 \to H_0 + \Delta H \to H_0 \to H_0 - \Delta H \to H_0$, each stage lasting $L/c$ (except the first, which lasts $L/2c$). Both have a period of $4L/c$.