Standing waves string

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. 

The wave described by eqn 1.6 is different from that discussed above. The displacement varies sinusoidally in space and time, but the positions of maximum and minimum displacement do not move. It is known as a standing wave, as opposed to the travelling wave illustrated in Fig. 1.1. Figure 1.2 shows a standing wave at three successive times. The points of zero displacement are called nodes, and those where the displacement is maximum, antinodes. Standing waves are formed in vibrating strings which are fixed at one or more points. They form the basis for musical instruments. 

Davis, M. (2007) Guitar strings as standing waves A demonstration. J. Chem. Educ. 84,1287-1289. 

The vibration therefore sets up standing waves with wavelengths restricted by integers (n), such that A = 21/n or l = n /2. As the string vibrates each point on the string goes up and down and passes periodically through zero. At special points, e.g. I 1/2 a I, n = 2 there is never any... 

A standing wave. The fundamental frequency of a guitar string is a standing wave with the string alternately displaced upward and downward. 

The standing wave produced by the vibration of a guitar string fastened at both ends. Each dot represents a node (a point of zero displacement). 

Standing wave a stationary wave as on a string of a musical instrument in the wave mechanical model, the electron in the hydrogen atom is considered to be a standing wave. 

The first few eigenfunctions and the corresponding probability distributions are plotted in Fig. 3.2. There is a close analogy between the states of this quantum system and the modes of vibration of a violin string. The patterns of standing waves on the string are, in fact, identical in form with the wavefunctions (3.24). 

FIGURE 4.17 A guitar string of length L with fixed ends can vibrate in only a restricted set of ways. The positions of largest amplitude for the first three harmonics are shown here. In standing waves such as these, the whole string is in motion except at its end and at the nodes. 

Figure 1.3 The standing waves of a vibrating string. The fundamental (first harmonic) has no nodes the second harmonic is twice the frequency of the fundamental and has one node, shown by a dotted vertical line. The number of nodes increases as the frequency and energy increase.

We may graphically examine the wavefunctions and the probability densities as we vary the quantum number, n. There is no reason to expect that the wavefunctions will be the same. In fact, the solutions look as shown in Fig. 7.2. A node is defined as a point at which ip and ip 2 are zero. We will not find the particle at this position. The wavefunctions, ip are identical to the standing waves generated by the vibrating string, an example with which we are all much more familiar and has been treated in the previous chapter. Note especially, that the n = 1 wavefunction for the particle-in-a-box is identical to the string plucked at its midpoint in Fig. 6.1. 

Superimposing the configurations produces the waveform of the guitar string s standing wave. 

The variation in the intensity of the electron charge can be described in terms of a three-dimensional standing wave like the standing wave of the guitar string. 

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