The wave equation February 23 2009 Laplace Equation Conclusion and The Wave Equation Overview Review material to date General approach for solving PDEs Other ideas about Laplace s Equation Derivation and physical meaning of wave equation Solution of the wave equation by separation of variables Introduction to the D Alambert solution of the wave equation Midterm Exam Wednesday March 4 Larry Caretto Mechanical Engineering 501B Seminar in Engineering Analysis February 23 2009 2 Solving PDEs Solving PDEs II Perform necessary operations if boundaries not homogenous Get appropriate separation of variables solution Diffusion define u x t v x t w x Want eigenfunctions to express initial or boundary condition v satisfies diffusion equation with zero boundary conditions w satisfies boundary Use homogenous boundary conditions to determine constants in solutions and eigenvalues General sum of all eigenfunctions used to fit initial or boundary condition Laplace equation use superposition Solution is sum of two or more solutions each of which has only one nonzero boundary Get equation in appropriate coordinate system rectangular or cylindrical 3 Midterm Exam Review Spherical Laplace Wednesday March 4 Covers material on diffusion and Laplace equations Includes material up to and including tonight s lecture and homework for Monday March 2 Open book and notes including homework solutions Laplace s equation in a sphere has solutions in Legendre polynomials Pn x 2 u 1 u sin 0 r r r sin u r Pn cos An r n Bn r 1 n n 0 Pn x 5 ME 501B Engineering Analysis 4 int n 2 m 0 1 n 2n 2m x n 2 m 2 n m n m n 2m 6 1 The wave equation February 23 2009 Review Hollow Cylinder Review Hollow Cylinder II Consider various boundary conditions Nonzero conditions on upper or lower surface only gives Bessel eignefunctions Nonzero conditions on inner or outer surface gives sine or cosine eigenfunctions Laplace s Equation in two dimensional cylindrical region 0 z L and Ri r Ro u r 0 u r L u Ri z 0 1 u 2u r 0 u Ro z u R z r r r z 2 I R m u r z Cm sin m z I 0 m r 0 m i K 0 m r m K 0 m Ri L m 1 L 2 K 0 m Ri Cm sin m z u R z dz I 0 m R0 K 0 m Ri I 0 m Ri K 0 m R0 0 7 8 Review Hollow Cylinder III Laplace s Equation in two dimensional cylindrical region 0 z L and Ri r Ro 1 u 2u r 0 u r L u N r r r r z 2 Eigenvalues m m R0 and eigenfunction R R J 0 m i Y0 m J 0 m Y0 m i 0 R R 0 0 Eigenfunction P0 mr radius ratio 0 5 9 0 25 radius ratio 0 9 0 35 0 20 40 60 80 100 10 Review Conclusions Eigenfunction expansion in P0 mr Y0 mR0 J0 mr J0R 0 mR0 Y0 mr Approach to solving Laplace equation is similar to that of diffusion equation m J 0 m Ri 2 ru N r P0 m r dr Ri sinh m L 2 J 02 m Ri J 02 m R0 Solution for uN r U a constant sinh m z J 0 m Ri P0 m r m 1 sinh m L J 0 m Ri J 0 m R0 u r z U 11 ME 501B Engineering Analysis 0 05 radius ratio 0 1 Review Hollow Cylinder IV Cm 0 05 0 15 Cm sinh m z Y0 m R0 J 0 mr J 0 m R0 Y0 m r m 1 R R J 0 m i Y0 m J 0 m Y0 m i 0 R0 R0 0 15 u r 0 u Ri z u Ro z 0 u r z Review Eigenvalues f Ri R0 0 25 Main difference is that second dimension y or r in Laplace equation gives closed boundary instead of open boundary in time Use separation of variables Have eigenfunction solution sine cosine Bessel or other in one dimension Use eigenfunction expansion to fit condition at one boundary 12 2 The wave equation February 23 2009 Review Conclusions II Vector Calculus Important results for Laplace equation See notes on vector calculus or chapters nine and ten in Kreyszig for background details not given here Results are independent of coordinate system but Cartesian used for examples Introduce gradient and divergence which are vector scalar functions Use superposition to solve Laplace equation with more than one nonzero boundary Additional cylindrical geometry considerations Complex Bessel functions when radial boundary is not eigenfunction solution Must include both Y0 and J0 when radial coordinate does not start at zero must have zero boundary at inner radius 13 Gradients Physical Gradients Gradient is a vector in written here in Cartesian space where we have f x y z Definition of gradient Del operator 14 Gradients of Laplace equation solutions often proportional to flux terms f f f j k z x y i j k x y z Heat flux and temperature gradient Diffusion flux and mass fraction gradient Velocity and velocity potential in ideal flow Current and electrostatic potential grad f f i grad f is magnitude and direction of maximum gradient df ds Grad f is perpendicular to line of constant f If we have a plot of constant potential the lines perpendicular to the potential are flux lines 15 16 Divergence Divergence converts vector v vxi vyj vzk into a scalar written as div v Definition of divergence Del operator v x v y v z x y z i j k x y z div v v Gauss divergence theorem n is vector normal to surface pointing outward div vdV v ndA 17 ME 501B Engineering Analysis Enclosed Volume Surface 18 3 The wave equation February 23 2009 div qdV q ndA Enclosed Volume Relation to Laplace Equation Surface Example of heat flux vector q W m2 q n is component of q normal to surface dA flowing outward Integrand in surface integral q ndA is heat flow watts flowing out through infinitesimal area dA Surface integral gives total heat flow through surface in outward direction The vector v may be gradient of a scalar representing a flux v k grad u div vdV div k grad u dV Enclosed Volume Enclosed Volume v ndA Surface For constant k div grad u dV udV 2 Enclosed Volume Enclosed Volume 1 v ndA k Surface 19 Interpretation of 20 2u 0 Complex Variable Basics When v k grad u is a flux that is the gradient of a scalar Laplace s equation for u says that the net inflow of v is zero 1 2udV v ndA 0 k Surface Enclosed Complex analysis gives insights to Laplace Equation in two dimensions Functions of complex variable z x iy f z u x y iv x y for example f z z2 x iy 2 x2 2ixy y2 f z u x2 y2 and v 2xy Volume Example of …