Waves in a String

Section 8.6 Partial Differential Equations Waves in a String... 

Figure 1-3 A standing wave in a string clamped at x = 0 and x = L. The wavelength k is equal to L.

We now demonstrate how Eq. (1-20) can be used to predict the nature of standing waves in a string. Suppose that the string is clamped at v = 0 and L. This means that the string cannot oscillate at these points. Mathematically this means that... 

A standing wave in a string 42 cm long has a total of six nodes (including those at the ends). What is the wavelength, in centimeters, of this standing wave ... 

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. 

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) ... 

A wave is a disturbance which travels and spreads out through some medium. Examples include ripples on the surface of water, vibrations in a string, and vibrating electric and magnetic fields (light waves). The wave disturbance can take many mathematical forms, but the simplest is the sinusoidal wave shown in Fig. 1,1. This illustrates how the displacement of the medium (y) varies with position (x) at three successive times. 

In non-dispersive systems, such as acoustic waves in a fluid or simple tension waves on a string, the wave speed does not vary with frequency. Thus the energy speed Cg is the same as the phase speed c, so that... 

Fourier series occur in various physical theories involving waves, because waves often behave sinusoidally. For example, Fourier series can represent the constructive and destructive interference of standing waves in a vibrating string. This fact provides a useful way of thinking about Fourier series. A periodic function of arbitrary shape is represented by adding up sine and cosine functions with... 

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