Waves string

Now, electron waves arc described by a wave equation of the same general form as that for string waves. The wave functions that are acceptable solutions to this equation again give the amplitude, <, this time as a function, not of a single coordinate, but of the three coordinates necessary to describe motion in three dimensions. It is these electron wave functions that we call orbitals. 

Like a string wave, an electron wave can have nodes, where the amplitude is zero. On opposite sides of a node the amplitude has opposite signs, that is, the wave is of opposite phases. Of special interest to us is the fact that between the two lobes of a /) orbital lies a no l plane, perpendicular to the axis of the orbital... 

Such function is known as wave function. It is important to note that electron waves are similar to string waves. When wave function gives the amplitude (< )) as a function of three co-ordinates it is known as orbital. In case of orbital lobes of opposite phases are... 

This simple example Illustrates the important kinematic properties of shock waves, particularly the concepts of particle velocity and shock velocity. The particle velocity is the average velocity acquired by the beads. In this example, it is the piston velocity, v. The shock velocity is the velocity at which the disturbance travels down the string of beads. In general, at time n//2v, the disturbance has propagated to the nth bead. The distance the disturbance has traveled is therefore n d -b /), and the shock velocity is... 

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. 

The quantity c = y/rjp is known as the jjjia e velocity, It is the speed a which waves travel along the string. Clearly, the left-hand side of Eq, (4) represents the one-dimensional Laplacian operating on the dependent variable. This expression can be easily generalized to represent wave phenomena in two or more dimensions in space. 

We have already seen (p. 2) that the individual electrons of an atom can be symbolised by wave functions, and some physical analogy can be drawn between the behaviour of such a wave-like electron and the standing waves that can be generated in a string fastened at both ends—the electron in a (one-dimensional) box analogy. The first three possible modes of vibration will thus be (Fig. 12.1) ... 

The nature of the resulting wave depends on the phase difference (2) is 0 degrees, or 360 degrees, then the two waves are said to be in phase, and the maximum amplitude of the resultant wave is A1 + A2. This situation is termed constructive interference. If the phase difference is 180 degrees, then the two waves are out of phase, and destructive interference occurs. In this case, if the amplitudes of the two waves are equal (i.e., if A = A2), then the two waves cancel each other out, and no wave is observed (Fig. 12.1). Standing waves, such as those seen when the string on a musical instrument vibrates, are caused when the reflected waves (from the bridge of the instrument) are in phase and thus interfere constructively. 

Using harmonic overtones, it is possible to sound a tone which will cancel one or more of its octaves reflected in the harmonic scales above and below it. This is easily demonstrated on a cello Suppose a tone, say the open string A, is sounded. The sound is a wave-vibration of air molecules caused by the string, which then acts as a resonator. 

To appreciate the node concept, it is useful to think of wave analogies. Thus, a vibrating string might have no nodes, one node, or several nodes according to the frequency of vibration. We can also realize that the wave has different phases, which we can label as positive or negative, according to whether the lobe is above or below the median line. 

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