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TABLE I

U(0)=U(1)=1 u(0) = 7, u(1) = 20 u(0)=1.5 u(l)=2

1 7 1.51 1 20 2.85714 2 1.333332 2 27 1.35 3.5 1.753 1.5 47 1.74074 5.5 1.571435 1.66667 74 1.57447 9 1.636368 1.6 121 1.63514 14.5 1.61111

13 1.625 195 1.61157 23.5 1.6206921 1.61538 316 1.62051 38 1.6170234 1.61905 511 1.61709 61.5 1.6184255 1.61765 827 1.6184 99.5 1.6178989 1.61818 1 338 1.6179 161 1.61809144 1.61798 2 165 1.61809 260.5 1.61801233 1.61806 3 503 1.61801 421.5 1.61804377 1.61803 5 668 1.61804 682 1.61803610 1.61804 9 171 1.61803 1 103.5 1.61804987 1.61803 14 839 1.61804 1 785.5 1.61803

1 597 1.61803 24 010 1.61803 2 889 1.61803

and

lim u(n + 1) =n-.oo

Thus,

U(n + 1) 1lim =-(1 + /5) = 1.618 -. (60)n- xo 8u(n) 2

Notice that it was not necessary to evaluate the Ciand C2 coefficients with the initial conditions of u(O)

= u(1) = 1 to find the limit. This means that no matterwhat u(0) and u(1) are, the limit will always be thesame, even though the numbers in the series are dif-ferent. This is a rather remarkable result-certainlynot intuitively obvious. Table I shows the first fewnumbers of three different Fibonacci series with theirlimit convergents. The convergents approach thetheoretical limit very rapidly.

ACKNOWLEDGMENT

The author is grateful to the Technical Paper Re-view Committee of the Ground Systems Group, HughesAircraft Company, for many helpful suggestions in thepreparation of this paper.

REFERENCES[1] H. Levy and F. Lessman, Finite Difference Equations. London:

Pitman, 1959.[2] K. S. Miller, An Introduction to the Calculus of Finite Differences

and Difference Equations. New York: Holt, 1960.[3] D. K. Cheng, Analysis of Linear Systems. Reading, Mass.:

Addison-Wesley, 1959.[4] E. Parzen, Modern Probability Theory and Its Applications. New

York: Wiley, 1960.[5] W. W. Harmon, Principles of the Statistical Theory of Communica-

tion. New York: McGraw-Hill, 1963.[6] D. Bergamini, Mathematics, Life Science Ser. New York: Time,

Inc., 1963.[71 R. G. Stanton, Numerical Methods for Science and Engineering.

Englewood Cliffs, N. J.: Prentice-Hall, 1961.[8] S. Goldberg, Introduction to Difference Equations. New York:

Wiley, 1958.

Summary Report on the New York University

Graduate Center at Bell Telephone

La boratories

ELIAS SCHUTZMAN, SENIOR MEMBER, I]'EE, AND JOHN N. SHIVE, SENIOR MEMBER, IEEE

Abstract-The New York University Graduate Center at BellTelephone Laboratories, Inc., was in continuous operation tor

nine full years after its establishment in the spring of 1957. Thisreport discusses the history preceding the creation of the GraduateCenter, the nature of the program, the curriculum of studies, andthe impact of the operation on the students, the faculty, and the two

Manuscript received June 14, 1968. This report was presented atthe Annual Meeting of the American Society for Engineering Educa-tion, Michigan State University, East Lansing, June 1967.

E. Schutzman is with the School of Engineering and Science, NewYork University, Bronx, N. Y.

J. N. Shive is with the Education and Training Center, Bell Tele-phone Laboratories, Inc., Holmdel, N. J.

organizations. A total of 783 employees of Bell Laboratories earnedmaster's degrees through studies undertaken at the GraduateCenter.

I. INTRODUCTION

N THE SPRING of 1957 representatives of NewYork University and Bell Telephone Laboratories,Inc., concluded an agreement which established

the New York University Graduate Center at BellTelephone Laboratories at Murray Hill, N. J. Thisaction launched a unique experiment in graduate engi-neering education, for it marked the institution of a

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IEEE TRANSACTIONS ON EDUCATION, DECEMBER 1968

full-scale daytime operation of a degree-grantingbranch of a university on the premises of an industrialorganization. Since, after nine years of operation, theGraduate Center was discontinued in 1966, it is ap-propriate to reflect on this experience, to give a sum-mary report of the activities of the Graduate Center,and to evaluate the performance in terms of the objec-tives of both the University and Bell Laboratories.A description of the activities at the Graduate Center,

the history preceding its establishment, and a sum-mary of progress made after two years of operationhave been given previously.",2 Certain salient featuresof those reports are included here to develop some per-spective.The first students were admitted to the Graduate

Division of the School of Engineering and Science ofNew York University to attend classes at the GraduateCenter during the summer of 1957. Prior to this time,all college-recruited members of the technical staff ofBell Laboratories at the B.S and M.S. degree levelswere required to participate in the three-year Communi-cation Development Training (CDT) Program estab-lished by the Company in 1948. It was considered byBell Laboratories that this program of formal class-room training and rotational department work assign-ments was necessary for young engineers who wishedto pursue research and development careers within theCompany. As a consequence, the curriculum andcourse work initially were directed toward the tech-nical needs of the Company. In most cases the courseswere taught by Bell Laboratories personnel. However,in some instances, individual faculty members fromnearby institutions were retained for instruction. Asthe program evolved, some courses became more funda-mental in character, while the content of others con-tinually changed to reflect the developing technologyof interest to the Company.

It became apparent by late 1956 that certain objec-tives of the CDT program might better be realized byhaving a university with its attendant resources par-ticipate in a cooperative arrangement with Bell Labora-tories. It was believed that an academic institution wasin a better position to offer a curriculum of fundamentalcourse work which would provide students at the B.S.degree level with a common body of basic knowledge.It was expected that this arrangement would haveother benefits as well. One of these was academic recog-nition for the participating student, since a graduatedegree would be awarded upon completion of the pro-gram. Also, the staff members of Bell Laboratorieswould be relieved of teaching obligations.

I S. S. Shamis, "The New York University Graduate Center atBell Telephone Laboratories," J. Engrg. Education, vol. 50, March1960.

2 S. S. Shamis and S. B. Ingram, "The NYU-Bell Labs program,"IRE Trans. Education, vol. E-2, pp. 58-61, April 1959.

After many discussions between representatives ofthe University and Bell Laboratories, a satisfactorycurriculum was formulated and an agreement wasreached which established the Graduate Center as abranch of the Graduate Division of the School of Engi-neering and Science.

II. THE GRADUATE CENTER PROGRAM

The Graduate Center was administered according tothe same scholastic standards and requirements asthose already in effect for the Graduate Division at theUniversity Heights campus of New York University.A resident director and assistant director (employedby the University), with the necessary supporting Uni-versity staff, were charged with the responsibility ofadministering the program, with the University main-taining complete control of admission requirements,scholarship standards, regulations governing the grant-ing of degrees, and content of courses. Qualified appli-cants, not employees of Bell Laboratories, were eli-gible to take graduate courses at the Center; theirnumber was limited by the available classroom spaceand the size of enrollment from Bell Laboratories dur-ing any semester. In general, the class section size wasnot allowed to exceed thirty students.A qualified applicant with the appropriate technical

background could select a program of studies leading toa master's degree in the field of electrical engineering,mechanical engineering, or engineering mechanics. Agroup of intensive summer courses, presented by thefaculty of the University, was made available to de-gree aspirants at the Graduate Center. Since the classwas formed through nation-wide recruiting, a variationin educational background existed among the membersof the entering class, and these summer courses were ofhelp to students in adjusting to their new educationalenvironment. Most students were able to satisfy un-dergraduate prerequisites by taking summer courses inadvanced calculus, electromagnetic theory, or passiveand active circuits at the Graduate Center. In thoserare cases when a student was deficient in more thantwo technical areas, the additional prerequisites weretaken during the regular academic year at the appro-priate undergraduate location of New York University.The student body at the Graduate Center consisted

of two main groups, composed of those who were enter-ing with bachelor's degrees and of those who were en-tering with master's degrees already earned. The num-ber in the former group far exceeded the number in thelatter. This report concentrates on the former group,those desiring their first graduate degree, aind in par-ticular, on the students in the field of electrical engi-neering, who represented most of the participants.The program at the Graduate Center was contin-

uously reviewed by representatives of the University and

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SCHUTZMAN AND SHIVE: NYU GRADUATE CENTER AT BELL LABORATORIES

TABLE I

Enrollment Withdrawals Master'sAcademic Semester Degrees

Regular Special Total Voluntary Requested Total Awarded

Fall 1957 233 28 261 14 8 22Spring 1958 249 62 311 11 11 22Fall 1958 279 48 327 9 7 16Spring 1959 266 40 306 3 11 14 97Fall 1959 305 72 377 10 11 21 4Spring 1960 287 69 356 6 11 17 78Fall 1960 348 58 406 5 11 16 5Spring 1961 337 47 384 3 16 19 154Fall 1961 314 50 364 5 5 10 2Spring 1962 307 55 362 1 2 3 134Fall 1962 259 47 306 2 7 9 13Spring 1963 236 52 288 1 6 7 140Fall 1963 158 63 221 4 1 5 2Spring 1964 155 41 196 1 1 2 69Fall 1964 88 56 144 5 1 6 3Spring 1965 84 49 133 2 0 2 65Fall 1965 17 55 72 1 0 1 1Spring 1966 16 48 64 0 0 0 16

Bell Laboratories, and the one that is outlined belowrepresents the most recent version for electrical engi-neering students. (The numbers in parentheses repre-

sent the number of credits for the particular course.)

First Year (24 credits)Fall SemesterFundamentals of Network TheoryFoundations of Mathematical AnalysisDynamics IAnalysis and Design of Active Circuits I

(3)(3)(3)(3)

Spring SemesterFundamentals of Electromagnetic Fields (3)Functions of a Complex Variable (3)Theory of Switching Circuits (3)Analysis and Design of Active Circuits II (3)

Second Year (18 credits chosen from the following)Probability and Statistics in Engineering (3)Time Series (3)Solid-State Electronics 1, 11, and III (9)Quantum Electronics (3)Magnetic Resonance in Solids (3)Introduction to Microwave Networks I (3)Microwave Measurements (3)Electromagnetic Wave Propagation (6)Transform Analysis of Networks (6)Feedback Amplifiers and Control Systems (6)Feedback Control Systems and Servomechanisms I and II (6)Network Theory (6)Network Synthesis (6)Methods of Noise and Random Process Analysis (6)Theory of Discrete Time Series (3)Dynamics II (3)

The School of Engineering and Science requires 36credits of graduate level courses for the master's de-gree; the degree candidate must receive grades of A or

B in course work of 24 credits. A candidate in the pro-gram accumulated 42 credits by the end of his secondyear, 6 credits beyond the minimum University re-

quirement. He attended classes three days per weekduring the first year (24 credits) and two days perweek during the second year (18 credits). In the main,the third year of the program consisted of technologicalcourses taught by Bell Laboratories personnel on a one-

day-per-week basis.

III. VITAL STATISTICS

Bell Laboratories hires most of its new employees byrecruitment on college campuses throughout the coun-try. The 783 candidates who received their master'sdegrees during the nine years of operation of the Grad-uate Center came from 130 undergraduate schoolslocated in 44 different states. Eight students receivedtheir undergraduate training at foreign schools. Of the783 M.S. degrees granted, 703 were in electrical engi-neering and the remaining 80 in either mechanical engi-neering or engineering mechanics.

Enrollment and withdrawal statistics are shown inTable I. The student enrollment during the first twoyears of the program consisted of both newly employedand previously employed members of the technicalstaff, in approximately equal numbers. From the fallof 1959 to the spring of 1963 (middle years of operation),a yearly input of approximately 175 B.S. degree leveland 50 M.S. degree level students were enrolled in theGraduate Center. The peak student enrollment oc-curred during the fall semester 1960-1961 (406 studentsregistered). During the last three years, the enrollmentsteadily declined, first because of the diversificationof the educational program of Bell Laboratories, andsecond because of the scheduled phasing-out periodprior to the termination of the operation of the Gradu-ate Center. The average number of students registeredfor the summer session (not indicated in Table I) was114.At no time did the percentage of registered students

requested to withdraw for reasons of poor scholarshipexceed 4.5 percent in any one semester. In most in-stances, this percentage was approximately 1.5 to 2.0percent. The decreasing numbers of withdrawals fromyear to year, indicated in column 7 of Table I, resultedfrom recruiting measures adopted by Bell Laboratoriesto improve the selection of employees better qualifiedfor the program. The information contained in Table

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IEEE TRANSACTIONS ON EDUCATION, DECEMBER 1968

500

400

EI

.i300

200

57-58 58-59 59-60 60-61 61-62 62-63 63-64 64-65 65-66ACADEMIC YEAR

Fig. 1. Statistical data concerning the New York University Graduate Center at Bell Telephone Laboratories, Inc., 1957-1966.

I is shown in graphical form in Fig. 1.Thus far, twenty employees of Bell Laboratories who

earned their master's degrees at the Graduate Centerhave already earned their doctorates. Twelve others areactively working for their doctorates, and it is expectedthat they will complete their work within a year.

IV. CONCLUSIONSThe program as constituted at the Graduate Center

offered the new employee student an attractive oppor-tunity since he was able to begin graduate study inearnest and at the same time earn a full salary. In addi-tion, he benefited from the advantages of exposure toa large research organization, was introduced early inhis career to significant engineering problems, and cameinto classroom contact with other high-caliber students.However, certain aspects of the program were diffi-

cult for the new man. In order to be successful, it wasnecessary for him to adjust to the demands of both theUniversity and the Company. Although concertedeffort was made to keep the students from being sub-merged in the work-study conflict, some neverthelessfound themselves in this situation. Company work re-quirements were sometimes interpreted in differentways by different supervisors in response to a widevariety of local departmental work pressures, whereasthe counseling provided by the University was usuallyuniform in content and emphasis. Several disadvantagesresulted from the off-campus setting. Choice of courseswas limited, as was the opportunity for consultationwith faculty members. Although of high quality, thestudent body was relatively uniform, not of the diversenature which is found at the University.Another important group to consider in this educa-

tional experiment is the faculty. Every attempt wasmade on the part of the Company to make the facultyfeel at home at the Graduate Center. Good facilitieswere made available to them in the form of secretarialand other services and very adequate office space. As aresult, faculty members provided the students withcomprehensive class notes in many of the courses. Ingeneral, the faculty enjoyed all of the privileges of BellLaboratories staff members when requesting librarybooks, periodicals, and reprints.

However, the faculty also suffered from some dis-advantages. In most cases they were exposed to a traveltime of at least one hour each way and a travel distanceof eighty miles round trip from the campus to the Grad-uate Center. Most graduate courses met for two one-hour-and-fifteen-minutes sessions on two separate daysduring the week. In most cases the maximum numberof days spent at the Graduate Center by a faculty mem-ber were these two days. Assignment to teach at theGraduate Center thus removed him from the main-stream of his campus activities, i.e., his research andhis contacts with other members of his department. Itwas difficult for him to establish with the students inhis class the kind of intimate relationship normallyexpected in graduate school. The informality of a cam-pus environment was missing.

In the early stages of the operation of the GraduateCenter an advisory committee was formed, the mem-bers consisting of representatives from both the Uni-versity and Bell Laboratories. Some of the difficultiesmentioned above became apparent during the first twoyears of operation. University representatives suggestedthat the best approach to meeting the educationalneeds of the Company would be to send their new em-

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ployees to the campus. In 1961, Bell Laboratories beganto send employees to various university campusesthroughout the country on a full-time basis. At first,the numbers involved in this diversification were small.However, as this part of the sponsored educationalprogram showed signs of success, the numbers increased.These diversions from the Graduate Center reduced thenumber of resident students and the efficiency withwhich the Center could be operated. Also, only alimited number of courses could be provided withoutrunning the risk of poorly subscribed course sectionsand the consequent high cost per student to the Com-pany.The contract for operation of the Graduate Center

provided that the arrangement could be terminated atthe option of either party with a suitably scheduledphasing-out period. After a careful study of all possibleways of providing the desired education for new em-ployees, notice of intention to terminate was given byBell Laboratories in 1964. The last master's degreeswere awarded in June, 1966.

This program of graduate level engineering educa-tion was conducted for nine years. To the University, itpresented an opportunity to perform a unique educa-tional experiment. To Bell Laboratories, it representeda needed stage in the evolution of a dynamic educationalprocess for new employees. It is acknowledged by bothinstitutions to have been a worth-while venture.

Short Papers and Classroom Notes_

Rectangular Waveguides,A Simple DevelopmentWILLIAM A. STIRRAT

Abstract-Fields within rectangular waveguides can be describedcompletely without employing differential equations. Maxwell'sequations are not used in this short development which, while pre-sented at an elementary academic level, produces a high degree ofvisualization.

INTRODUCTION

The background received from the first two years of undergrad-uate work is, through the following development, all that is necessaryto acquire a clear visualization of waveguide action and to developthe equations involved. Although students at the graduate level havehad to first develop Maxwell's equations before studying waveguidesand cavity resonators, they acquire less insight than that providedby the more elementary treatment.

THEORY

Basic to this development is the assumption [1 ] that the electricE-field is perpendicular to the magnetic H-field, that both fields areperpendicular to the direction of propagation, and that the E/H ratiois everywhere the intrinsic impedance of the medium. The analysisuses only rays and wavefronts, as in optics.

To assume that the walls of the waveguide or cavity are perfectconductors is realistic and conventional. The electric field parallel tosuch a wall must go to zero at the wall. Since such a wall acts likea short-circuited inductor, the alternating magnetic field normal tothe wall must go to zero at the wall. These boundary conditions areconventional.

A- and B-Type Fields

Fields which go to zero at a particular wall will be classed asA -type in relation to that wall. Fields which need not go to zero at

Manuscript received April 24, 1968; revised May 18, 1968.The author is witb the U. S. Army Electronics Command, Fort Monmouth, N. J.

07703.

a particular wall will be classed as B-type with respect to that wall.Electric fields normal to and magnetic fields parallel to a wall areB-type fields with respect to that wall. The walls are assumed to beperfect conductors. The type that each electric and magnetic fieldcomponent (subscript indicating field direction) is with respect toeach wall is shown in Table I.

TABLE IFIELD TYPE WITH RESPECT TO WALL

Field Axis Normal to the Wallx ~y z

E2 B A AE A B AE, A A BH. A B BHY B A BHz B B A

ReflectionFig. 1 shows fields propagating along a ray and reflecting from a

wall, as in optics. The wavefront is shown perpendicular to the direc-tion of propagation. The fields are propagating at a velocity which,in air, is C and, in any dielectric, is the same as the velocity of light.The angle of incidence 0, equals the angle of reflection. In this exam-ple the incident E-field Ei points up normal to the paper. As with allfields of the A type, this E-field reverses polarity at the point of re-flection to meet the requirement that at point 1, where E and thereflected field Er coincide, the total field shall be zero.

The orientation of the E-field with respect to the H-field alwaysfollows a right-hand rule (fingers point from E to H, thumb in thedirection of propagation) as shown in Fig. 1. The vertical componentof H is seen to be an A-type. Before and after reflection the H-fieldhorizontal component, a B-type, has its orientation determined bythe total incident H-field orientation, and by the orientation ofA-type E- and H-field components. B-type fields require and, asdemonstrated in this example, undergo no change in reflection.

243