Thomas W. Cairns


Emeritus Professor of MathematicsKeplinger Hall U335

Military Service

Prof. Cairns was a commissioned officer in the Army of the United States/Army Security Agency. Two months after he received his commission, the Korean armistice was signed making him technically a Korean veteran. He spent two years on active duty, principally attached to the National Security Agency (NSA), and nine years total in the army reserve.


Upon finishing graduate school Prof. Cairns came to The University of Tulsa and is serving his 53rd year on the faculty. He is a Professor of Mathematical Sciences. From 1967 to 1977, he was Chair of the Department of Mathematics, during which time the program in computer sciences was instituted.

He had a background in collegiate varsity sports and in 1976 started the women's varsity volleyball program at The University of Tulsa and coached it for 17 years. He has been active in promoting women's equity in sports and science and has been given some credit for being a prime force behind the Title IX compliance at TU.

In addition to the time he spent in the army attached to NSA, he worked there on occasion in the capacity of a WAE Expert. He was also engaged at times as a summer employee at the Amoco Exploration and Production Laboratory in Tulsa, working with their geophysicists on problems in petroleum seismology. He wrote their program implementing the Cooley-Tukey FFT algorithm which was used there for many years. The FFT was a huge boost for processing of petroleum seismic records. He did similar work with Cities Service R&D.

In more recent years, Prof. Cairns became active in undergraduate research in mathematics and also more generally as a coordinator of the Tulsa Undergraduate Research Challenge (TURC). During his years coordinating TURC, TU received national recognition because of its part in this university producing Goldwater Scholars in numbers exceeded by only two other universities nationally. He has had many undergrads give papers at mathematics meetings on a variety of topics in applied mathematics.

In the last few years, along with a colleague who is an exercise physiologist, Prof. Cairns developed, with some National Science Foundation (NSF) support, a general education course in sport science. The course meets a general curriculum science requirement for non-science majors and the subject matter is mainly biomechanics based on Newton's laws.

He has also had students working on the fluid mechanics of a volleyball in air, specifically fitting a differential equation of motion to volleyball paths video taped in the gym and, using cutting edge motion analysis systems, the kinetics and kinematics of volleyball spikers.

Education and Degrees Earned

  • Ph.D., Mathematics, Oklahoma State University (1960)
  • M.S., Mathematics, Oklahoma State University (1954)
  • B.S., Physics, Oklahoma State University (1953)

Areas of Academic Specialty

Prof. Cairns teaches two kinds of courses in sport science. Each semester SI-1004 is offered mainly for students who use it to meet a general education requirement for a laboratory science. The SI stands for Scientific Inquiry and the course was developed with the help of NSF support for innovative teaching of science. SI-1004 features the use of the Aerial Performance Analysis System (APAS) which is a professional motion analysis package.

The content consists of explaining how physical laws, especially Newton's laws of motion, affect the way sports are played and influence performance levels. Examples are: why a curveball curves, why a javelin lifts, how body mechanics are implemented using bone-muscle levers, why a sprinter starts in a four point stance.

Prof. Cairns offers upon demand special topics courses in sport science or biomechanics to one to four students at a time. These also involve the use of APAS applied to a specific problem of common interest. The students are usually expected to present the results of their work in a professional meeting.

Areas of Research Focus

Prof. Cairns and his students research aerodynamics of the paths followed by volleyballs in motion. The classical equation of motion for objects moving through air is sixth order highly nonlinear differential equation that contains two dimensionless parameters that are specific to the object: the drag coefficient CD and the lift coefficient CL. Lift refers to the consequences of the ball spinning. The basic problem is to determine the values of these parameters for a volleyball as a function of speed and, in the case of CL, the spin rate. If the ball is not spinning them CL = 0.

At the request of Prof. Cairns, CD was the subject of a wind tunnel study by Hahn and McCulloch who were aeronautical engineering students at the University of Michigan. That study produced good results and, in fact, the definitive values of the drag coefficient for all relevant ball speeds.

Even without having accurate values for CL these results were useful because one of the principal weapons in volleyball is the so-called float serve which is not spinning.

A second group of students at the University of Michigan followed up with a study of a spinning volleyball.

This study was not successful because of the inherent experimental problems associated with spinning the ball in the wind tunnel and measuring the forces acting at the same time. Cairns' approach to this problem is to shoot volleyballs across the gym, videotape the paths and compute the ball locations at 1/60 second intervals using the Ariel Performance Analysis System. Then the paths are matched to the solution of the equation of motion in order to produce a formula for CL. This is a work in progress.

Prof. Cairns presented preliminary results at a meeting of the International Sports Engineering Association (ISEA) which was well received and, in fact, featured in Science Magazine. One of the reasons for the interest generated at that meeting is that the volleyball is relatively light compared to its size and this means that all sorts of interesting things can happen to it during flight. As the study of ball paths go, it's a kind of missing link.

Previous Teaching Experience

  • The University of Tulsa (1959-present)

Previous Relevant Work Experience

  • Consultant to cities service R&D, Amoco R&D, NSA

Professional Affiliations

Courses Taught at TU

  • Sport Science (SI 1004)

Awards & Recognition

  • TU Mr. Homecoming


  • Cairns, T. W. and Bednar, J. Bee. A comparison of differential correction algorithms in digital filter design, IEEE Transactions on Circuits and Systems, 1976.

  • Osher, William J., and Cairns, T., Computer simulation in the study of atrial fibrillation. Journal of the Oklahoma State Medical Association, January, 1967 , 6-10.

  • Cairns, Thomas W., (2004). Modeling lift and drag forces on a volleyball. The Engineering of Sport 5, 97-103.

  • Cairns, Thomas W., Farrior, D., Farley, A., Marshall, M., Perez, G., Van Lierop, K., Sigley, A. (2000). Modeling the path of a volleyball. International Journal of Volleyball Research, 3 (1), 2-7.

  • Cairns, Thomas W., On the fast Fourier transform on finite Abelian groups., (1971). IEEE Transactions on Computers, C-20, 569-571.

  • Shanks, J. and Cairns, T. (1968). The use of a digital convolution device to perform recursive digital filtering and the Cooley-Tukey algorithm. IEEE Transactions on Computing, 1968.

  • Bednar, J. Bee and Cairns, T., Two–dimensional differential correction: A comparison of nonlinear and linear formulations. IEEE Transactions on Circuits and Systems, CAS-25, 5, 316-319 (1978).

  • Cairns, Thomas W. and Van Lierop, Kristien, (2000). Volleyballs and aerodynamics: A review. International Journal of Volleyball Research, 3 (1), 8-14.