Philip M. Morse

STOCHASTIC PROPERTIES OF WAITING LINES

The stochastic properties of waiting lines may be analyzed by a two-stage process: first solving the time-dependent equations for the state probabilities and then utilising these transient solutions to obtain the auto-correlation function for queue length and the root-mean-square frequency spectrum of its fluctuations from mean length. The procedure is worked out in detail for the one-channel, exponential service facility with Poisson arrivals, and the basic solutions for the m-channel exponential service case are given. The analysis indicates that the transient behavior of the queue length n(t) may be measured by a “relaxation time,” the mean time any deviation of n(t) away from its mean value L takes to return (1/e) of the way back to L. This relaxation time increases as (1 − ρ)−2 as the utilization factor ρ approaches unity, whereas the mean length L increases as (1 − ρ)−1. In other words, as saturation of the facility is approached, the mean length of line increases; but, what is often more detrimenta...

The Transmission of Sound Inside Pipes

Methods developed in a previous paper are used to obtain an exact solution for the transmission of sound inside a rectangular pipe with absorbing material on its inner walls. Plots are given whereby it is possible to determine the phase velocity and attenuation of the simpler type waves in terms of the specific acoustic impedance of the walls and the frequency of the wave. Several examples are given of the effect of the absorbing surfaces on the distribution of the sound pressure amplitude over a plane perpendicular to the axis of the pipe. The effect of the reactive part of the materials acoustic impedance on the attenuation of the sound is also discussed. Formulas are developed for the transmission of sound in tubes of circular cross section. The plots for rectangular pipes can also be used in calculations of the acoustical properties of rectangular rooms.

Linear Acoustic Theory

Acoustical motion is, almost by definition, a perturbation. The slow compressions and expansions of materials, discussed in thermodynamics, are not thought of as acoustical phenomena, nor is the steady flow of air usually called sound. It is only when the compression is irregular enough so that over-all thermodynamic equilibrium may not be maintained, or when the steady flow is deflected by some obstacles so that wave motion is produced, that we consider part of the motion to be acoustical. In other words, we think of sound as a by-product, wanted or unwanted, of slower, more regular mechanical processes. And, whether the generating process be the motion of a violin bow or the rush of gas from a turbo-jet, the part of the motion we call sound usually carries but a minute fraction of the energy present in the primary process, which is not considered to be acoustical.

Edward Uhler Condon, 1902–1974

The middle third of the twentieth century was the era of hegemony of physics in American science. During that whole period Edward Uhler Condon was a leader in physics, in research of his own, in stimulating research in others, in applying physics, and in calling attention to the effects on all of us of its indiscriminate and irrational application. When he made his first contribution to theoretical physics in 1926, the word physics was not in the vocabularies of most Americans and the revolutionary concepts of quantum mechanics and relativity were just being worked out in Europe; by 1960 the applications of electronics and solid state physics had begun to change our lives irreversibly, and the implications of nuclear physics were manifest to everyone. Ed Condon contributed to each part of this explosive evolution.

The beginnings of operations research in the United States

This paper, drawn from the authors autobiography, gives an account of the events leading up to the creation of the first operations research group in the United States, and describes how it approached its first problem, improving antisubmarine warfare operations off the Atlantic coast of the United States.

Theorie der Streuung langsamer Elektronen an Atomen

ZusammenfassungZur Erläuterung des Ramsauereffekts wird die Beugung von Elektronenwellen an Atomen berechnet mit Hilfe einer Reihenentwicklung nach Kugelfunktionen, ähnlich wie sie in der Optik zur Erklärung der Farben kolloidaler Lösungen bentzt wird. Das Atom wird idealisiert als Kern mit einer Elektronenschale. Austausch ist nicht berücksichtigt. Die Wirkungsquerschnittskurven dieses Modells sind Funktionen zweier Parameterβ undr0,β bestimmt den Kurventypus. Die Kurven sind „quasiperiodisch“ in diesem Parameter und die Perioden stimmen mit denen des periodischen Systems überein.r0 bestimmt sowohl die Querschnitts- als auch die Geschwindigkeitsskale. Eine Methode zur Berechnung vonβ undr0 für ein beliebiges Atom wird angegeben. Die theoretischen Kurven stimmen mit den experimentellen nach Größe und Form sehr gut überein. Auch die Winkelabhängigkeit wird gut wiedergegeben.

Measures of Library Effectiveness

Mathematical models of library operations are presented, allowing managers to estimate measures of effectiveness for a library. These models describe the amount of use made of resources by a user in a visit, the distribution of book circulation in a collection, the dependence of circulation on time, and the effect of multiple copies on user satisfaction. Predictions are made on the basis of the models of the consequences of breaking a central library into branch libraries. The effect, in terms of frustrated use, of removing the least-used books from a collection is discussed, as are strategies for for duplication. The emphasis is on facilitating getting results from models; for this purpose graphic techniques supplement the mathematical formulas.

Search Theory and Browsing.

Search theory is applied to the activity of scanning the library shelves for books of possible interest, or a catalog (card or computerized) for items of immediate utility. The result is a formula quantifying the fact that browsing follows the law of diminishing returns. From the formula one can obtain methods for optimizing the chance of success for a given amount of search effort. These methods are utilized to discuss how a library (or catalog) might be arranged to aid the average browser. In particular, if space limitations require that a given section of the library be divided into an open-shelf browsing fraction and a remainder, less accessibly shelved, the theory indicates what size the browsing fraction should be to optimize the success of the average browser. The effectiveness of division according to book circulation versus book shelf age, in regard to efficiency for browsing of the open-shelf fraction, is calculated and plotted against size and mean shelf age of the undivided section.

Some Aspects of the Theory of Room Acoustics

An exact solution for the decay of sound in a rectangular room is obtained; assuming that each wall is uniformly covered with absorbing material, which may differ from wall to wall. It is concluded, from recent experimental measurements, that the boundary conditions for the sound field are correctly expressed in terms of the effective normal impedance of the wall material. The sound is analyzed into its component normal modes of vibration, and the reverberation times and frequencies of the different normal modes are calculated as functions of the wall impedances and their phase angles. Curves are given for these quantities for a wide range of the parameters involved. The effect of the absorbing material in distorting the sound field is shown, and several other interesting points are brought out in the discussion: that waves which travel “parallel” to a wall are absorbed by the wall, but are not absorbed as much as are waves striking at more oblique angles; that it is sometimes possible to increase the rev...

Mathematical problems in operations research

During the past war a new field of application of the scientific method was developed, called operations research, or operational analysis. A number of mathematicians contributed to this development and some of their work should be of general interest to mathematicians, though reasons of military security have precluded publication of much of it. I t is my intention to discuss a number of problems in this field which can now be talked about, and to indicate why further work in operations research is of some importance at present. The term operations research has heretofore been used to connote various attempts to study the operations of war by scientific methods. From a more general viewpoint, however, operations research can be considered to be an at tempt to study those operations of modern society which involve organizations of men or of men and machines. The study is behavioristic, not subjective, and the aim is to be able to predict the changes produced in the operation by changes in external conditions. The subject can be said to cover, for example, the quantitative aspects of city planning (including traffic analysis), those parts of the study of telephone traffic handling which are called systems engineering, and certain aspects of efficiency engineering. Its methods might be applied to a scientific study of peacetime operations, such as those of railroads or of other transportation systems, if this were desired. In each case one seeks to study the dependence of certain measures of effectiveness of an operation upon certain operational parameters which are subject to the control of the director of the operation. I t is important to appreciate what operations research is not, as well as what it is : it is not efficiency engineering or historical analysis nor is it a branch of applied statistics. I t views operations from a much more active and experimental point of view than do these other subjects. I t is obvious that operations research is a branch of engineering, rather than of pure science, since it is concerned with the application of scientific method to immediate and pressing practical problems. At least in the initial stages it must be developed in close connection with these practical problems and in personal contact with the administrators who have control over the operation under study. The