Waves sinusoidal

The crimp imparted to the tow has a sawtooth or sinusoidal wave shape. Because the filaments are usually crimped as a group, the crimp in parallel fibers is in lateral registry, ie, with the ridges and troughs of the waves aligned, as shown in Figure 14. 

The second component is caused by the different harmonic quantities present in the system when the supply voltage is non-linear or the load is nonlinear or both. This adds to the fundamental current, /,- and raises it to Since the active power component remains the same, it reduces the p.f of the system and raises the line losses. The factor /f/Zh is termed the distortion factor. In other words, it defines the purity of the sinusoidal wave shape. 

Figure 8 HRTEM image of the ineommensurately modulated phase observed at room temperature in Ti5oPd43Cr7- The modulations can be described by a sinusoidal wave with wave-vector 0.31 [IlOJbcc (courtesy Schwartz et al. )...

Continuous-Wave Transmission. Anadrill, a subsidiary of Schlumberger, markets a tool which produces a 12-Hz sinusoidal wave downhole. Ten-bit words representing data are transmitted by changing or maintaining the phase of the wave at regular intervals (0.66 s). A 180° phase change represents a 1, and phase maintenance represents a 0. 

In Fig. 1.12, a typical experimental set-up of a bath-type reactor is shown. An electric signal with sinusoidal wave of a chosen ultrasonic frequency is generated by a function generator. The signal is amplified by a power amplifier. Then it is... 

The Fourier transform (FT) relates the function of time to one of frequency—that is, the time and frequency domains. The output of the NMR spectrometer is a sinusoidal wave that decays with time, varies as a function of time and is therefore in the time domain. Its initial intensity is proportional to Mz and therefore to the number of nuclei giving the signal. Its frequency is a measure of the chemical shift and its rate of decay is related to T2. Fourier transformation of the FID gives a function whose intensity varies as a function of frequency and is therefore in the frequency domain. 

Some examples of FID signals and their corresponding frequency domain spectra are shown in Figure 9.41. The FID signal related to a single resonance peak (Figure 9.41(a)) is seen to consist of a decaying sinusoidal wave whose frequency corresponds to that of the resonance frequency. Two... 

Although a liquid sheet may leave the nozzle with some perturbations, the principal cause of the instabilities is the interaction of the sheet with the high-velocity air streams whereby rapidly growing waves are imposed on the sheet. Disintegration may occur when the amplitude of these waves reaches a critical value. Each full sinusoidal wave is initially distorted to yield two half-waves of very similar forms. The constant stretching of the half-waves increases... 

In order to describe the material properties as a function of frequency for a body that behaves as a Maxwell model we need to use the constitutive equation. This is given in Equation (4.8), which describes the relationship between the stress and the strain. It is most convenient to express the applied sinusoidal wave in the exponential form of complex number notation ... 

If the acoustic pressure, P, is replaced with - P sin w t (the form appropriate for a sinusoidal wave of pressure amplitude P and circular frequency w (= 2n f), Eq. A.25 reduces to that first derived by Noltingk and Neppiras. 

The TA Instruments CSL2 rheometer can perform low frequency oscillatory measurements as well as steady-state viscosity determinations, even though it has a simple mechanical system. The sinusoidal wave form is generated mathematically in the computer rather than with an electromechanical drive system. The stress is controlled, and the resulting strain is determined and stored in memory. The computer analyzes the wave form and calculates the viscosity and elasticity of the specimen at the frequency of the test. As of this writing (1996), the oscillation software covers a frequency range of 10-4 -40 Hz. This range could be increased as faster software and computers become available. 

The output of this circuit is generated from a series of square-wave pulses. Square waves are easier to generate than sine waves and can accurately be produced by a digital logic IC. This circuit generates a series of square-wave pulses, skewed in time and summed. This resultant summed waveform resembles a stepped representation of a sinusoidal-wave. This waveform is passed through a filter, and the final signal is a clean sine wave. 

A mathematical description of an ultrasonic wave must describe the dependence of the particle displacement on distance and time, and the reduction of its amplitude with distance traveled through the material. For plane sinusoidal waves the following equation is appropriate ... 

In practice ultrasound is usually propagated through materials in the form of pulses rather than continuous sinusoidal waves. Pulses contain a spectrum of frequencies, and so if they are used to test materials that have frequency dependent properties the measured velocity and attenuation coefficient will be average values. This problem can be overcome by using Fourier Transform analysis of pulses to determine the frequency dependence of the ultrasonic properties. 

Fig. 1.1 A sinusoidal wave, showing the displacement (y) as a function of position (x), at three successive times, in (a), (b), and (c).

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. 

The above discussion of the correspondence principle was not entirely satisfactory. It may be true that for very high quantum numbers, the wavefunction represents a probability distribution indistinguishable from that predicted in classical mechanics. However, classical mechanics does not need probability distributions, as it deals with precisely known trajectories. How can the wave picture be compatible with these To answer this question, we must look at a wavefunction constructed by the superposition of sinusoidal waves with different lengths. We use the earlier equations in this chapter to write... 

It is shown in Fig. 2.9, for a case where p and p2 are fairly similar. The periodic variations of amplitude are known as beats . They come from the interference of two sinusoidal waves which make up yf in eqn 2.37 at some points these are in phase, and at other points out of phase. This phenomenon... 

Fig. 2.9 The wavefunction given by eqns 2.37 and 2.38, representing the superposition of two waves of different length. Note the heats , periodic variations in amplitude arising from the Interference of the two sinusoidal waves.

Fig. 2.10 A wave packet, obtained by superposing a set of sinusoidal waves in such a way that they interfere destructively outside a certain region.

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