Introduction to Research
Introduction to Research
That post title is a little joke, because I wasn't introduced to research - I was thrust into it. My vision of entering into a research career was that I would undergo an orientation period that would involve lectures on the types of research that was being conducted at the Center, with tours of the wind tunnels and other facilities, and then work together with an experienced researcher for several years to develop my capabilities to do independent research. What actually happened was that Dr. Katzoff sat down with me, explained the problem, handed me some books on basic aerodynamics and some research papers relating to his problem, and sent me off to solve it.
What a shock! Once more I had gotten in over my head. However, there was no reasonable option but just to dig in and do my best. I had two major problems to overcome: first, to learn the basics of fluid dynamics and aerodynamics; and second, to solve Dr. Katzoff’s problem. I spent as much time as I could on studying the aerodynamics textbooks, but I had to focus my primary effort on the problem at hand.
At the time it seemed almost inconceivable that I would be handed a difficult research problem immediately on entry onto the staff. But now, when I can imagine the situation from Dr. Katzoff’s point of view, it doesn’t seem unreasonable at all. He had formulated the problem in mathematical terms, and since I had an advanced degree in math, he had every reason to expect that I could work through the math to a solution. And he didn’t pressure me. He made it clear that he understood that it would take some time for me to come up to speed.
What he didn’t realize was that, despite those two math degrees, I had almost zero knowledge of the kind of math that was required to analyze this problem. For example, the most advanced applied math course that I had taken was Ordinary Differential Equations. This problem required a knowledge of partial differential equations. That was just one of the many shortcomings in my education. My basic physics background had given me only a meager glance at flow physics, which by its nature is more difficult than the physics of objects.
Without going further into the technical material that I had to learn, I’ll just state that I worked my brain intensively over those first weeks. One other problem that I had to overcome was my anxiety, which would cause me often literally to tremble when I would sit down at my desk in the morning and start in on my studies. It would sometimes take me an hour to get my mind quieted to the point that I could focus it on my work.
During this period of study, I was enlisted by Dr. Katzoff for a few activities that were not related to our specific research problem. On one occasion he mentioned that LRC's official German translator had asked him for help with the term faltung, which baffled her. I volunteered that it was the German term for the convolution integral – which I thought was common knowledge. But (I learned later) that little contribution seemed to impress, him and added to his faith in me as some kind of expert on applied math – a faith that I in no way deserved.
He once told me that there was a saying among engineers that you could walk into a meeting knowing just a few basic formulas and walk out a hero. Something like that happened when he took me with him to a presentation at NACA's DC headquarters. A university professor was proposing a new way of modeling helicopter aerodynamics. I assumed that my presence was just part of an orientation process to familiarize me with NACA, as my knowledge of helicopter aerodynamics was zero. But during the discussion during the presentation I mentioned that the result of one of the integrations seemed to be inconsistent with the assumptions. I was surprised that anyone even noticed the remark, because the professor's concept was pure nonsense, and when the door had closed behind him the panel of experts all had a good laugh, the managers that had collected us apologized for the inconvenience, and we all returned to our home research centers. But once more (again unknown to me) Dr. Katzoff was convinced that I was a real mathematician. How little he knew how little I knew!
I anticipated that it would require months for me to bring my education up to the point that it would match that of a technical graduate properly prepared to do research, so I decided to forgo that way and approach the problem in reverse. I collected the most recent publications that dealt with situations that were in some way similar to the one I was faced with, and studied them. Of course, I didn’t understand them. But I found out what I needed to learn to understand them. So I studied that material; but I couldn’t understand all of it. So went back further, and so on, until I did understand what I was reading, and then I started working back forward again until I could understand the most up-to-date publications. At this point I knew as much about the mathematics of this particular subject as the engineers who were specialists in wind tunnel interference problems.
This project taught me several basic principles about learning and research. First, the initial dread that I had of having to comprehend so much difficult material in such a short time gradually dissipated – for a surprising reason. I found that the material was difficult compared to, say, basic statistics; but compared to the abstract math that I had been studying, it was child’s play. I began to develop some respect for those two years of graduate studies that had stretched my mental capacities.
Nonetheless, the sheer volume of the math that I needed to learn was overwhelming. And with math, one can’t jump directly into advanced concepts. As I mentioned with regard to my college curriculum, math studies are cumulative in that each course requires as a prerequisite a knowledge of the prior course material.
Dr. Katzoff’s idea of a way to approach our wind tunnel interference problem was by a device known as “contour integration” and the “calculus of residues”. These methods are rarely covered even in texts on advanced applied math, and they require a knowledge of functions of complex variables. I had only studied functions of real variables, and that on an abstract level.
I was plugging away at it one day when a curious thing happened. Dr. Katzoff’s desk was opposite mine in such a way that we faced each other across the two desks. I was studying math books, and he was reading a list of abstracts of recently published technical papers. Suddenly he looked up with a kind of confused expression, and said, “This doesn’t make any sense. The title of this article is, ‘On the Roots of (a certain algebraic equation)’. The roots of that equation are obvious” (The roots are the solutions of the equation). He continued: “I think I’ll get that paper, and see just what’s going on.” He picked up his phone, called the library, and two days later he had the paper. He just glanced at it and smiled, because he saw immediately that there had been a typographical error in the abstract. A variable had been omitted. When it was properly inserted, the equation became a highly nonlinear equation, whose roots had to be calculated numerically.
He handed the paper to me. When I had studied it briefly, I found that the roots of that equation were required for the solution of a problem in heat diffusion and radiation. That problem was not physically related to our problem, but its boundary conditions were the same. The problem had been solved by means of a Dini series. I had never heard of that kind of series, but it was closely related to Fourier series, which I had just been studying. I found a section on Dini series in a math text, studied up on it, and soon saw how it could be used to solve our problem.
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