by Ed Prism


The purpose of the transformation is to convert a differential equation into an algebraic equation in order to simplify its solution.
This technique exploits what is known as the "pinch" effect.
Along with the specular and Rayleigh-fading signal components, additive bandpass zero-mean Gaussian noise is also present in the receiver.
Is it possible for a class to be infinite and yet essentially different in multiplicity from another infinite class, such as that of the positive integers?


Simulators have become recognized as an essentail tool for research on and development of automatic systems, and the present effort is directed toward extending the application of this tool for research and development of semi-automatic systems. At the same time we obtained results that are directly applicable to many practical situations in which sound waves are plane or nearly plane. However, we note that even a perfectly square driving pulse is finally distorted as a consequence of the limited transducer bandwidth. We shall now indicate how all these results can be extended, at least within certain limits, to encompass waves and vibrations of much more general form, including pulsed wave trains, square waves, and individual transients such as are obtained in shock excitation.

Frequency limits
Let us examine these spectral characteristics more closely with reference to Fig.1. For convenience in observing the action of the excitation, a time scale was chosen such that 10 sec in the computer were equivalent to 1 msec in the problem.


Fig. 1

Whether in its pristine state, in solution or colloidal, a liquid is motivated, as are gasses and solids, by the nature and behavior of its molecules. Assuming that under these circumstances the specific distortions due to fading are unpredictable on any specific pulse, it is then impossible to coherently extract all the available signal energy. Similar derivations to those above were also carried out for "exact" coherent detection with either FSK or phase reversal keying, a model in which it is assumed that coherent matched-filter detection is available on the combined signal. Fig. 2 shows the analog results for three different initial conditions.


Fig. 2

In practice, the frequency of a standard cavity is compared with the frequency of the test cavity which eliminates the influence of fluctuations of the ambient temperature or humidity on the compressibility of the gas. This is largely true of equations of the parabolic type as well. If two such signals, of equal strength, are added together for transmission, the transmitted signal is simply a carrier whose phase appears to shift among the four phase positions 45, 135, -135, and - 45 degrees. This is a fundamental equation of state for fluids which accounts for the phenomena of condensation and evaporation which are characteristic of the transition between gaseous and liquid phase.

  Some properties of wave motion
It will be instructive to study the velocity of sound waves in gases and liquids, as related to properties of the medium. What is observed when several paths exist with time differentials smaller than the reciprocal signal bandwidth? Up to this point, the system has been assumed to be linear. Most systems will exhibit limiting and other nonlinear phenomena, which may or may not be ignored. From Fig. 3, it can be estimated that the maximum frequency that will be encountered is on the order of 70 radains/sec. The boundary conditions as well as the initial conditions must be satisfied in the process.


Fig. 3

  Dual-diversity selection combining
One of the simplest, most unitary forms of information is the recording of a choice between two equally probable simple alternatives, one or the other of which is bound to happen- a choice, for example, between heads and tails in the tossing of a coin. Since specific projects are inevitably classified for security reasons, this section will present only basic ideas and will not deal with any particular application.
For example, the velocity of a sound wave is a function of the density and compressibility of the medium. Formulation of mathematical models for such systems leads to ordinary differential equations. This approach must be used when the solution for a dependent variable involves other dependent variables (or other derivatives of the first variable) that are being solved for at the same time.
The resulting computer diagram is given in Fig. 4.


Fig. 4

The reader is referred to more complete texts on the subject for detailed treatment of other formulas and for analysis of the resulting error.
In the past there were muted sanctuaries where anyone suffering from sound fatigue could go into retirement for recomposure of the psyche. We have sketched the boundaries of the field from the point of view of the physical phenomena involved.
No knowledge of the internal structure is assumed.
Therefor, the technique of sonics, like those of electronics, will be used by engineers and scientist of many different backgrounds.

March, 1958