.. I- , ., -: .. - NASACONTRACTOR REPORT Q> N h -I pz U DIFFRACTION OF A PULSE BY A THREE-DIMENSIONALCORNER by Lu Ting and Fanny K'wg Prepared by I. NEW YORK UNIVERSITY New York, N. Y. 10453 for NATIONALAERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, D. C. MARCH 1971 ". TECH LIBRARY KAFB. NM 1. Report No. I 2. GovernmentAccession No. 1 3. Recipient'sCatalog No. r NASA CR-1728 4. Titleand Subtitle DIFFRACTION OF A PULSE BY A ~-CORNER 7. Author(.) 8. PerformingOrganization Re ort No. Lu Tingand Fanny Kung NYU ReDort"70-66 9. Performing Or anization Name andAddress 10. Work UnitNo. New Y0r% University University HeLghts 11. Contract or Grant No. New York, N.Y. 10453 NGL-33-016-119 13. Type of Report and Period Covered 2. SponsoringAgency Name and Address CONTRACTOR REPORT National Aeronautics & Space Administration Washington, D. C. 20546 14.Sponsoring Agency Code " 5. SupplementaryNotes 6. Abstract Forthe diffraction ofa pulse bya three-dimensionalcorner, the solution is conicalin three variables 5 = r/(Ct), 8 and so. The three-dimensionaleffect is confinedinside the unit sphere 5 = 1, aiththe vertex of thecorner as the center. Theboundary data on the unitsphere is provided by theappropriate solutions for the diffraction of a pulse bya two-dimensionalwedge. The solution exterior to the Zornerand inside the unit sphere is constructed by theseparation of the variable 5 from 8 and q. The associatedeigenvalue problem is subjected tothe differential equation in potential theory with an irregular boundaryin 8 - n plane. A systematicprocedure is presentedsuch that theeigenvalue problem is reducedto that ofa sys ternof linearalgebraic 2quations.Numerical results for the eigenvalues and functions are obtainedand applied to constructthe conical solution for the diffractio of a planepulse. For the diffraction ofa generalincident waveby zooa'ners oredges, two theorems are presented so thatthe value at the rrertexof the corner or along the edges can be determinedwithout the Zonstruction of thethree-dimensional non-conical diffraction solutions. Relevantnumerical programs for the analysis are presented in the appendix. 17. Key Words (Selected by Author(s)) 18. DistributionStatement Sonic Boom Diffraction Unclassified - Unlimited planepulse I 19. SecurityClassif. (of this report) 20. SecurityClassif. (of this page) 21- No. of Pager Unclassified Unclassified 91 For sale by the National Technical InformationService, Springfield, Virginia 22151 TABLE OF CONTEXTS SUMMARY 1 INTRODUCTION 2 FORMULATION 3 THEEIGENVALUE PROBLEM 5 NUMERICALSOLUTIONS OF THEEIGENVALUE PROBUX 10 CONSTRUCTION OF THECONICAL SOLUTION 3.6 ANUMERIC& EXAMPLE 19 GENERALINCIDENT WAVE 21 "MEAN-VALUE"THEOREMS FOR DIFFRACTION BY ACONICAL, SURFACE AND APPLICATIONS 24 CONCLUSION 31 ACKNOWLEDGMENT 32 REFERENCES 33 FIGURES 34 APPENDIX 46 PRINTOUT 50 TABLES 84 iii LIST OF FIGURES FIGURE -PAGE 1. Incidenceof a planepulse on a corner. 34 2. Eigenvalueproblem for the corner. 35 3. A two-dimensionalcorner. 36 4. Plane'pulse incidence head on to face OAD, the reflectedplane pulse and the diffracted fronts around edges OA, OD, and vertex 0. 37 5. Two-dimensionalconical solutions around edges OA and OD. 38 6. Over pressurealong constant 8 lineson face OAD (cp=3rr /4). 39 7. Overpressure along constant cp lines onface OAB (9~/2) andalong constant 8 linesonface OBD (cp=3rr/4). 40 8. sonic Typical boom41 signature. 9. Diffraction by a cone. 42 C om parison of two cones. two 10. of Comparison 43 11. Incidentof a planepulse on a block at instant t > L/C (OB=L, OA=OD;W; also shown are thediffracted wave fronts around edges OA, OD, BA' , BD' and around vertices 0 and B). 44 12. Incident of a planepulse on a bl.ockwith extended fictitioussurfaces at instant t > L/C (also shown are thediffracted wave frontsaround edges OA,OD and vertex 0). 45 iv DIFFRACTION OF A PULSE BY A THREE-DIMENSIONAL CORNER by Lu Tingand Fanny Kung New York University, New York, N.Y. SUMMARY Forthe diffraction of a pulseby a three-dimensionalcorner, e.g., the cornerof a cube,the solution is conicalin three variables 5 = r/(Ct), 8 and to. The three-dimensionaleffect is confinedinside the unit sphere 5 = 1, or the sonic sphere r = Ct withthe vertex of the corner as the center. Theboundary data on the unit sphere is providedby the appropriate solutions forthe diffraction of a pulse by a two-dimensional wedge.The solution exterior to thecorner and inside the unit sphere is constructedby the separationof the variable 5 from 8 and q. The associatedeigenvalue problem is subjectedto the same differentialequation in potential theory for the spherical angle variables 8 and cp, butwith an irregular boundary in 8 - Cp plane. A systematicprocedure is presentedsuch that the eigenvalue problem is reducedto that of a system of linearalgebraic equations. Numerical resultsfor the eigenvalues and functions are obtainedand are appliedto constructthe conical solution €or the diffraction of a planepulse. For the diffraction of a generalincident wave bycorners or edges, the solutions are no longerconical. Two theoremsare presented so thatthe value at the vertex of the corneror along the edges can be determined without the con- structionof the three-dimensional non-conical diffraction solutions. Relevantnumerical programs for the analysis are presentedin the appendix. 1. Introduction The problem of diffraction and reflection of acoustic waves or electro- magnetic waves bywedges, corners and other two-dimensional or axially sym- metric obstacleshas received extensive investigations. A surveyof these investigationscan be found in Reference 1. The presentinvestigation is motivatedby the study of the effect of sonic boom onstructures. The pressure wave createdby a supersonicairplane is three-dimensionalin nature. However, theradius of curvature of the wave front is usually much largerthan the length scale of a structure. Therefore,the incident waves canbe approximated by progressing plane waves withthe wave formusually in the shape of the letter N and are referred to as N-waves. Fortwo-dimensional structures in the shape of a rectangularblock, the diffraction of a planepulse by the first corner is givenexplicitly bythe two-dimensional conical solution of Keller andBlank r2]. The solutionfor each subsequent diffraction by the next corners can be obtained bythe use of Green's function for a wedge [l, pp. 108-1151. Forright angledcorners, the diffraction. solutions can be obtained by the solution ofan Abel typeintegral equation [ 31. By means ofthe integral of Duhamel, the solution for the diffraction of a planepulse by a two-dimensional structure was employedto construct the solution for the diffraction of an N-wave bythe same structure [4]. Forthe three-dimensional problem of the diffraction of sonic boom by structures,the first step is theconstruction of the solution for the diffractionof a planepulse by a cornerof a structure. The formulation ofthe differential equation and the appropriate boundary conditions for theproblem in three-dimensional conical variables is presentedin the next section. 2 2. Formulation Forthe acoustic disturbance pressure p, thegoverning differential equation is thesimple wave equation, in the region outside a trihedronsimulating the corner of a structure. As shown in Fig. 1, two edges, OA and OB, ofthe trihedron are inthe x-yplane and are bisected by the negative x-axis with half angle QI and the third edge OD, which is thenegative z-axis with the vertex 0 as theorigin. Let t=O bethe instant when thepulse front hits the vertex. Theboundary condition on the three faces of the trihedron is ap/an = 0 (2 1 The three-dimensionaldisturbance due to the vertex is confinedinside thesonic sphere r = Ct where r = (2 + + z2)%. Outsidethe sonic sphere,the pressure distribution is giveneither by the regular reflection ofthe plane pulse from the surfaces of the trihedron or by thediffraction of thepulse by the edges. When thepulse front is parallelto the edge, thediffracted wave is given by a two-dimensionalunsteady conical solution. When thepulse front is not parallel tothe edge, the diffracted wave is givenby a steadythree-dimensional conical solution. For both cases, solutions are givenin two conicalvariables in [Z] bymeans ofBusemann’s conicalflow method. Due tothe absence of a time scaleand a lengthscale, the disturbance pressure p nondimensionalizedby the strength of the incident pulse should be a functionof the three-dimensional conical variables, x/(Ct), y/(Ct) andz/(Ct) or in terms ofthe spherical coordinates by 5 = r/(Ct), 9 and.cp. The simple wave equationfor p(5,8,Cp) becomes, insidethe unit sphere, 5 = 1, andexterior to thetrihedron. Theboundary conditions are: ap/a = 0 onsurface OAB, 8 = n/2, - (TT-a)< Cp < n-a (41 ap/acp= o surfaceon OBD, cp = -nW, n/2 < 8 < n (5b 1 and p = F(8,cp) onunit sphere 5 = 1 outsideof the trihedron. The jump acorssthe sonic sphere is inverselyproportional to the squareroot of the area ofthe ray tube, (dS)’. Since all the raysreach thesphere come fromthe origin where dS = 0, the jump acorss thesonic 0 sphere is zero. The pressure is continuousacross the sphere 5 = 1; and F(8,cp) is definedby the solutions outside the sonic sphere in two conica variablesgiven by [Z]. 3 To construct the solution by the method of separation of variables, the usual trial substitution p(g,e ,cp) = 2 (c) G(p,rp) is introduced where p = cos0and eq. (3) becomes 62 (1-C2) Z"(5) + 2(l-C2) Z'(C) - XQ+l) Z(C) = 0 (6 1 for 1 > 5 B 0, and for the domain in p - cp planewith lcp[ < n-a when 0 2 p 2 -1, andwith I@[s l-r when 1 2 p > 0. Sincethe pressure p andalso G are singlevalued, G should be periodic in cp when 1 2 p > 0, i. e., Therange of cp is thereforerestricted from -n to TT andthe two ends are connectedby the periodicity condition. Thereplacement of the variable 0 by p and theuse of the constant of separation Xa+l) are motivated by the intention of representing G(p,B) bythe spherical harmonics.