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 Copy 322 
 RM L54A29a 
 
 NACA 
 
 RESEARCH MEMORANDUM 
 
 ON SLENDER-BODY THEORY AT TRANSONIC SPEEDS 
 
 By Keith C. Harder and E. B. Klunker 
 
 Langley Aeronautical Laboratory 
 Lar^ley Field, Va. 
 
 UNIVERSHY OF FLO«DA OSSIFICATION cmr.B. TO «usc-:.^ 
 
 DOCUMENTS DEPARTMENT /^urHORm- J.V.CROWLEY DATE; 9-7-55 
 
 1 20 MARSTON SCIENCE LIBRARY WL 
 
 P.O. BOX 1 1 701 1 CHANGE NO. 3075 
 
 GAINESVILLE, Fl 32611-7011 USA 
 
 CLASSIFIED DOCUMENT 
 
 This material contains Information affecting the National Defense of the United States within the meaning 
 of the espionage laws, Title 18, U.S.C., Sees. 793 and 7&4, the transmission or revelatioQ of which in any 
 manner to Ein unauthorized person is prohibited by law. 
 
 NATIONAL ADVISORY COMMITTEE 
 FOR AERONAUTICS 
 
 WASHINGTON 
 
 March 24, 1954 
 
 CONWeiNTIAL OT 
 
/W 0^> ''^l ^?DH^S 
 
 NACA RM L5i4-A29a CONFIDENTIAL 
 
 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 
 
 RESEARCH MEMORANDUM 
 
 ON SLENDER-BODY THEORY AT TRANSONIC SPEEDS 
 By Keith C. Harder and E. B. Klunker 
 
 SUMMARY 
 
 The basic ideas of the slender-body approximation have been applied 
 to the nonlinear transonic- flow equation for the velocity potential in 
 order to obtain some of the essential features of slender-body theory 
 at transonic speeds. The results of the investigation are presented 
 from a unified point of view which demonstrates the similarity of 
 slender-body solutions in the various Mach nvimber ranges. The primary 
 difference between the results in the different flow regimes is repre- 
 sented by a certain function which is dependent upon the body area 
 distribution and the stream Mach niimber. The transonic area rule and 
 some conditions concerning its validity follow from the analysis. 
 
 INTRODUCTION 
 
 Slender-body theory originated with Monk's paper (ref. l) in 192^+ 
 in which the forces on slender airships were calculated for low-speed 
 flight. In 1938 Tsien (ref. 2) pointed out that Munk's airship theory 
 also applied to the flow past inclined, pointed bodies at supersonic 
 speeds. The subject gained new importance in 19^6 with the appearance 
 of Jones's paper (ref. 3) in which it was shown that the basic ideas of 
 the slender-body approximation could be used to calculate the forces on 
 slender lifting wings at both subsonic and supersonic speeds provided 
 that proper account was taken of trailing- vortex sheets. Since Jones's 
 paper, the subject has received wide treatment in the literature. In 
 an important paper in 19^9 j Ward (ref. h) developed a general unifying 
 theory for the flow past smooth, slender, pointed bodies at supersonic 
 speeds which contains as special cases the lifting planar wings of Jones 
 and the slender nonlifting bodies treated by Von Karman (ref. 5)' The 
 corresponding problem at subsonic speeds has been examined by Adams and 
 Sears (ref. 6) who also extended the slender-body concepts to shapes 
 which are not so slender. Lighthill (ref. 7) has given a method for 
 calculating the flow past bodies with discontinuities in slope. Keime 
 (ref. 8) has developed solutions for slender wings with thickness and 
 various lifting configurations have been treated by Heaslet, Spreiter, 
 Lomax, Ribner, and others (refs. 9 to li<-). 
 
 CONFIDENTIAL 
 
CONFIDENTIAL 
 
 NACA RM L5i<-A29a 
 
 The slender-body theory presented in references 2 to Ik has been 
 based upon the linearized equation for the velocity potential. In the 
 present paper^ the basic ideas of the slender-body approximation are 
 applied to the nonlinear transonic equation for the velocity potential 
 in order to obtain some of the essential features of slender-body theory 
 at transonic speeds. The attempt has been made to present the results 
 from a unified point of view which demonstrates the similarity of the 
 slender-body solutions in the various Mach number ranges. 
 
 The authors wish to acknowledge the invaluable aid and advice of 
 Dr. Adolf Busemann of the Langley Laboratory during the writing of this 
 paper. 
 
 SLENDER-BODY APPROXIMATION 
 
 Slender-body theory deals with that class of shapes whose length 
 is large compared with any lateral dimensi-dn. For such shapes at both 
 subsonic and supersonic speeds, the flow in planes normal to the stream 
 direction can be approximated by solutions of Laplace's equation. The 
 justification is that for very slender wings or bodies the variation 
 of the geometrical properties in the stream direction is small and, 
 consequently, the rate of change of the longitudinal component of the 
 velocity in the stream direction is also small. The various slender- 
 body solutions have all been developed on the basis of the linearized 
 potential equation. However, a similar development can be made on the 
 basis of the nonlinear transonic equation. 
 
 The simplest differential equation for the disturbance potential 
 which is generally valid at transonic speeds (ref. I5, for example) is 
 
 E 
 
 w 
 
 (7 
 
 1)m2$x]<] 
 
 XX 
 
 + $ 
 
 $ 
 
 rr 
 
 ^98 
 
 = 
 
 (1) 
 
 where x, r, and -d 
 Mach number, and 7 
 and constant volume, 
 coordinates | and 
 
 less potential by 
 
 equation becomes 
 
 are cylindrical coordinates, M is the stream 
 is the ratio of specific heats at constant pressure 
 With the introduction of the dimensionless 
 r\ by X = Z| and r = bi^, and of the dimension- 
 2 
 = — 0(^^11;^)^ "the transonic potential- flow 
 
 l/f 
 
 (7 
 
 DM^d) 0, 
 
 <^r 0. 
 
 ee 
 
 h^-h^-f--^ = ^ 
 
 (2) 
 
 CONFIDENTIAL 
 
NACA RM L54A29a CONFrDEWTIAL 
 
 where Z is a characteristic length and b is a characteristic width 
 such as the largest lateral dimension of the configuration. For suffi- 
 ciently small values of the width parameter b/z, it appears that the 
 terms involving derivatives in the stream direction can be neglected 
 to obtain the result that the flow satisfies Laplace's equation 
 
 <^rr + ^<^r + ^^^^ = (5) 
 
 in the cross-flow plane. Equation (3) represents the slender-body 
 approximation to equation (l). Some conditions will be determined 
 subsequently which are necessary in order for solutions of equation (3) 
 to be approximate solutions of equation (l). 
 
 The boundary conditions for the flow about a bod;^'' in a uniform 
 stream are the vanishing of the disturbance velocities at infinity and 
 
 Sn dx dx 
 
 on the body where n is in the direction normal to the body contour in 
 
 the cross-flow plane. For flows satisfying Laplace's equation in the 
 
 cross-flow plane, the surface boundary condition can be integrated 
 
 (ref. k, for example) to give 
 
 where dt is an element length in the direction of the tangent to any 
 contour C in the cross-flow plane enclosing the body, S(x) is the 
 cross-sectional area distribution of the body, and the prime denotes 
 differentiation with respect to the indicated argument. 
 
 The slender-body solution of equation (l) can be represented by a 
 solution of equation (3) plus a function of integration. Since equa- 
 tion (3) is independent of Mach number, the form of the solution is 
 identical with the known slender-body solutions for subsonic and super- 
 sonic flow. The slender-body analyses of references k and 6 have 
 established that the solution can be represented by a distribution of 
 sources and higher order singularities on the axis; an equivalent form 
 is given by a distribution of sources in the region of the cross-flow- 
 plane interior to the surface boundary. The slender-body solution is 
 then expressed as 
 
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NACA RM L54A29a CONFIDENTIAL 
 
 where cp denotes the solution of Laplace s equation in the cross-flow 
 plane with | appearing as a parameter introduced by the geometry of 
 the cross section at ^. The function 9, being independent of the 
 stream Mach number, can be evaluated for incompressible flow past the 
 shape \inder consideration. 
 
 A necessary condition for equation (6) to be an approximate solu- 
 tion of equation (l) is that the terms neglected in equation (l) be 
 small compared with those retained. By assuming for the moment that 
 log p is the only singular term in $, the ratio of the term 
 
 1 - M^ - (7 + 1)M^xJ'^xx' which is neglected in the slender-body 
 
 approximation, to any of the remaining terms in equation (l) is of the 
 order 
 
 (f)^ J(l - m2) P(log I) + 0(1)1 + (if [o(log2 f) + O(log f) + 0(1) 
 
 where 0( ) denotes order of and the nonsingular terms are denoted 
 by 0(1). This ratio can be made smaller than any prescribed value e 
 by restricting the solution to the interior of a cylinder of some 
 radius, say r/z. For given values of M and €, the radius of this 
 cylinder increases with decreasing b/l and the ratio R/b approaches 
 infinity as b/z approaches zero. Moreover, for given values of e 
 and R/z, larger values of b/z are permitted as M approaches 1. 
 
 From equation (6) it is apparent that S'(x/z) and its deriva- 
 tives must be finite in order to satisfy the requirement of small dis- 
 turbances. Moreover, an additional restriction on the asymmetry of 
 the body is sometimes required (ref. k), particularly for lifting con- 
 figurations - namely, the radius of curvature of the body boundary in 
 the cross- flow plane must be 0(b) where the boundary is convex outward. 
 The restrictions on body shape imposed by the function g(x/z) will be 
 considered subsequently after the nature of this function is established. 
 
 The function g(x/z) is determined from considerations involving 
 
 the complete transonic differential equation (eq. l) and, consequently, 
 
 is dependent upon the stream Mach number. Examination of equation (6) 
 
 r f^n R 
 at T ~ < T shows that the contribution of the asymmetric term of 
 
 the source distribution to the potential can be made smaller than any 
 
 prescribed value by making b/Rp sufficiently small ( — = =— 5 5— ). 
 
 \Po ^o ^0/ 
 
 To this order of approximation, then, the flow field external to Rq 
 is axisymmetric . Moreover, there is an axisymmetric flow which approxi- 
 mately matches the pressure and flow direction of the slender-body 
 
 CONFIDENTIAL 
 
CONFIDENTIAL NACA RM L5l+A29a 
 
 solution at r = Rq . The potential Oq for this associated axi symmetric 
 flow satisfies equation (l) and is expressed symbolically as 
 
 ^o-? 
 
 'o'u/i) ,„„ r , „ /„/, „/,n 
 
 at ° I 
 
 log J + go(x/l,r/l) (7) 
 
 where the characteristic width b and length I are taken the same as 
 for the asymmetric shape. In equation (7)^ Sq{x/i) is the area dis- 
 tribution of the associated body of revolution which has the same outer 
 flow (r 5"Rq) as the asymmetric shape. If this axisymmetric flow 
 satisfies the slender-body conditions, then g^ is independent of r 
 for r < R. In order for the radial derivatives to match at r = Rq, 
 So(x/l) must be equal to s(x/z); that is, the associated body of 
 revolution must have the same axial distribution of cross-sectional 
 area as the asymmetric shape. In order for the pressure distributions 
 to match at r = Rq, go(x/0 must be the same as g(x/z ) . Thus, 
 g(x/z) is the same function as that for a body of revolution having 
 the same axial distribution of cross-sectional area. 
 
 In the preceding discussion the region of validity of the slender- 
 body approximation to 4)q was tacitly assumed to be at least as large 
 as that for $. This condition is certainly true since the singular 
 terms in the two solutions are the same. 
 
 In the slender-body approximation the term ll - M - (7 + l)'l>xj^^ 
 is required to be small compared with any of the other terms in the 
 transonic differential equation. If this condition is to be satisfied 
 in the neighborhood of weak shock waves where g'(x/z) would be required 
 to have a jump proportional to the pressure rise and g"(x/z) would be 
 infinite, the quantities (which were included in the terms denoted 
 as 0(1)) 
 
 g"(x/z)[l - m2 - (7 + Dm^^oJ (8) 
 
 and 
 
 g"(x/l)|^ Jj fip-^,i;^ix/l)lQgd(cos ^ - ^ COS ^ij + (sin ,3 - ^ sin ^1) Pi dp^ diS^ (9) 
 
 CONFIDENTIAL 
 
NACA RM L5l<-A29a 
 
 CONFIDENTIAL 
 
 must be bo\inded there. Thus, at shock vaves the coefficients of g"(x/z) 
 in expressions (8) and (9) must vanish. In the first of these expres- 
 sions, which is axisyTnmetric, the coefficient of g (x/z) vanishes for 
 a local Mach number of 1. Moreover, the average value of the local 
 Mach number ahead and behind a weak normal shock wave is 1. With this 
 interpretation of shock waves, then, the coefficient of g (x/z) vanishes 
 and the solution should represent the flow in the neighborhood of weak 
 normal shocks. Also, since g(x/z) is the same function for both a 
 slender configuration and its associated body of revolution, their 
 shock-wave systems would have the same strength and location. In order 
 for expression (9) to be bounded in the neighborhood of a shock wave, 
 the double integral must vanish. Since this double integral gives rise 
 to asymmetric shapes and is zero for chordwise locations where the body 
 is axisymmetric, the additional restriction is obtained that the body 
 cross section must be circular in the vicinity of shock waves. 
 
 The slender-body solutions in the various Mach number ranges are 
 similar in that they are all represented by equation (6) although the 
 function g(x/z) differs for the various speed ranges. Analytic 
 expressions for g(x/z) have been obtained for supersonic and subsonic 
 flows by considering general solutions of the complete linearized equa- 
 tion satisfying the boundary condition of vanishing disturbance veloc- 
 ities at infinity. These solutions were then expanded in the neighbor- 
 hood of the body to evaluate g(x/z). Ward (ref. k) has determined 
 this function for supersonic flows as 
 
 g(x/z) 
 
 2rt 
 
 s'(x/z) 
 
 ■-x/z 
 
 ^°^(^) - J^ ="(si'-<l - ^i)^^i 
 
 The corresponding result at subsonic speeds was obtained by Adams and 
 Sears (ref. 6) as 
 
 g(x/z) 
 
 _ 1 
 2rt 
 
 S'(x/Z) i,.fl-M2\ 1 
 
 [_ 2 ^°^\ k J ^J 
 
 
 |^)^s"U,)log(|,-f)d,, 
 
 -x/z 
 
 s"(li)log(^ - |i)d|i 
 
 where the body extends from x = to x = Z. Although an analytic 
 expression for g(x/z) at transonic speeds is not known, it has been 
 established that the stream Mach number enters the solution only through 
 g(x/z) and that the only geometric property of the body influencing 
 
 CONFIDENTIAL 
 
COKFIDENTIAL 
 
 NACA RM L5UA29a 
 
 this function is the area distribution. The transonic similarity rule 
 for bodies of revolution (ref. I5 or I6) shows that g(x/z) can be 
 expressed in the form 
 
 , , . s (x/z) 
 ;(x/l) = — —^ log 
 
 4rt 
 
 (. . 1)M=@= 
 
 + f(x/Z;K) 
 
 where the similarity parameter is 
 
 K = 
 
 1 - M^ 
 
 (7 + 1)m2| 
 
 er 
 
 AERODYNAMIC FORCES 
 
 Since the slender-body solutions are all represented by equation (6), 
 formal expressions for the aerodynamic forces can be determined which are 
 valid throughout the Mach number range. Consequently, many of the 
 essential features of slender-body theory at transonic speeds can be 
 obtained without resorting to detailed calculations. 
 
 Lift 
 
 The most significant difference between the slender-body solutions 
 at subsonic, transonic, and supersonic speeds is that the function g(x/z) 
 differs in these various speed ranges. However, the term in the pressure 
 arising from the function g(x/z) makes only a uniform contribution to 
 the pressure at any value of x and, therefore, cannot affect the lift 
 distribution or the lift. Thus, within the slender-body approximation, 
 the lift distribution depends only upon the function 9 and, conse- 
 quently, is independent of the stream Mach number. Several investigators 
 (Robinson and Young (ref. I7) and Heaslet, Lomax, and Spreiter (ref. 9), 
 for example) have previously noted that the linearized slender -body 
 theory gave consistent results, even at a Mach number of 1, for planar 
 systems. 
 
 According to slender-body theory, the lift distribution can be 
 obtained completely from solutions of Laplace's equation in the cross- 
 flow plane. Since this equation is linear, the lift is proportional 
 to the angle of attack even at transonic speeds. Ward has obtained 
 an especially simple form for the drag due to lift in which 
 
 CONFIDENTIAL 
 
NACA RM L5i+A29a 
 
 CONFIDENTIAL 
 
 Dl = I aL 
 
 where a is the angle of attack measured from zero lift and L is 
 the lift. 
 
 Drag 
 
 By computing the momentum change of the fluid passing through a 
 cylinder enclosing the "body, the drag is determined as 
 
 q q. \ I 
 
 ") 
 
 s'(|)g'(|)d| + 
 
 cp^dt 
 on 
 
 (10) 
 
 where the body extends from ^ = to | = 1, C ' denotes the contour 
 of the body at the stern which in the case of wings or wing-body com- 
 binations includes the trailing- vortex wake, q is the stream dynamic 
 pressure, and D^^ is the base drag. Equation (lO) is valid throughout 
 the Mach number range provided the appropriate forms of the func- 
 tion g(x/z) are employed. The line integral is zero for nonlifting 
 configurations if the body is closed or if the body ends in a cylindrical 
 section whose elements are parallel to the stream. The effect of Mach 
 number (excluding the variation of base drag with Mach number) is con- 
 tained in the term involving g(x/z). 
 
 When the subsonic form of g(x/z) is used in equation (lO), the 
 correct result is obtained that the drag of nonlifting configurations 
 is zero. By using the supersonic form of g(x/z), the drag varies with 
 Mach number like [s'(l)] log(M - l). For pointed bodies, or for 
 bodies which end in a cylindrical section, the supersonic slender-body 
 theory indicates that the drag is independent of Mach number. For 
 "bodies which do not satisfy these conditions, the supersonic result 
 indicates that the drag approaches infinity as the Mach number 
 approaches 1. These results from linear theory cannot be considered 
 satisfactory at transonic speeds since they give a discontinuity in 
 the drag as the Mach number is increased through 1; whereas experimental 
 data show that the drag starts to increase rapidly at a subsonic Mach 
 number and varies smoothly through 1. However, the few known solutions 
 of the nonlinear transonic- flow equation are in good agreement with 
 experiment in this regard. Consequently, the drag rise of slender shapes 
 should be correctly approximated by equation (lO) when the transonic 
 form of g(x/z ) is employed in the drag equation. 
 
 CONFIDENTIAL 
 
10 CONF-IDENTIAL NACA RM L5l4-A29a 
 
 Transonic Area Rule 
 
 The body shape enters into the function g(x/z) only as a function 
 of the cross-sectional area distribution throughout the Mach number 
 range. This property of the slender-body solutions leads to an important 
 result even though the analytic expression for g(x/z) is not known at 
 transonic speeds. Examination of equation (lO) shows that the body 
 cross-sectional shape enters into the slender-body drag expression only 
 through the contour integral evaluated at the stern of the configura- 
 tion. For a fixed geometry at the base, then, the drag of nonlifting 
 configurations depends only on the axial distribution of the body cross- 
 sectional area and is independent of the cross-sectional shape. Thus, 
 within the slender-body approximation, the drag of a nonlifting configu- 
 ration is the same as that of the associated body of revolution having 
 the same streamwise distribution of cross-sectional area provided the 
 base geometry is fixed. It is in this sense that an equivalent body of 
 revolution is associated with a wing-body combination. This result, 
 often referred to as the area rule, is especially significant at 
 transonic speeds where larger values of the width parameter b/l are 
 permitted than in other speed ranges . 
 
 The property of the dependence of the drag upon the distribution 
 of cross-sectional area has previously been obtained by Ward (ref . k) 
 and Graham (ref. l8) for supersonic flow and has been observed experi- 
 mentally by Whitcomb (ref. 19, for example) at transonic speeds. The 
 importance of this result was first noted by Whitcomb who demonstrated 
 that the area rule could be used as a basis for the design of low-drag 
 wing-body combinations at transonic speeds. 
 
 From the preceding development, the transonic area rule is subject 
 to the restrictions of slender-body theory with the additional condition 
 that the base geometry be fixed. These restrictions imply that modifi- 
 ca Lions to the equivalent body of revolution must be performed in the 
 region between shock waves. However, the seriousness of violating this 
 condition is not well understood at the present time. For example, 
 schlieren photographs seem to indicate that, even in cases where the 
 presence of the wing affects the strength of the shock, the average 
 strength of the shock may be close to that for the equivalent body of 
 revolution. 
 
 Langley Aeronautical Laboratory, 
 
 National Advisory Committee for Aeronautics, 
 Langley Field, Va., January l8, 195^- 
 
 CONFIDENTIAL 
 
NACA RM L5UA29a CONFIDENTIAL 11 
 
 REFERENCES 
 
 1. Munk, Max M. : The Aerodynamic Forces on Airship Hulls. NACA 
 
 Rep. l8i+, 192i+. 
 
 2. Tsien, Hsue-Shen: Supersonic Flow Over An Inclined Body of Revolu- 
 
 tion. Jour. Aero. Sci., vol. 5, no. 12, Oct. 1938, pp. U80-i4-85. 
 
 5. Jones, Robert T.: Properties of Low-Aspect-Ratio Pointed Wings at 
 Speeds Below and Above the Speed of Sound. NACA Rep. 835, 19^6. 
 (Supersedes NACA TN 1032.) 
 
 k. Ward, G. N. : Supersonic Flow Past Slender Pointed Bodies. Quarterly 
 Jour. Mech. and Appl. Math., vol. II, pt . 1, Mar. I9U9, pp. 75-97- 
 
 5. Von Karman, Th.: The Problem of Resistance in Compressible Fluids. 
 
 Atti del V Convegno della Fondazione Alessandro Volta, R. Accad. 
 d' Italia, vol. XIV, 1936, pp. 5-59- 
 
 6. Adams, Mac C, and Sears, W. R.: Slender-Body Theory - Review and 
 
 Extension. Jour. Aero. Sci., vol. 20, no. 2, Feb. 1953, PP. B5-98* 
 
 7. Lighthlll, M. J.: Supersonic Flow Past Slender Bodies of Revolution 
 
 the Slope of Whose Nferidian Section Is Discontinuous. Quarterly 
 Jour. Mech. and Appl. Math., vol. I, pt. 1, Mar. 1914-8, pp. 90-102. 
 
 8. Keune, Friedrich: Low Aspect Ratio Wings With Small Thickness at 
 
 Zero Lift in Subsonic and Supersonic Flow. KTH-Aero TN 21, Roy. 
 Inst, of Tech., Div. of Aero., Stockholm, Sweden, 1952. 
 
 9. Heaslet, Max A., Lomax, Harvard, and Spreiter, John R.: Linearized 
 
 Compressible -Flow Theory for Sonic Flight Speeds. NACA Rep. 95^, 
 1950. (Supersedes NACA TN 182U.) 
 
 10. Spreiter, John R.: Aerodynamic Properties of Slender Wing-Body 
 
 Combinations at Subsonic, Transonic- and Supersonic Speeds. NACA 
 TN 1662, 1914-8. 
 
 11. Lomax, Harvard, and Heaslet, Max A.: Linearized Lifting-Surface 
 
 Theory for Swept-Back Wings With Slender Plan Forms. NACA TN 1992, 
 191^9. 
 
 12. Spreiter, John R.: Aerodynamic Properties of Cruciform-Wing and Body 
 
 Combinations at Subsonic, Transonic, and Supersonic Speeds. NACA 
 TN 1897, 1914-9. 
 
 CONFIDENTIAL 
 
12 CONFIDEIWIAL NACA RM L5l+A29a 
 
 15- Heaslet, Max A., and Loraax, Harvard: The Calculation of Pressure 
 on Slender Airplanes in Subsonic and Supersonic Flow. NACA 
 TN 2900, 1953. 
 
 1^+ . Ribner^ Herbert S.: The Stability Derivatives of Low- Aspect -Ratio 
 
 Triangular Wings at Subsonic and Supersonic Speeds. NACA TN 1423, 
 19^+7- 
 
 15. Busemann, Adolf: Application of Transonic Similarity. NACA TN 2687, 
 
 1952. 
 
 16. Oswatitsch, K., and Bemdt, S. B.: Aerodynamic Similarity at 
 
 Axisymmetric Transonic Flow Around Slender Bodies. KTH-Aero 
 TN 15, Roy. Inst, of Tech., Div. of Aero., Stockholm, Sweden, 
 1950. 
 
 17. Robinson, A., and Young, A. D.: Note on the Application of the 
 
 Linearized Theory for Compressible Flow to Transonic Speeds. 
 Rep. No. 2, College of Aeronautics, Cranfield, Aero. Res. Council 
 (British), Jan. 19i4-7. 
 
 18. Graham, Ernest W.: Pressure and Drag on Smooth Slender Bodies in 
 
 Linearized Flow. Rep. No. SM-13417; Douglas Aircraft Co., Inc., 
 Apr. 20, 19ij-9- 
 
 19. Whitcomb, Richard T.: A Study of the Zero- Lift Drag-Rise Charac- 
 
 teristics of Wing-Body Combinations Near the Speed of Sound. 
 NACA RM L52HO8, 1952. 
 
 CONFIDENTIAL 
 
 NACA-Langley - 3-24-54 - 325 
 

 
 
 
 
 
 
 
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