The Sine Wave

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 5.162. The increase in KC at T=58 gives very high peaks in FO (curve A). The sine wave disturbance of F3 is seen in curve D.

It should be recognized that the discrete Fourier coefficients G(x, y, co) are represented by complex numbers. The real part Re(G(x, y, to)) of the complex number represents the amplitude of the cosine part of the sinusoidal function and the imaginary part Im(G(x, y, co)) represents the amplitude of the sine wave. 

The data collected are subjected to Fourier transformation yielding a peak at the frequency of each sine wave component in the EXAFS. The sine wave frequencies are proportional to the absorber-scatterer (a-s) distance /7IS. Each peak in the display represents a particular shell of atoms. To answer the question of how many of what kind of atom, one must do curve fitting. This requires a reliance on chemical intuition, experience, and adherence to reasonable chemical bond distances expected for the molecule under study. In practice, two methods are used to determine what the back-scattered EXAFS data for a given system should look like. The first, an empirical method, compares the unknown system to known models the second, a theoretical method, calculates the expected behavior of the a-s pair. The empirical method depends on having information on a suitable model, whereas the theoretical method is dependent on having good wave function descriptions of both absorber and scatterer. 

The m/z values of peptide ions are mathematically derived from the sine wave profile by the performance of a fast Fourier transform operation. Thus, the detection of ions by FTICR is distinct from results from other MS approaches because the peptide ions are detected by their oscillation near the detection plate rather than by collision with a detector. Consequently, masses are resolved only by cyclotron frequency and not in space (sector instruments) or time (TOF analyzers). The magnetic field strength measured in Tesla correlates with the performance properties of FTICR. The instruments are very powerful and provide exquisitely high mass accuracy, mass resolution, and sensitivity—desirable properties in the analysis of complex protein mixtures. FTICR instruments are especially compatible with ESI29 but may also be used with MALDI as an ionization source.30 FTICR requires sophisticated expertise. Nevertheless, this technique is increasingly employed successfully in proteomics studies. 

Newman and Lerner (N2) have used an arrangement where the signal picked up by a microphone attached to the flat surface below the orifice plate is amplified and fed to a loud speaker. The amplified bubble signal is then fed to one pair of fixed contacts of a double-pole double-throw switch of which the other pair of fixed contacts is connected to an audio-frequency generator. The movable contacts of the switch are connected to the vertical and ground terminals of an oscilloscope. This arrangement permits the observation of either the bubble signal or the sine wave as a function of the internal linear time-base of the oscilloscope. 

The signal, amplified to a good sound level at the loudspeaker, is fed to the vertical input of the oscilloscope. A stationary trace is obtained on the oscilloscope. The frequency of the sine wave of the oscilloscope is varied until a single trace is obtained. This frequency is then equal to the frequency of bubble formation. 

When activated, the instrument delivers a sustained high-frequency AC waveform. Current density is high at the implement and local heating causes tissue destruction. The sine wave continues until the switch is released. 

Verify a True Peak measurement instrument by first establishing an initial calibration nsing a sine wave. Then verify the pulse response of the instrument nsing a nonsymmetrical duty cycle pulse and compare the results to the sine wave calibration. Follow these steps ... 

Switch the pulse/funetion generator from the sine wave to the pulse signal as shown in Figure S-3. 

When the tracer input to a system is varied in a sinusoidal fashion, the specific form of response observed will depend on the frequency, k, of the sine wave. Applying eqn. (7) and using Table 9 (Appendix 1)... 

The results of the Fourier analysis show that the magnitude of the sine wave at 1 kHz is 7.463 V, and the magnitude of the sine wave at 2 kHz is 1.393. These results are similar to the results obtained using Probe. This method also gives us the magnitude of the frequency components too small to see on the Probe graph, the phase of each frequency component, and the total harmonic distortion. From the data above, an equation for the output voltage is ... 

Run the simulation and plot the sine wave and square wave on separate plots. To add plots to the Probe window, select Plot and then Add Plot to Window from the Probe menus. 

EXEHCISE 0-1 Design a 60 Hz sync circuit. The input to the circuit is a 12 V amplitude, 60 Hz sine wave. The output of the circuit should be a 1 ms 5 V pulse that occurs when the sine wave crosses zero with a positive slope. 

The electron is not allowed outside the box and to ensure this we put the potential to infinity outside the box. Since the electron cannot have infinite energy, the wave function must be zero outside the box and since it cannot be discontinuous, it must be zero at the boundaries of the box. If we take the sine wave solution, then this is zero at =0. To be zero at x=a as well, there must be a whole number of half waves in the box. Sine functions have a value of zero at angles of nn radians where n is an integer and so... 

In actual calculations on crystals, it is impractical to include all lO atoms and so we use the periodicity of the crystal. We know that the electron density and wave function for each unit cell is identical and so we form combinations of orbitals for the unit cell that reflect the periodicity of the crystal. Such combinations have patterns like the sine waves that we obtained from the particle-in-the-box calculation. For small molecules, the LCAO expression for molecular orbitals is... 

Although all three simulators correctly predicted the frequency and amplitude of the sine wave, only the IsSpice simulation predicted the DC offset in the output waveform. The reason for this is the simulations for each used the LM124 model that came with the simulation... 

It follows from Table 5 that both SF and Sc can be considered as step functions, applied at the moment (t = 0) the sine wave train is started. By elimination of AE2 from eqns. (76) and (79), with substitution of eqn. (74), an integral equation in Ajii2 = AjfF- 2 = — Ajrc, 2 is obtained... 

A time-dependent quantity X t) responding to the sine wave modulation of some low level perturbation dKcos a>t and frequency a>/2n, is generally written in linear conditions as ... 

S. Brukenstein et al. proposed in 1973 [5] this technique as an alternative to the sine-wave modulation technique. Their approach was based on the Benton s analysis of fluid flow near an impulsively started disk from rest [61]. They assumed that the... 

The motivation for the sine-wave representation is that the waveform, when perfectly periodic, can be represented by a Fourier series decomposition in which each harmonic component of this decomposition corresponds to a single sine wave. More generally, the sine waves in the model will be aharmonic as when periodicity is not exact or turbulence and transients are present, and is given by... 

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