The field of flight dynamics has been studied since the early days of aviation. As mentioned previously, G. H. Bryan in England worked out the complete equations of airplane motion the same year as the Wright brothers' first flight (ref. 4.1). By the 1950's, it was difficult to find any field of research in flight dynamics that had not been thoroughly explored. The gust-alleviation program, however, revealed one problem that had received very little attention. This problem was the lateral response (that is, rolling and yawing response) due to flight through atmospheric turbulence.
Modeling the Turbulence
As mentioned in the section on gust alleviation, the scale of turbulence in the atmosphere is ordinarily so large that the gusts may be considered constant across the wing span. This circumstance made it possible to use a single vane in the nose of the airplane to sense the gusts with the possibility (in theory at least) of getting 98 percent complete alleviation. In considering lateral response, however, such an assumption is obviously incorrect, because if the gusts were constant across the span, the airplane would receive no rolling disturbance due to turbulence.
The method of calculating lateral response of an airplane to discrete gusts was the subject of a report by R. T. Jones (ref. 5.3). He showed that the conventional equations of motion could be used to calculate the response to a rolling gust by assuming that a rotational motion of the atmosphere about the longitudinal axis acted on the airplane through the so-called stability derivatives associated with rolling of the airplane, that is, the rolling moment due to rolling velocity and the yawing moment due to rolling velocity. These derivatives were already used in the standard lateral stability equations and were well known as a result of previous theoretical and experimental research. The effect of a rotational motion of the atmosphere about a vertical axis could likewise be calculated with the aid of the yawing moment derivatives. What was needed was a way to determine the rolling and yawing motions of the atmosphere associated with random turbulence.
The turbulence in the atmosphere was ordinarily characterized by what is known as the point spectrum of turbulence. This spectrum is what would be obtained by measuring the velocity fluctuations along a straight line as it is traversed at a relatively high speed. Measurements of the point spectrum of vertical velocity, for example, were made by recording the motion of a vane to measured angle of attack and correcting the readings  continuously for the inertial motion of the airplane at the location of the vane. This method was used in the measurements by H. L. Crane and R. G. Chilton described previously (ref. 12.5).
To extend this information to obtain velocity fluctuations across the wing span, the usual assumption made is that the turbulence is isotropic; that is, the spectrum of turbulence is the same no matter what direction the atmosphere is traversed. It seems reasonable, for example, that an airplane flying through a region of turbulence from north to south would experience the same amount of disturbance as an airplane flying through the same region from east to west. Using the assumption of isotropy, H. S. Ribner presented an analysis of response to turbulence using a two-dimensional Fourier series to represent the disturbances (ref. 14.1). This approach was also described in a textbook by Bernard Etkin (ref. 14.2). Franklin W. Diederich, in a doctoral thesis at California Institute of Technology under R. W. Liepmann, calculated the spectrum of lift on a wing flying through turbulence by using a correlation function approach. This thesis was later published as an NACA Technical Report (ref. 14.3). Diederich also showed how to calculate rolling and yawing moments on the wing, but the complete calculations were not carried out. I considered the approach used by Diederich to be easier to comprehend than that used by Ribner, but both should give the same answer if used correctly.
Calculation of Response
I assigned an engineer in my division, John M. Eggleston, to work with Diederich and complete the calculations of yawing and rolling moments (ref. 14.4). These calculations were made for wings of rectangular, elliptic, parabolic, and triangular planform and involved rather complex mathematical manipulations. Eggleston continued this work to make a complete matrix formulation of the airplane response and added the effects of the fuselage and tail to those of the wing by a method that I suggested (ref. 14.5). Eggleston calculated the effects of the rolling moments, yawing moments, and side forces separately and then added these contributions to get the response spectrum of the resulting motion. An important consideration in these calculations is that the lateral motion is only statistically, rather than uniquely, determined in flight through any given velocity pattern along the fuselage centerline because different velocity distributions across the span may be encountered for a given distribution along the fuselage centerline. The calculations can therefore produce only the power spectrum of the response for a given point power spectrum of turbulence.
I continued my analysis to determine the relation between the spectra of effective rolling gusts and yawing gusts and the point spectrum of turbulence. Again, the results of Eggleston and Diederich's calculations (ref. 14.4) were used, but the effective atmospheric rolling and yawing motions were calculated and applied directly as inputs to the lateral equations of motion. I worked out some of the same examples used by Eggleston. Much to my surprise, the results turned out to be identical to those of Eggleston. These results were obtained with much shorter calculations, and to me, gave a clearer picture of the physical process involved (ref. 14.6). Later, a technical report was published presenting both Eggleston's matrix calculations and my shorter method (ref. 14.5) together with a simple 45-step procedure for calculating the power spectra of the lateral responses for specific cases.
The relation between the spectrum of rolling gusts and the point spectrum of turbulence used in the analysis is shown in figure 14.1. Ordinarily, a power spectrum is presented as the square of the dependent variable plotted against the independent variable on log-log paper. This type of plot is shown in part (a) of the figure. A squared variable is compatible with the statistical significance of the...
...quantity in which only the amplitude and not the phase relationship is obtained from the analysis. Log-log paper is useful in showing a large range of the variables. This method of presentation, however, obscures a clear impression of the actual shapes of the curves. The same quantities shown by taking the square root of the ordinate and plotting it as a function of reduced frequency on linear scales is shown in part (b) of the figure. This plot shows the expected results that at very long wavelengths, the amplitude of the effective rolling gust is small because the gradient of vertical velocity along the span is small. At high frequencies, where the wavelength is small compared to the span, the effective  rolling gust is again decreased because the fluctuations are averaged along the span. Between, the effects reach a maximum. In a region indicated by the sloping dotted line, the disturbances approach an approximation obtained by assuming a uniform gradient across the span equal to the slope of the central region of a sinusoidal gust with a wavelength corresponding to the frequency. The only rather puzzling aspect of the curve is the slight tendency of the curves to turn up at very low frequencies. Possibly this effect is caused by rare encounters with widely spaced large-amplitude rolling gusts, first in one direction and later in the other, which would introduce a harmonic of very low frequency in the spectrum.
At the time these studies were made, about 1958, I wanted to conduct some flight tests to verify the predicted relationship between the measured point spectrum of turbulence and the resulting lateral response of the airplane. These methods might have also provided an alternate method to measure the spectrum of atmospheric turbulence. Unfortunately, flight testing of high-speed airplanes at Langley was discontinued at that time. To my knowledge, such flight tests have never been carried out, and they remain one of the few uncompleted phases of flight dynamics.
The work of Franklin W. Diederich at Langley deserves special mention. His work, which involved both aeroelasticity and response to turbulence, came at a time when high-speed computers were just becoming practical and when closed-form solutions of complex problems were becoming less necessary. As a result, his work to produce widely useful charts and formulas appears to be largely forgotten, even by engineers working in the fields to which he made such valuable contributions. Now, when similar problems are encountered, a computer simulation of the specific problem under consideration is made with such techniques as finite-element methods or step-by-step solutions. The programs for making such calculations are generally available to the engineers involved. Though some familiarity with the physics of the problems is advisable, a detailed knowledge of classical theorems from aerodynamic theory or of methods of manipulating complicated mathematical functions is usually not required.
The work on response to random turbulence, including the subject of gust alleviation, has also received little attention from the aeronautical industry because the main structural design conditions on airplanes come from rare encounters with discrete gusts in thunderstorms. Such events have been studied through operational experience rather than by research programs designed to study the details of response to turbulence. As a result, much very interesting research work on this subject now lies buried in old technical reports that are occasionally unearthed by scholars, but rarely used in practice.