# An Introduction to Hydrodynamics and Water Waves by Bernard LeMehaute By Bernard LeMehaute

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This type of phenomenon is observed in pipes or tunnels, where a fluid stops or starts or balances because of a gate movement. Local inertia has to be taken into account in hydraulic engineering applications such as surge tanks, water hammer, and locks. 2 In the second case, the velocity maintains the same magnitude, but changes its direction. In this case the inertia force is due to the centrifugal acceleration. For example, in a periodic gravity wave in infinite depth, the magnitude of the velocity at a given point is a constant but its direction revolves continuously at all points (Fig.

The volume of influx, during a time dt, into the considered volume at x is q dt or hu dt, where q is the discharge, h the depth, and u the horizontal velocity component. The efflux out of the volume at x + dx is or + u ah + h au (3-4) az¢ + ay 2 + az¢ az 2 = 0 which can be written '\1 2 ¢ = 0. This is the well-known Laplace equation which has been subjected to extensive research in mathematical physics. 1). Since this last system of coordinates is rarely used, the continuity h ----t-Figure 3-3 Translatory wave.

Av _ au) = ! (ov' _ ou') 2 OX oy 2 ox' oy' u' and v' being the velocity components along the x' axis and the y' axis, respectively. 1 The Continuity in a Pipe The Continuity Principle The principle of continuity expresses the conservation of mass in a given space occupied by a fluid. The simplest, well-known form of the continuity relationship in elementary fluid mechanics expresses that the discharge for steady flow in a pipe is constant; that is, p VA = constant, where A is the cross-sectional area of the pipe and V is the mean velocity.