Waves traveling, standing

Both standing wave and traveling wave patterns can be used in linear accelerators. If traveling waves are used, the phase velocity of the waves must be made equal to the velocity of the particles accelerated as the particle velocity increases, the phase velocity must also increase. But, phase velocities in simple waveguides always are greater than the velocity of light, and loading must be introduced to reduce the phase velocity to the desired value, This can be accomplished by the introduction at intervals of washer-shaped irises,... 

The picture of the electron in an orbit as a standing wave does, however, pose the important question of where the electron, regarded as a particle, is. We shall consider the answer to this for the case of an electron travelling with constant velocity in a direction x. The de Broglie picture of this is of a wave with a specific wavelength travelling in the x direction as in Figure 1.4(a), and it is clear that we cannot specify where the electron is. 

Figure 1.3 (a) A standing wave for an electron in an orbit with n = 6. (b) A travelling wave, resulting... 

We see that for this special case the composite wave is the product of two functions one only of the distance x and the other only of the time t. The composite wave (x, t) vanishes whenever cos kx is zero, i.e., when kx = jr/2, 2)71/2, 5tc/2,. .., regardless of the value of t. Therefore, the nodes of P(x, i) are independent of time. However, the amplitude or profile of the composite wave changes with time. The real part of P(x, /) is shown in Figure 1.3. The solid curve represents the wave when cos.cot is a maximum, the dotted curve when coscot is a minimum, and the dashed curve when cos cot has an intermediate value. Thus, the wave does not travel, but pulsates, increasing and decreasing in amplitude with frequency co. The imaginary part of I (x, t) behaves in the same way. A composite wave with this behavior is known as a standing wave. 

In a bath-type sonochemical reactor, a damped standing wave is formed as shown in Fig. 1.13 [1]. Without absorption of ultrasound, a pure standing wave is formed because the intensity of the reflected wave from the liquid surface is equivalent to that of the incident wave at any distance from the transducer. Thus the minimum acoustic-pressure amplitude is completely zero at each pressure node where the incident and reflected waves are exactly cancelled each other. In actual experiments, however, there is absorption of ultrasound especially due to cavitation bubbles. As a result, there appears a traveling wave component because the intensity of the incident wave is higher than that of the reflected wave. Thus, the local minimum value of acoustic pressure amplitude is non-zero as seen in Fig. 1.13. It should be noted that the acoustic-pressure amplitude at the liquid surface (gas-liquid interface) is always zero. In Fig. 1.13, there is the liquid surface... 

The electromagnetic field in the cavity is a standing wave which can be described as a superposition of two travelling waves propaga-... 

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. 

Fig. 1.2 (lustration o< a standing wavs, showing how the displacement 0) varies with position (x) at three successive times in (a), (b), and (c). The standing wave can be regarded as a superposition of two travelling waves, of equal amplitude and moving in opposite directions. 

Of course, according to the Section ELD, this calculation should really be carried out at wavelengths X — s instead of traveling times T—s. The Sagnac-type experiments are also standing-wave systems. Then the magnitude of shift of the interference fringes with the above oo... 

In the spirit of the standing-wave picture of Sagnac-type experiments, this theory needs to recalculate the result of the Michelson-Morley experiment as well. In the M-M experiment there is a new unknown hidden parameter cp, which denotes the speed of light in the direction perpendicular to the earth s velocity. The traveled path of light in the perpendicular arm Xp = 2Tcp [dim X = meter], [where cp is speed of light perpendicular to the velocity... 

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