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An extensive interior ballistics study of a two-stage light gas gun was performed which resulted in a detailed measurement of the projectile velocity as a continuous function of time. The two-stage gun consisted of a 88.9-mm i.d. pump stage and a 28.6-mm i.d. launch stage. Five main parameters of gun operation, propellant mass, piston mass, pump gas, pump gas pressure, and projectile mass were varied in this study. Projectile velocities were measured using a very precise velocity interferometer for any reflecting surface (VISAR) technique. The measurements showed that the initial acceleration of the projectile is discontinuous due to the shock nature of the applied pressure on the projectile upon rupture of the burst diaphragm. These shock accelerations are understood easily via simple shock-tube theory. 9 figures, 3 tables.

This work describes the application of original computer graphics methods to examine the effect of inclusionswithin the solid rocket motor thrust chamber. The adoptance of a dynamic three-dimensional triangular mesh with aself-intersection removal algorithm, as the core of the burning surface regression module, allows the aforementionedtargetto bereached.Manyphenomena notmeasurablein an analyticalclosed form, likedifferent-shapedvoidswithinthe grain, can be investigated. Indeed, both local heterogeneities and surface imperfections could generate unexpectedvariations in the combustion chamber pressure pattern and early thermal protections exposure. The aforementionedtechniques are developed and tested on a ballistic simulator developed using computer graphics methods to improvemesh handling. Simulation results are carried out and discussed.

Analyses of internal ballistics and criteria for reaction propagation in nonelectrical tube explosive lines were performed based on explosive loading density, average particle size, and typical geometry using an approach based on heat transfer and decomposition kinetics in the explosive particles in quasi steady state. The results indicate the existence of a critical or maximum average explosive particle size for a given explosive loading density beyond which sustained propagation may cease. Large variations usually exist in the explosive loading density, so that fine particle sizes below 20 μm are preferable. Fine particle size also may increase the adhesion to the inner tube wall for mitigation of powder migration problems. Computer simulations of detonating end tip to nondetonating end tip transfer were performed using the CTH hydrodynamic code developed by the Sandia National Laboratories. Preliminary results indicate that the hot gas from the detonating end tip is more effective for initiating pyrotechnics in the nondetonating end tip and that a larger gap between the end tips is essential for the success of hot gas initiation

Optical aberrations over a cylindrical turret with a flat window were measured using a two-dimensional wavefront sensor and a Malley probe, as a function of laser beam elevation angle. Topology of the flow around the cylinder was extensively studied using hot wires. It was found that, depending on the window elevation angle, the flow either had a weak separation bubble followed by a reattached boundary layer or a strong separation with a large recirculation region behind the cylindrical turret. Optical aberrations were shown to be consistent with flow topology. It was found that optical aberrations were high around an elevation angle of 100 deg and at large lookback elevation angles.

Microdebris particles generated by hypervelocity impacts between complex projectiles and ballistic missile targets can affect sensitive components of such aerospace systems. Current models assume an isotropic dispersion of this microdebris about the postimpact debris cloud center of mass. Data collected from recent tests indicate that a nonisotropic microdebris dispersion model is needed. The results are presented of a study aimed at improving the capabilities of an existing model that characterizes the microdebris environment created in a high-speed impact. Following a review of test data collection techniques, a nonisotropic microdebris dispersion model is developed as a series of modifications to the existing microdebris size distribution model. Relevant test data are then used to evaluate the proposed nonisotropic model. It is found that the modified model is more correct in accounting for the microdebris fragment population created in a high-speed impact than the current isotropic dispersion model.

The effect of lift and drag forces and aerodynamic moment on the flight of a spinning projectile in the atmosphere is investigated. The influence of the spin rate of the projectile on the range and height of flight is also studied, taking into consideration the progressive decay in the spin rate during the flight. The governing equations of motion of the projectile have been solved numerically, and the results are presented graphically. The effect of various parameters such as angle of projection, initial velocity, and spin on the flight of projectiles are plotted. From the results it can be concluded that the lift and drag forces and the aerodynamic moment, which are dependent on the spin rate of the projectile, influence the range and height of the trajectory significantly. Only planar motion of the projectile is considered

Theoretical and experimental studies are conducted on the quantitative evaluation of a friction force at the interface between a projectile and the launch tube wall in a ballistic range. From a mechanical balance relation applied to the projectile, the friction force is expressed theoretically as a function of the net acceleration of the projectile. With this formulation, experimentally measured in-bore projectile velocity proe les obtained using a velocity interferometer system for any ree ector are well interpreted. If the initial projectile motion is signie cantly suppressed by the friction force while the burning rate of the propellant remains signie cant, the base pressure becomes much higher than that with a negligibly small friction force, thereby enhancing the net acceleration. However, if the projectile-tube e t is excessively high, the main function of the friction force is to decrease the muzzle velocity.

A concept for propelling a projectile to hypervelociti es is described, and the feasibility is demonstrated by numerical simulations. In theory, the concept employs imploding blast waves to accelerate a projectile as it travels down a launch tube. The launch tube can be open to the atmosphere or sealed and maintained at some low pressure to minimize internal drag. The launch tube has a liner that contains a suitable explosive or energetic material. The explosive is configured with inert annular rings to prevent upstream detonation. A suitable trigger detonates each explosive ring sequentially as the projectile passes. The resulting blast wave causes an elevated pressure on the afterbody of the projectile. The acceleration continues until the projectile exits the launch tube.

We consider supersonic flight of a blunt slender body of revolution through a large-scale cloud of heated gas (thermal) floating in the stratified atmosphere. An efficient numerical method to determine the body motion is proposed.

A transient ballistics and combustion model is derived to represent the closed vessel experiment that is widely used to characterize the high pressure burning rates of solid propellants. The model is applied to explain why burning rates deduced from the closed vessel are in basic agreement with those measured from an equilibrium strand burner in the case of homogeneous propellants, but differ significantly in the case of nitramine composite propellants. Thermal profile time lag effects become small at high pressure because the burning rates become high. However, the development of the burning surface structure of those nitramine propellants which exhibit shifts in pressure exponent causes the mass burning rate to lag and then exceed the equilibrium value. It is necessary to consider this mechanism in applications dealing with high pressures and pressurization rates.

HE inclusion of small quantities of various lead compounds in the double-base propellant system results in increased burning rate at low pressure, followed by a plateau burning region, where the burning rate remains almost independent of variation in pressure, and the postplateau region, in which the burning rate/pressure relationship is similar to that of an unleaded propellant (Fig. 1). This phenomenon of independence of burning rate to pressure, generally known as "platonizatio n," implies a low value of pressure index n in the plateau region. A low value of n permits the design of lighter rocket motors because of lower safety factors. The magnitude of n is one of the important factors in determining the suitability of a propellant for rocket propulsion applications. The importance of low-pressure ballistics was realized as early as in 1939 with the widespread use of rockets. Earlier studies centered around low-energy nitrocellulose propellants with a view toward minimizing nozzle erosion, burning rate, and pressure index. However, the real impetus to the platonization of double-base propellants came from the observation that the use of lead stearate as a lubricant in the extrusion of large rocket charges actually had modified their burning characteristics. This paper reviews the literature on platonization of double-base propellant with different inorganic and organic compounds of lead and other metals, and presents various theories to explain the mechanism of the plateau effect.

An analytical model is developed for the linearized velocity-coupled combustion response function. The model treats elements of response to perturbations in pressure, in composition due to the heterogeneity of composite propellants, and in crossflow velocity. The effects of AP particle size are accounted for in terms of effects on controlling ballistics properties and in terms of fluctuations in propellant composition. There are two facets of the crossflow problem: the effect of crossflow velocity on the various response elements, and the response to velocity perturbations. Both are dealt with in this paper. Important trends derived from series of parametric computations are described.

the spatial distribution of particle mass. Assumption of uniform number distribution gives an over-estimation of particle damping. For noncircular cylindrical rocket chambers, steady state internal ballistics must be considered to determine the particle distribution. Particle size growth should in general have a stabilizing effect. Quantitative prediction must await better knowledge of how particles grow within the combustion chamber.

2 Sturm, E. J. and Reichenbach, R. E., "Aluminized Composite Solid-Propellant Burning Rates in Acceleration Fields," AIAA Journal, Vol. 7, No. 11, Nov. 1969, pp. 2087-2093. 3 Manda, L. J., "Compilation of Rocket Spin Data, Vol. II: Literature Survey," Rept. 3001-2, CR-66641, 1968, NASA. 4 Anderson, J. B. and Reichenbach, R. E., '76-Inch Diameter Centrifuge Facility," TN 66T-4, Sept. 1966, Dept. of Aeronautics, Naval Postgraduate School, Monterey, Calif. 5 Sturm, E. J., "A Study of the Burning Rates of Composite Solid Propellants in Acceleration Fields," Ph.D. thesis, AD832206, March 1968, Naval Postgraduate School, Monterey, Calif. 6 Green, L., Jr., "Erosive Burning of Some Composite Solid Propellants," Jet Propulsion, Vol. 24, No. 1, Jan-Feb. 1954 DD 9-15. ' 7 Wimpress, R. N., Internal Ballistics of Solid Fuel Rockets, 1st ed., McGraw-Hill, New York, 1950. 8 Lawrence, W. J. and Deverall, L. L, "The Erosive Burning Behavior of Selected Composite Propellants," Proceedings from the 4th ICRPG Combustion Conference, Stanford Research Inst., Menlo Park, Calif., Oct. 1967, pp. 459-466. 9 Crump, J., "Aluminum Combustion in Composite Propellants," Proceedings of the Second ICRPG Combustion Conference, Aerospace Corp., El Segundo, Calif., 1965.

The angular motion of a spin stabilized solid rocket is computed, considering the effect of jet damping about axes normal to the spin axis. The approximate formula used is chosen for computational accuracy and to fit the rocket geometry. Results are compared with measurements of body angular acceleration made during the rocket's flight in space. The comparison indicates that the jet damping is a very strong effect in this case, and that the approximate formula gives adequate accuracy for computing the reduction in conical precession motion due to jet damping. It is probable that the same formula can be used to estimate jet damping for other solid rockets. I. Introduction T HE term jet damping refers to the change in the angular rates of a thrusting rocket due to the torques exerted by the gases flowing out the nozzle. The jet damping effect is especially important in spin stabilized rockets where it acts to reduce the amplitude of the conical free precession motion and, therefore, to reduce the motion to a steady spin. It is important to know what the cone angle is at staging, as this strongly influences the attitude error of the next stage. In rockets operating in air, jet damping is usually second in importance to aerodynamic moments. In vacuum, however, the jet damping can be the most important torque acting on the rocket. Several simplified theories have been proposed to give estimates of the rate of damping for a particular rocket.1"5 In general, these theories use one-dimensional gas flow in the rocket and characterize the thrust chamber and nozzle by certain approximate dimensions. Proper verification of these approximate theories has been difficult to obtain because of the lack of experimental data on the attitude of a spinning rocket in vacuum flight. In the present paper, a formula for pitch jet damping is derived from the general rocket equations in a body-fixed coordinate system assuming one-dimensional flow. The formula is selected to use quantities that are known accurately and to be compatible with the geometry of the rocket considered. The formula, a modification of the one presented in Ref. 4, is evaluated to give the predicted rate of cone angle damping for a particular frequently used spin-stabilized rocket, the Allegany Ballistics Laboratory! X-248 with a satellite payload. The resulting prediction of jet damping is then compared with measurements of the angular motion made in actual flight.

References 1 Ishii, S., Niioka, T., and Mitani, T., "An Analytical and Experimental Study for Solid Propellant Combustion in an Acceleration Field," Combustion Science and Technology, Vol. 8, No. 4, 1974, p. 177. 2 Ishii, S., Niioka, T., and Mitani, T., "Combustion of Composite Propellants in Acceleration Field," TR-354, 1974, National Aerospace Laboratory, Miyagi, Japan. 3 Crowe, C. T. and Willoughby, P. G., "Effect of Spin on the Internal Ballistics of a Solid Propellant Motor," AIAA Paper 66-523, Los Angeles, Calif, 1966. 4 Enskog, D, "Kinetische Theorie der Warmeleitung, Reibung und Selbstdiffsion in gewissen verdichteten Gasen und Flussigkeiten," Svenska Vetenskapsakademiens Handi, Vol. 63, No. 4, 1922, pp. 44S.

A technique is developed which permits propellant gas temperature to be extracted from pressure measurements taken within the gun tube. Basically similar to acoustic thermometry, the wave speed of a signal propagating at the sonic velocity is determined. In this application, the speed of the expansion wave which propagates back up the gun tube following shot ejection is derived from pressure data taken at discrete axial locations within four calibers of the muzzle of a 20mm gun. The propellant gas temperature is calculated from the measured speed of this wave and the equation of state.

The effect of the oxidizer size and size distribution on the pressure and temperature sensitivity for a nonaluminized composite solid propellant was investigated using a new mathematical model called the Petite Ensemble Model. Calculations were made using the new model, taking into account the oxidizer size and size distribution in determining the burning rates, rate exponents, and temperature sensitivities. Results were obtained for both unimodal and bimodal propellants with oxidizer particle sizes ranging from ultrafine (0.7 /*) to coarse (400 /*). For unimodal propellants, it was determined that the burning rate was higher for the smaller oxidizer sizes, as would be expected. Moreover, the rate exponent was larger for the small and large oxidizer particles with a minimum exponent obtained for some intermediate oxidizer diameter. Temperature sensitivity was found to be higher for propellants with larger oxidizer particles. However, it was found that the temperature sensitivity was constant for small particles. It was shown that for large particles, the temperature sensitivity and rate exponent decreased as the oxidizer distribution was widened.

An experimental and theoretical investigation of the performance of a ballistic compressor is described. Comparison of the computed ideal behavior of the ballistic compressor with experimental test pressures show that the ideal theory overestimates the generated pressure by as much as 30%. An assessment of the factors responsible for the nonideal behavior of the ballistic compressor is given. These factors include: valve losses, external and internal (blow-by) leakage, friction of the inertia! piston, nonideal behavior of the driver and test gases, shock wave formations, and heat losses. The valve losses and both the internal and external leakages have been measured experimentally. A parametric study shows that the choice of equation of state of the test gas is not very critical. The heat loss experienced by the test gas was estimated to be l°7o of the total energy imparted to the test gas. Shock-wave formations have negligible effect on the performance of the ballistic compressor for the mass of the inertial piston (1.34 kg) used.

Micrometeroid impact effect on candidate skin materials, determining change in optical properties

Micrometeoroid environment effect on metal surface thermal properties, using simulated space vehicle placed in solar space environment chamber

T-burner experiments have provided the most fruitful technique for obtaining the acoustic admittance of burning solid propellants The objections to the findings of these experiments are: 1) the admittance values are questioned on the grounds that acoustic losses in the appaiatus are not well enough understood for adequate corrections to be made, and 2) relative acoustic losses are great enough that some operational propellants, known to produce oscillations in motors, burn stably in the T-burner The first objection is largely removed by demonstrating that three different methods for operating the burner and analyzing the data give results in close agreement Besides confirming the results of the other two, one method described here also provides a means to study more stable propellants, thus removing the second objection to the T-burner technique

Surface ignition of solid propellants has been represented by several analytical models, each involving obvious compromises with regard to scope of applicability. These models are distinguishable primarily in terms of site of the exothermic reaction governing ignition. Early research with nitrocellulose led to development of a theory involving chemical heat generation in the condensed phase. Two subsequent theoretical models were developed to explain ignition of the solid fuel ingredient of a composite propellant in an oxidizing atmosphere, and these two models were then extended on a heuristic basis to encompass a composite propellant in an inert atmosphere in which the oxidizing gas was produced by decomposition of the solid oxidizer. These two models are distinguished by whether the oxidation occurs at the surface or in the gas film above the surface. This report reviews the solid, heterogeneous, and gas-phase ignition theories and reviews the nature and implications of the assumptions involved. It is concluded that, while possessing certain drastic simplifications in common, the various quantitative models differ so conspicuously in their assumptions regarding external initiating stimulus as to make quantitative comparisons or tests of validity impossible.

HYDRODYNAMIC aspects of the gun environment have been central to the theme of interior ballistics theory for over a century. Decoupled from a nontrivial description of propellant ignition and flamespread, the problem has been amenable to numerical solution for more than a decade. More recently, however, the occurrence of several catastrophic gun ammunition malfunctions has been linked to ignition-indu ced two-phase flow dynamics. A number of approaches have been pursued intensely to provide a solution to the combined twophase flow, ignition and flamespread interior ballistics problem. A detailed review of four such modeling efforts has been provided by Kuo. 1 Work reported herein relates to efforts aimed at reconciling experimental data with numerical predictions obtained using a computer code developed by Gough.2 Manipulation of poorly known input parameters or statements of constitutive physics was performed both to identify those parameters that most affect the calculated results and to draw useful inferences about the physical processes themselves. Considerable success has been demonstrated in attempts to simulate the performance of a Navy 5-in./54-caliber case gun; however, similar agreement with data obtained in an Army 155-mm howitzer was unattainable without substantial modifications to available input data. An attempt has been made to provide a physically motivated explanation for this difference in the level of agreement for the two configurations. Contents The Gough model, also known as the NOVA code, is a twophase flow treatment of the gun interior ballistic cycle, formulated under an assumption of quasi-one-dim ensional flow. Gas and particles are treated as interpenetrat ing media, with conservation equations derived using formal averaging techniques. Constitutive laws include a covolume equation of state for the gas and an incompressibl e solid phase. Compaction of an aggregate of particles is allowed, with intergranular stress represented as a function of propellant bed porosity. Interphase drag and heat transfer are represented by reference to empirical correlations for fixed and fluidized beds. Functioning of the igniter is included by providing as input an experimentall y determined mass injection rate as a function of position and time. Grain temperature follows from the convective heat-transfer correlation and unsteady heat conduction in the solid, with ignition based on a surface temperature criterion. Propellant combustion is then assumed

Effort has recently been devoted to finding a means to alleviate the effects of the reflection at the walls of the range tank of the bow and wake shock system originating with a hypersonic projectile. (Author)

The preliminary photoelastic data of Ordahl and Williams for calculating the maximum stress in a family of slotted solid propellant rocket grains are amplified and improved in accuracy. The effect of changes in the shape of the inverse star point on the maximum stress at the star point also is investigated, and the experimental data are compared with Wilson's analytical results that used conformal mapping. Finally, an empirical formula covering the range of parameters tested is derived for possible use in integrated internal ballistics computations. The design data are useful in the elastic and viscoelastic stress analysis of free or case-bonded, internally or externally pressurized rocket grains.

(International) on Combustion, The Williams and Wilkins Co., Baltimore, Md., 1953, pp. 893-906. 22 Green, L., Jr., "Observations on the Irregular Reaction of Solid Propellant Charges," Jet Propulsion, American Rocket Society Journal, Vol. 26, No. 8, Aug. 1956, pp. 655-659. 23 Price, E. W., "Review of Combustion Instability in Solid Propellant Rocket Motors," TN 5008-4, Feb. 1961, U.S. Naval Ordnance Test Station, China Lake, Calif. 24 Wimpress, R. N., Internal Ballistics of Solid-Fuel Rockets, 1st ed., McGraw-Hill, New York, 1950. 25 Berl, W. G., "Combustion Instability of Solid Rocket Propellants and Motors," Review Paper 2, July 1962, Solid Propellant Information Agency, The Johns Hopkins Univ., Applied Physics Lab., Silver Spring, Md. 26 Berl, W. G., Hart, R. W., and McClure, F. T., "Solid Propellant Instability of Combustion, 1964 Status Report," TG 3718A, July 1964, The Johns Hopkins Univ., Applied Physics Lab., Silver Spring, Md. 27 Anderson, A. B. C. and Hunt, M. H., "Operational Characteristics of the DynaGage (General Motors Gage) in Investigations of Pressures in Statically Fired Rockets," NOTS 150, Sept. 1948, U.S. Naval Ordnance Test Station, Inyokern, Calif. 28 Swanson, C. D., "Resonance Burning in Rocket Grains," NOTS TM No. 439, May 1951, U.S. Naval Ordnance Test Station, Inyokern, China Lake, Calif. 29 Price, E. W., "Axial Mode, Intermediate Frequency Combustion Instability in Solid Propellant Rocket Motors," AIAA Paper 64-146, New York, 1964. 30 Roberts, A. K. and Brownlee, W. G., "Nonlinear Longitudinal Combustion Instability: The Influence of Propellant Composition," AIAA Paper 69-480, New York, 1969. 31 Brownlee, W. G., "Nonlinear Axial Combustion Instability in Solid Propellant Motors," AIAA Journal, Vol. 2, No. 2, Feb. 1964, pp. 275-284. 32 Dickinson, L. A., Capener, E. L., and Kier, R. J., "Research on Unstable Combustion in Solid Propellant Rockets," SRI Project PRU-4865, Jan. 1965, Stanford Research Institute, Menlo Park, Calif.

Thin plate ballistic limit for perforation by cylindrical projectiles, using computational procedures and graphs

A comprehensive experimental program has been conducted in the ballistics range facilities of the U. S. Naval Ordnance Laboratory to determine various aerodynamic characteristics of wakes behind hypervelocity bodies. The effects of Mach number, Reynolds number, and wall-to-stagnation enthalpy ratio on base flow geometry and transition from laminar to turbulent wake flow were investigated. It was found that the wake transition characteristics can be described in terms of two wake transition Reynolds numbers. The transition Reynolds numbers increase with both increasing flight Mach numbers and increasing wallto-stagnation enthalpy ratios. In a certain Reynolds number region the transition Reynolds numbers are independent of the body Reynolds number. The growth of laminar and turbulent wake behind cones of various half-angles and spheres was investigated. It was found that the higher the pressure drag coefficient, the steeper the growth of the turbulent near wake. The behavior of the edge of the turbulent wake was studied and described in terms of statistical parameters. It was found that the edge roughness increases at the same rate as the average wake width, and the small-scale structure of the hypersonic turbulent boundary layer persists for some distance in the near wake.

Tests have been conducted in the Naval Ordnance Laboratory's ballistics ranges to investigate the drag, stability, and wake characteristics of slender cones and spheres at very high speeds. Drag coefficients were found to be linear functions of the mean squared angle of attack. Zero angle of attack drag coefficients were correlated using a mean viscous interaction parameter. Stability derivatives were found to be independent of the mean squared angle of attack. Good agreement was obtained between the experimentally determined drag and stability coefficients and theory. The width of the turbulent viscous core behind both cones and spheres was found to be proportional to the cube root of (xCDA). The constant of proportionality was the same for cones and spheres and varied with flight Mach number. Wake transition Reynolds numbers were found to be independent of the freestream body Reynolds number for a constant Mach number, but strongly dependent on flight Mach number.

Theme A has arisen in recent years for testing of re-entry vehicle heat shield materials in combined ablation and erosion environments. Such testing can be accomplished for very short time intervals (0.05 sec) in ballistics range facilities. For longer times measured in seconds, however, ablation and erosion tests currently can only be accomplished separately in different wind tunnel facilities. In state-of-the-art ablation facilities, the expansion nozzles are so short that direct drag acceleration of particles in the flow is relatively ineffective. Conversely, in wind tunnel facilities designed for maximum drag acceleration of particles by use of very long nozzles, the Mach number increases to such a degree that stagnation point heating rates are well below the range of interest for ablation testing. An analytical study has been made for a facility configured to produce ablation and erosion environments simultaneously. The proposed approach utilizes a conventional arc-heated ablation facility and a separate light-gas particle acceleration nozzle for generation of the erosion flux. These two components are combined by means of an inertial transfer of particles across a mixing layer formed between the two flows, similar to the process proposed as part of the RHEA (Re-entry Heating Energies Analyzer) facility concept.

An experimental program was conducted to test the validity of extrapolating the PANT (Passive Nosetip Technology) boundary-layer transition correlation, based on wind-tunnel/calorimeter-model results, to actual nosetip materials exposed to actual reentry environments. Pre-ablated ATJ-S graphite nosetips were flown on specific ballistics range trajectories through both air and nitrogen (with and without ablation). Surface temperature contours were measured via electrooptical pyrometry, from which transition zone presence and location were inferred. Significant discrepancies were noted between predicted and experimentally observed transition zone behavior, as influenced by Reynolds number and wall-temperature effects. A question was raised concerning characterization of a surface microroughness distribution, for transition purpose, by its median value. In addition, significant surface roughness effects on laminar-flow heat-transfer rates were noted.

These n — r equations can be solved for // + ) . . . , ju, the solution values substituted in (11), and then putting t = t$ for i = r + 1, . . . , n yields the required values of £*"(fo). Thus, the steps in this method are exactly parallel to those in the adjoint method except that here one additional computation (equivalent to a matrix X vector multiplication) is necessary after the simultaneous linear equations have been solved. It hardly seems worth the effort of introducing the adjoint system to avoid this simple step, especially in an exposition of principles. The first use of the adjoint system in problems having a superficial resemblance to that considered here was by Bliss4 hi his work in ballistics during WW I, but in, these applications it serves a much more useful purpose. A simple example of this kind is that in which the £*(£) are variations from a normal trajectory due to abnormal initial conditions and an expression for the final value of just one of the £*(0 is required, in terms of arbitrary initial^ variations of all of the £*(£). The coefficients in this expression can be obtained with only one integration of the adjoint equations, whereas if the same expression were to be obtained by integrating the equations of variation, n integrations would be required.

References 1 Wilson, L. N., "Body shape effects on axisymmetric wakes: transition," AIAA J. 4, 1741-1747 (1966). 2 Clay, W. G., Labitt, M., and Slattery, R. E., "Measured transition from laminar to turbulent flow and subsequent growth of turbulent wakes," AIAA J. 3, 837-841 (1965). 3 Lees, L., "Hypersonic wakes and trails," AIAA J. 2, 417428 (1964). 4 Wilson, L. N., "Far wake behavior of hypersonic spheres," General Motors Defense Research Labs. TR 66-19 (1966); also AIAA J. (to be published). 5 Behrens, W., "Flow field and stability of the far wake behind cylinders at hypersonic speeds," PhD Dissertation, California Institute of Technology (1966). 6 Webb, L., Hromas, L., and Lees, L., "Hypersonic wake transition," AIAA J. 1, 719-721 (1963). 7 Hromas, L. and Lees, L., "Effect of nose bluntness on the turbulent hypersonic wake," Ballistics Systems Div. TDR-62354 (1962).

Rastogi, R. P., Kishore, K., and Deepak, D., "Heterogeneous Combustion of Aluminized Propellants," AIAA Journal, Vol. 14, Feb. 1976, pp. 229-234. Wooldridge, C. E. and Muzzy, R. J., "Internal Ballistics Considerations in Hybrid Rocket Design," Journal of Spacecraft and Rockets, Vol. 4, Feb. 1967, pp. 255-262. Marxman, G. A., Wooldridge, C. E., and Muzzy, R. J., "Fundamentals of Hybrid Boundary Layer Combustion," Progress in Astronautics and Aeronautics, Heterogeneous Combustion edited by H. G. Wolfhard, I. Glassman, and L. Green Jr., Vol. 15, 1964, Academic Press, New York, pp. 485-522. Seifert, H. S., "Hybrid Rocket Theory and Design," Jet, Rocket, Nuclear, Ion and Electric Propulsion: Theory and Design, Vol. 7, ed. by W.H.T. Loh, Springer-Verlag, New York, 1968, pp. 332-335. The value for/?5 was omitted in the typesetting.