Water waves speed

Gas entrained in the fluid and the flexibiflty of the pipe wall both result in lowering of the wave speed. For deaerated water, the wave speed is about 1250 m/s. Detailed methods of analysis and evaluation of hydraulic transients may be found in the flterature (25). 

Example 10 Response to Instantaneous Valve Closing Compute the wave speed and maximum pressure rise for instantaneous valve closing, with an initial velocity of 2,0 m/s, in a 4-in Schedule 40 steel pipe with elastic modulus 207 X 10 Pa, Repeat for a plastic pipe of the same dimensions, with E = 1.4 X 10 Pa. The liquid is water with P = 2.2 X 10 Pa and p = 1,000 kg/m. For the steel pipe, D = 102,3 mm, b = 6,02 mm, and the wave speed is... 

Some wave phenomena, familiar to many people from the human senses, include the easy undulation of water waves from a dropped stone or the sharp shock of the sonic boom from high-speed aircraft. The great power and energy of shock events is apparent to the human observer as he stands on the rim of the Meteor Crater of Arizona. Human senses provide little insight into the transition from these directly sensed phenomena to the high-pressure, shock-compression effects in solids. This transition must come from development of the science of shock compression, based on the usual methods of scientific experimentation, theoretical modeling, and numerical simulation. 

In most solids, the sound speed is an increasing function of pressure, and it is that property that causes a compression wave to steepen into a shock. The situation is similar to a shallow water wave, whose velocity increases with depth. As the wave approaches shore, a small wavelet on the trailing, deeper part of the wave moves faster, and eventually overtakes similar disturbances on the front part of the wave. Eventually, the water wave becomes gravitationally unstable and overturns. 

For a shock wave in a solid, the analogous picture is shown schematically in Fig. 2.6(a). Consider a compression wave on which there are two small compressional disturbances, one ahead of the other. The first wavelet moves with respect to its surroundings at the local sound speed of Aj, which depends on the pressure at that point. Since the medium through which it is propagating is moving with respect to stationary coordinates at a particle velocity Uj, the actual speed of the disturbance in the laboratory reference frame is Aj - -Ui- Similarly, the second disturbance advances at fl2 + 2- Thus the second wavelet overtakes the first, since both sound speed and particle velocity increase with pressure. Just as a shallow water wave steepens, so does the shock. Unlike the surf, a shock wave is not subject to gravitational instabilities, so there is no way for it to overturn. 

It is clear that sound, meaning pressure waves, travels at finite speed. Thus some of the hyperbolic—wavelike-characteristics associated with pressure are in accord with everyday experience. As a fluid becomes more incompressible (e.g., water relative to air), the sound speed increases. In a truly incompressible fluid, pressure travels at infinite speed. When the wave speed is infinite, the pressure effects become parabolic or elliptic, rather than hyperbolic. The pressure terms in the Navier-Stokes equations do not change in the transition from hyperbolic to elliptic. Instead, the equation of state changes. That is, the relationship between pressure and density change and the time derivative is lost from the continuity equation. Therefore the situation does not permit a simple characterization by inspection of first and second derivatives. 

FIGURE 4.1 As a water wave moves across an otherwise calm tank, its maximum amplitude and wavelength can be determined. Its speed is found by dividing the travel distance of a particular wave crest by the time elapsed. 

Some water waves reach the beach at a rate of one every 3.2 s, and the distance between their crests is 2.1 m. Calculate the speed of these waves. 

D. D. Joseph, Domain perturbations The higher order theory of infinitesimal water waves, Arch. Ration. Mech. Anal. 51,295-303 (1975) D. D. Joseph and R. Fosdick, The free surface on a liquid between cylinders rotating at different speeds, Arch. Ration. Mech. Anal. 49, 321-81 (1973). 

Fig. 3. Maximum horizontal water particle speed at the bottom on Cable and Anchor Reef, as calculated from wave-recorder records, and the square of the wind speed measured at Eatons Neck during a winter storm.

Consider a small-amplitude sinusoidal plane wave of length A and amplitude a propagating along an air-water interface in the positive x direction with wave speed c, as sketched in Fig. 10.4.1. The water is taken to be incompressible with viscosity and other dissipative effects neglected so that the wave amplitude at the interface remains unchanged. The vertical displacement of the disturbed surface, f, may be written... 

Equation (10.4.18) is the sought after dispersion relation for surface waves on deep water. It may also be written as a dependence of wave speed on wavelength ... 

The wavelength A is seen from Eqs. (10.4.11) and (10.4.12) to be that length associated with the transition from the dominance of capillary waves to gravity waves, or vice versa. For water, as noted earlier, A ,j = 1.7 x 10" m to which the corresponding minimum wave speed is 0.23 ms. ... 

Consider the water wave shown here, (a) How could you measiue the speed of this wave (b) How would you determine the wavelength of the wave (c) Given the speed and wavelength of the wave, how could you determine the frequency of the wave (d) Suggest an independent experiment to determine the frequency of the wave. [Section 6.1]... 

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