Periodicity of waves

FIGURE 6.10. Periodicities of waves and their representation. Shown, for the same unit cell in each case, are various waves F hkl) with different periodicities, (a) 300, (b) 050, (c) 350, and (d) 352. Note that, for E(352), the wave repeats 3 times in the a direction, 5 times in the b direction, and 2 times in the c direction. 

The postulate of Nagaoka and de Broglie, and the discovery of electron diffraction suggested that the appearance of integer quantum numbers relates to the periodicity of wave motion, which is also characterized by integers, and that the behaviour of quantum particles should be described by the general wave equation, which in one dimension reads ... 

Table 2. Correlation coefficients between the periods of waves for variations.

Fig. 18. Periods of waves 1-5 as a function of the peak positions of the waves.

With the surf similarity parameter = tana/y/Hs/Lop expressed in terms of the deepwater length Lop = gT /27t (Tp = peak period of wave spectrum) the following stability formula is obtained in terms of the characteristic size D of the container ... 

In the band 0.05-1 Hz all models of ground motion are dominated by a peak called the micro-seismic peak. This peak is in fact made up of two peaks. The first peak, typically between 10 and 16 s period, corresponds to the natural period of waves generated by storm winds in mid-ocean. This peak is sometimes called the primary... 

Theoretically, the asymptotic fonn of die solution for the electron wave fiinction is the same for low-energy projectiles as it is at high energy however, one must account for the protracted period of interaction between projectile and target at the intennediate stages of the process. The usual procedure is to separate the incident-electron wave fiinction into partial waves... 

The partial wave decomposition of the incident-electron wave provides the basis of an especially appealing picture of strong, low-energy resonant scattering wherein the projectile electron spends a sufficient period of time in the vicinity... 

Consider the wave packet populating just one vibrational level. This occurs for only a short period of time (the length of the femtosecond pulse). Then we can think of vibration occurring in a classical fashion. The wave packet travels along the vibrational level until it reaches the other extremity when it may be reflected and continue to travel backwards and forwards along the level. Because of the strongly anharmonic nature of the vibration the wave packet is broadened, as shown, as r increases. 

Fig. 42. Torsion pendulum and typical damped sine wave output. P is the period of the motion and M2 are successive ampHtudes (241).

Successive reflections of the pressure wave between the pipe inlet and the closed valve result in alternating pressure increases and decreases, which are gradually attenuated by fluid friction and imperfect elasticity of the pipe. Periods of reduced pressure occur while the reflected pressure wave is travehng from inlet to valve. Degassing of the liquid may occur, as may vaporization if the pressure drops below the vapor pressure of the liquid. Gas and vapor bubbles decrease the wave velocity. Vaporization may lead to what is often called liquid column separation subsequent collapse of the vapor pocket can result in pipe rupture. 

The response of a target is a function of the ratio of the blast wave duration and the natural period of vibration of the target (T/T ). Neither of these parameters can be closely defined. 

Since the recognition in 1936 of the wave nature of neutrons and the subsequent demonstration of the diffraction of neutrons by a crystalline material, the development of neutron diffraction as a useful analytical tool has been inevitable. The initial growth period of this field was slow due to the unavailability of neutron sources (nuclear reactors) and the low neutron flux available at existing reactors. Within the last decade, however, increases in the number and type of neutron sources, increased flux, and improved detection schemes have placed this technique firmly in the mainstream of materials analysis. 

Along with, and closely connected to, the developments in precise impact techniques is the development of methods to carry out time-resolved materials response measurements of stress or particle velocity wave profiles. With time resolutions approaching 1 ns, these devices have enabled study of mechanical responses not possible in the early period of the 1960s. The improved time-resolutions have resulted from direct measurement of stress or particle velocity, rather than from improved accuracy and resolution in measurement of position and time. In a continuation of this trend, capabilities are being developed to provide direct measurements of the rate-of-change of stress. With the ability to measure such a derivative function, detailed study of new phenomena and improved resolution and accuracy in descriptions of known rate-dependent phenomena seem possible. 

Table 3.3 summarizes the history of the development of wave-profile measurement devices as they have developed since the early period. The devices are categorized in terms of the kinetic or kinematic parameter actually measured. From the table it should be noted that the earliest devices provided measurements of displacement versus time in either a discrete or continuous mode. The data from such measurements require differentiation to relate them to shock-conservation relations, and, unless constant pressures or particle velocities are involved, considerable accuracy can be lost in data processing. 

An explosion is defined by StrelUow and Baker " as an event in wliich energy is released over a sufficiently small period of time and in a sufficiently small volmne to generate a pressure wave of finite amplitude traveling away from tlie source. Tliis energy may have been originally stored in tlie system as chemical, nuclear, electrical, or pressure energy. However, tlie release is not considered to be explosive unless it is rapid and concentrated enough to produce a pressure wave tliat can be heard. 

Changing a size 400 system to a size 50 system, for example, effectively speeds-up the decay into a plane wave state but otherwise leaves the period unchanged. Likewise, although patterns starting from two different initial states consisting of either 10 or 20 randomly positioned nonzero sites typically have different appearances, the periodicity of the resulting global structures is exactly the same. 

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