Sine waves

Instruments are available based on sine waves, square waves tind pulses singly or in combinations. 

We now define the effect of a translational synnnetry operation on a fiinction. Figure Al.4.3 shows how a PHg molecule is displaced a distance A X along the X axis by the translational symmetry operation that changes Xq to X = Xq -1- A X. Together with the molecule, we have drawn a sine wave symbolizing the... 

Figure Al.4.3. A PH molecule and its wavefiinction, symbolized by a sine wave, before (top) and after (bottom) a translational synnnetry operation.

The interaction of electromagnetic radiation with matter can be explained using either the electric field or the magnetic field. For this reason, only the electric field component is shown in Figure 10.2. The oscillating electric field is described by a sine wave of the form... 

The quantity x k) in Equation (8.20) is the experimentally observed absorption, like that in Figure 8.32, after subtraction of the smoothly declining background. What is left is a sum of sine waves of which we require the wavelengths which can be related to Rj, provided the phase factor 6j k) is known. This process of obtaining wavelengths from a superposition of... 

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

In EIS, a potential is applied across a corroding metal in solution, causing current to flow The amount of current depends upon the corrosion reaction on the metal surface and the flow of ions in solution. If the potential is apphed as a sine wave, it will cause harmonics of the current output. The relationship between the apphed potential and current output is the impedance, which is analogous to resistance in a DC circiiit. 

The test voltage must be as close to a sine wave as practicable and frequency as noted in column 3 of Tables 14.1 and 14.2, for series II and Table 13.2 for series I voltage systems, and applied for one minute as follows ... 

Figure 14.20 A typical free vibration record (sine wave) illustrating natural frequency of vibration and level of damping of an object F f>...

Inverters do not output a pure sine wave but synthesize the output wave with pulses. Because of the pulses, harmonics are presented to the motor and, hence, the somewhat higher losses. Common systems are either 6 pulse or 12 pulse. This definition comes from the number of... 

Rgure B-3S. The relationship between peak, peak-to-peak, and average amplitudes for a sine wave. 

Now presume that P achieves a minimum for a particular sine wave, say the m " wave. Thus,... 

Figure 13-3. Typicai pressure wave in the suction and discharge piping shown as composite of (A) waves that are muitipies of fundamentai sine wave, (B) waves (1), (2), and (3) representing muitipie frequencies of puisations that act as exciting forces for vibration. Note Verticai axis is pressure. (Used by permission Hicks, E. J. Oil and Gas Journal, p. 38, Juiy 24, 1978. PennWeii Pubiishing Company. Aii rights reserved.)...

Assume that the shock specification refers to the maximum deceleration (a ) of a half sine wave impact. The mean deceleration will be taken equal to 0.66 a. ... 

The early system was transmitting 1.5 bits/s (4 sine waves to identify one bit). Later systems went to 3 bits/s. Now with a 24-Hz carrier frequency, 6 bits/s can be transmitted. 

We have built a fluidic puiser system that can generate approximately 100 psi peak to peak with 500 gal/min mud flowrate. It is to be used down to 15,000 ft. The surface detector needs a 5-psi peak to peak sine wave for proper phase detection. The following oil-base mud is used ... 

The system transmits 5 bits/s with a phase-shift-keying system. Four sine waves are necessary to define the phase with a negligible chance of error. Assume a perfect pipe, drill collar ID same as the drill pipe and no wave reflections at the drill pipe ends. 

The most recent development in the starting of squirrel-cage induction motors is the introduction of the electronic soft-start. This principle has been derived from variable-frequency speed controllers using switched Thyristor or power transistor bridges. The supply sine wave is chopped so that a reduced voltage and frequency is applied to the motor.These are gradually increased so that the motor speed rises in a controlled manner, with the starting current limited to any chosen value. 

High/low control This may be on/off in certain applications. Control is crude, with the burner moving to high fire at a point somewhere below the nominal set point and returning to low fire somewhere above it. The result is a sine wave control pattern with a band rather than a point of control. This is, however, still an adequate control mode for many purposes. 

Audible sound has a frequency range of approximately 20 Hertz (Hz) to 20 kilohertz (kHz) and the pressure ranges from 20 x 10 N/M to 200 N/M. A pure tone produces the simplest type of wave form, that of a sine wave (Figure 42.1). The average pressure fluctuation is zero, and measurements are thus made in terms of the root mean square (rms) of the pressure variation. For the sine wave the rms is 0.707 times the peak value. 

Figure 43.10 is the superposition of two sine waves having different frequencies. These curves are represented by the following equations ... 

In the above equation, the first two terms are the undamped free vibration, while the third term is the undamped forced vibration. The solution, containing the sum of two sine waves of different frequencies, is itself not a harmonic motion. 

Ax = Amplitude of each discrete sine wave ft) = Frequency... 

Another modification to the slow strain-rate test involves the superimposition of a low amplitude sine wave ripple on the slow uniform extension (Fig. 8.47). In effect this produces higher strain rates (which appear to be more damaging for hydrogen embrittlement), while still giving a long test duration, with adequate time for the accumulation of hydrogen in the steeps. 

In this procedure, a constant sine wave a.c. potential of a few millivolts is superimposed upon the usual d.c. potential sweep. The applied d.c. potential is measured in the usual way and these results are coupled with measurements of the alternating current. 

The basis of all control Icd-potcntial techniques is the measurement of the current response to an applied potential. There exist a multihide of potential excitations, including a ramp, potential steps, pulse trains, a sine wave, and various combinations thereof. The present chapter reviews those techniques that are widely used. 

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