String oscillations

The rules presented operate not only in many musical instruments and the formation of standing waves in them (in the resonator of a guitar, for instance) but are also used in models of an ideal black body, in quantum mechanics, in physics of soUd-state properties, etc. All of these are discussed in other sections of the book. 

A transverse wave runs along an elastic cord at a speed o = 15 m/sec. The period of oscillations of the cord points is 7 = 1.2 sec, amplitude A = 2 cm. Define (1) wavelength, (2) phase of oscillation cp, displacement velocity and acceleration (f of the point at a distance jc = 45 m from the source of waves at time instant t = 4 sec and (3) phase difference Acp of the oscillation of two points lying on the cord at a distance Xj = 20 cm and X2 = 30 cm from the source of waves. 

Solution (1) According to definition, wavelength is A = vT. Substituting the values we obtain X = 18 m. (2) The wave equation is ( = A cos o)(t - (x/v)). The phase of the point oscillations with coordinate x in the time instant t stays under cos sign (p = co(t -(x/v)) or (p = (2X/T) (t —(x/o)). Calculation gives (p = 5.24 rad or 300°. We can find the displacement from the equation substituting the A and rp values appears equal to if = 1 cm. The velocity of the point can be found by the time derivation of substimting all the values derived we arrive at =9 cm/sec. The acceleration is the second time derivative on the displacement 

Solution Let us choose an axis x directed perpendicularly to the wall and the origin at a distance / from the reflected wave soitrce. Then the eqiration of the wave will be written as 

As in a point with coordinate x the reflected wave wiU come back covering twice the distance I - x) and the reflected wave will lose a phase %I2 at reflection the reflected wave equation can be written as 

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We need establish only one of the extreme profiles for the string. Then, as cosftot) oscillates between -i-l and—1, the string oscillates between the two extremes. In other words, sin(x) and — sin(x) are the same solution at different times. (They differ by a phase factor.)... 

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. 

It is well-known, for instance, that if one applies tension at properly timed intervals to pull a stretched wire or string, it begins to oscillate laterally. Lord Rayleigh performed an experiment of this kind by attaching a stretched wire to a prong of a tuning fork when the latter... 

The guitar string can oscillate on both sides of the 12th-fret node even when the string is plucked on only one side of the node. This occurs because waves can pass through the node. 

This prediction was based on the notion that a blackbody is composed of tiny oscillators that produce a continuum of waves, like those you get when you pluck the strings of a violin. But the spectrum physicists predicted for blackbody radiation—an infinite amount of high-energy radiation—and the experimental data did not fit. They were not even close. And this was the problem that Max Planck was working on in 1900. 

Figure 6.2. The temporal oscillations observed at x = 0.5 for the vibrating string shown in Fig. 6.1.

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