String, vibrating

The differential equation of the vibrating string is the one-dimensional wave equation, which is a partial differential equation, second order in each of t and x  

The coefficients C and D can be found via the boundary conditions by the orthogonality properties of the triangular functions. 

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The eigenfunctions of the vibrating string can be shown to have the property of orthogonality, as anticipated, i. e. 

Years ago, people thought that electrons travel around the nucleus in definite circular patterns, or orbits. You may have seen pictures like this. Now we know that electrons don t follow a perfect circle around the nucleus but are more likely to go around in certain places. A famous physicist named Erwin Schrbding-er said the electron is like a vibrating string. If you took a picture of all of the places that electrons go, it would look like a cloud, like the drawing below. The electrons do orbit in shells, which are regions of space around the nucleus. If you think of the nucleus as a beehive, then the electrons would be the bees swarming around it. 

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. 

Granato, A. V. and Liicke, K. (1966). The vibrating string model of dislocation damping. In Physical acoustics IV-A (ed. W. R Mason), pp. 225-76. Academic Press, London. [252]... 

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. 

When more terms are added to the wave equation, corresponding to complex losses and dispersion characteristics, more terms of the form y(n -l, m - k) appear in (10.10). This approach to numerical simulation was used in early computer simulation of musical vibrating strings [Ruiz, 1969], and it is still in use today [Chaigne, 1992, Chaigne and Askenfelt, 1994],... 

In the field of acoustics, the state of a vibrating string at any instant of time to is normally specified by the displacement y(t0,x) and velocity y(to,x) for all x [Morse, 1981], Since displacement is the sum of the traveling displacement waves and velocity is proportional to the difference of the traveling displacement waves, one state description can be readily obtained from the other. 

Lindsay, 1973] Lindsay, R. B. (1973). Acoustics Historical and Philosophical Development. Dowden, Hutchinson Ross, Stroudsburg. Contains Investigation of the Curve Formed by a Vibrating String, 1747, by Jean le Rond d Alembert. 

The wave equation for the ideal (lossless, linear, flexible) vibrating string, depicted in ... 

Stiffness in a vibrating string introduces a restoring force proportional to the fourth derivative of the string displacement [Morse, 1981] ... 

The principle of the resonator is the same as that of a vibrating string and in the illustrated example one end of the line is maintained at ground potential (the node) and the other is the antinode. The permittivity of the surroundings determine the length of the line for a particular resonance frequency and, in this case, also serves to capacitively couple energy into and out of the resonator. 

Figure 8.3 Schematic depiction of a vibrating string as a metaphor for a carbon-12 resonance, a) Tromba marina, b) Vibrating string anchored at endpoints. [From Filippo Bonanni, Antique Musical Instruments and Their Players (New York Dover, 1964).]...

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