Waves, water

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. 

A stone dropped in a pond pushes the water downward, which is countered by elastic forces in the water that tend to restore the water to its initial condition. The movement of the water is up and down, but the crest of the wai c produced moves along the surface of the water. This type of wave is said to be transverse because the displacement of the water is perpendicular to the direction the wave moves. When the oscillations of the wave die out, there has been no net movement of water the pond is just as it was before the stone was dropped. Yet the wave has energy associated with it. A person has only to get in the path of a water wave crashing onto a beach to know that energy is involved. The stadium wave is a transverse wave, as is a wave in a guitar string. 

Fig. 5.35a-h Flow regimes in the pipe of 25 mm at f/os = 36 m/s (a) Uis = 0.016 m/s, disturbance waves with motionless droplets (b) Uis = 0.027 m/s, disturbance waves with moving droplets (c) U s = 0.045 m/s, disturbance waves and liquid film on the upper tube part (d) Uis = 0.17 m/s, disturbance air-water waves and liquid film on the upper tube part (e) Uis = 0.016 m/s, small air-water clusters (f) Ui = 0.027 m/s, air water clusters fe) Uis = 0.045 m/s, huge air-water clusters (h) Uis = 0.17 m/s, huge air-water clusters that block the tube cross-section. Reprinted from Hetsroni et al. (2003b) with permission... 

Examples of wave patterns, (a) Floats produce standing water waves. (Z>) X rays generate wave interference patterns, (c) Protruding atoms on a metal surface generate standing electron waves. 

Such sign changes in the orbital are analogous, e.g., to water-wave displacements above or below the mean surface level. The overall sign of the orbital may be reversed without physical effect, so only sign differences within the orbital are significant. 

Gent, P. R. (1977). A numerical model of the air flow above water waves. Part II. J. Fluid Mech. 82, 349-369. 

An easily visualised example of a transverse wave is that obtained when a stone is dropped into a pool of water. The disturbance, or water wave, can be seen spreading across the surface in the form of circular crests of increasing radius. Any objects in the pool (e. g. 

This is not really very complicated and it applies equally well to water waves or electromagnetic radiation. What is almost needlessly complicated is the variety of units commonly used to express k and v for electromagnetic radiation. One problem is tradition, the other is the desire to avoid very large or very small numbers. Thus, as Figure 9-7 shows, we may be interested in electromagnetic wavelengths that differ by as much as a factor of 1016. Because the velocity of electromagnetic radiation in a vacuum is constant at 3 X 108 meters sec-1, the frequencies will differ by the same factor. 

TNT spheres have also been employed for generating shock waves in water. They have been fired above (Ref 37) and just below (Ref 6) the surface, half-submerged (Ref 54), and at various depths ranging from 200 to 14000 ft (Refs 46 78). The theory of expl water wave generation has been reviewed (Refs 28 29), and measurements made (Refs 55,102 171). Studies have been made underwater in an effort... 

B.M, LeMehaute et al, Contributions to the Mino Lake Experiements. Vol L Predictions of the Water Waves and Run-Up Generated by TNT Explosions in Mino Lake , Rept NESCO-S-256-2-Vol 1, Natl Engrg Science Co, Pasadena... 

Particles (baseball) versus Waves (water wave)... 

Equation (10.125) is valid only for waves in shallow water, i.e., for waves of great length and moderate amplitude relative to their depth. For so-called deep-water waves, as might be encountered in the ocean, for example, but still presuming small amplitudes, a more accurate equation is... 

Since an electron has wave character, we can describe its motion with a wave equation, as we do in classical mechanics for the motions of a water wave or a stretched string or a drum. If the system is one-dimensional, the classical wave equation is... 

But according to the present approach, they behave as quantum states in Hilbert space. These latter can be diffracted and modulated interference patterns. This latter is taken as the signal of wave behavior in the standard textbooks. But, as water waves in a pond designed to mimic a two-slit setup... 

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