Perturbation Methods ALI HASAN NAYFEH Copyright Q 2004 WILEY-VCH Valag GmbH & Co. KGaA Subject Index Acoustic, 77, 78 Aerodynamic, 110 Airfoil theory, 98, 99, 235, 303; see Supersonic airfoil theory Airy s equation, 312, 336 Airy s functions, 49, 336, 378 Algebraic equation, 2,57,74,95, 327 Algorithm, 168,171,199,209 for canonical systems, 212-214 generalized, 202-206 simplified, 206-208 Anomaly, 79 Aperiodic motion, 189 Astrophysics, 235 Asymptotic expansion, 23,78 of Airy s functions, 337 of Bessel s function, 16 defiition of, 12 divergent, 16 elementary operation on, 18-19 uniform, 17-19 uniqueness of, 14 see UlSO Asymptotic series Asymptotic matching principle; see Matching Asymptotic partitioning, 327-331 Asymptotic sequence, 12,14,16,18,19 factorial, 12 fractional powers in, 136, 137 logarithms in, 137,144 Asymptotic series, 10-12 defiition of, 11 versus convergent series, 15-16 see Asymptotic expansion Attenuation, 46, 78 Averaging,, 159-227 generalized, 168-171,191, 211,223-225,231 Krylov-Bogolinbov,, 165-168 Krylov-Bogolinbov-Mopolski, method Of, 174-179,183,194,211,212, 223,224,246,248 Struble s, 171-174,176, 183, 194,223 using, canonical variables, 179-189 Lagrangian, 216-222 Lie series and transforms, 200-246 von Zeipel transformation, 189-200 van der Pol s, 164-165 see ulso Smoothing Beam, niration of, 105,106,155,226,306 Bearing, slider, 54,125-128 Bellman, equation of, 22 Benard problem, 77 Bending of, shells and tubes, 54,353; see nlso Unsymmetrical bending of plates Benney s technique, 3842 Bernoulli s equation, 83 Besselfunctions, 1,5,6,15,312,315,383 asymptotic expansion of, 16,312-314, 329-331 integral representation of, 16, 314 zeros of, 21 Bethe-Salpeter equation, 372 Blunt body problem, 99 Boltzmann s equation, 236 Born approximation, 362 Born expansion, 308,361-367, 373 renormalization of, 367-372 Boundary conditions, loss of, 31, 34, 37, 38,54,111,114,122 transfer of, 27 Boundarylayer,18,23,79,111,112,147, 233 location of, 114-116, 122 Prandtl, 34 417
418 SUBJECT INDEX problem with two, s, 128-133 stability of, 353 Branch point, 82 Bretherton's equation, exercises involving, 227,306 treated, by variational approach, 217-221 by multiple scales, 266-269, 298-300 wavewave interaction for, 219-221,266-269 Brownian motion, 236 Buckling, 233 Canonical, averaging, variables, 179-189 equations, 180,199 Jordan, form, 327 mixed, variables, 199 system, 190,201 transformation, 181,187,191 variables, 181.183,184,195,202,216, 224 Caustic, 234,375-380 Change, 111,113 of characteristics, 89 see also Sharp change; Type change of Characteristic, parameters, 89,94 wave speeds, 91 Characteristic exponent, 58,62,66 Characteristics, expansion in terms of, 57, 86-94,303, 374 Circuit, electronic, see van der Pol oscillator Cluster expansions, 363 Compatability relationship, 217,222 Composite expansion, 114,144,384,385 construction of, 121 for earth-moon-spaceship problem, 139 for equation with variable coefficients, 125 for simple example, 121-122 for slider bearing, 128 for thermoelastic waves, 136 for unsymmetrical bending of plates, 133 Composite expansions, applied to turning point problems, 348-350, 144-154,317,318 Composite solution, 113 Conservation form of equations, 226 Coordinate, optimal, 50-51 parabolic, 79 perturbations, 1,4-7,21, 309-314,379 role of, systems, 23,49-52,54, 380 see olso Strained coordinates Correlation, 363 Cosmological expansion, 233 Cycle, limit, 35 Cylinder, elliptic, 346 a solid, expanding, 83-86 Cy-lindrical, functions, 358,378,380 jet, 99 Degenerate, 72, 74 Derivative-expansion procedure, 302 applications of, 243-269 description of, 230,236-240 limitations of, 269-270 see ulso Multiple scales, Detuning, 250 Diagram, 366,367 bare, 364 connected, 370,371,372 double, 365 dressed, 365 Diffraction, 346,380 Diffusion equation, 38 Dirichlet problem, 38 Discontinuity, see Singularity Dispersion relationship, 178,217,218, 219, 220,222,227,266,301 Dispersive waves, 78,234, 303 long nonlinear, 3842 see ulso Bretherton's equation; Klein- Gordon equation; Thermoehstic waves; Wave-wave interaction Divisor, small, 196 Domain, effect of, on nonuniformity of expansions, 38,42 infinite, 24-31 Duffing equation, 50,s 1 exercises involving, 54,104,105,223, 224,304 with slowly vary coefficients, 286 straightforward expansion for, 24-25 treated, by averaging using canonical variables, 182-183 by derivativeexpansion procedure, 243-245 by generalized multiple scales, 286-291 by Krylov-Bogoliubov method, 167
SUBJECT INDEX 4 19 by Krylov-Bogoliubov-Mitropo~~ method, 175-176 by Lindstedt-Poincard method, 58-60 by renormalization, 95-96 by Struble's method, 171-174 by two-variable expansion procedure, 271-273 by von Zeipel's procedure, 192-194 Dyson equation, 371, 381 Earth-moon-spaceship problem, 233,302, 303 exercises involving, 53, 107 illustrating limitations of strained coordinates, 102-103 straightforward expansion for, 4345 treated, by Lighthill's technique, 82-83 by composite expansions, 153-154 by matched asymptotic expansions, 137-139 by multiple scales, 295-298 Eccentricity, 64, 233, 346 Edge layer, 111 Eiconal equation, 374, 377, 379 Eigenvalue, 56 Eigenvalue problem, linear, 68-71 quasilinear, 71-76 Elastic, 46, 58,90, 353 waves, 89-94 Elliptic equation, 37,42,98, 189,234, 303, 360 Energy level, 56, 58 Entropy layer, 79 Euler-Lagrange equation, 216, 217, 218, 220,222 Faa de Bruno operators, 192 Feynman diagrams, 308,361-372 night mechanics, 233 Floquet theory, 60,62,64,66 Flow, down an inclined plane, 38-42,99 hypersonic, 79 jet, 99 past a body, 33-34, 113 through a channel, 54,56,216 see also Flow past a sphere; Supersonic airfoil theory Flow past a sphere, exercise involving, 158 straightforward expansion for, 28-31 treated by matched asymp totic expansions, 139-144 Flutter, 234 Foci, 253 Fokker-Planck equation, 235, 381 Fourier, 45,167,178 Frequency, 56,58,96, 165,252 Fresnel diffraction, 380 Frobenius,, 5,310 Gauge, function, 7,8 transformation, 232 Gaussian, 363, 367,369,372 Generalized expansion, see Composite expansion Generalized version of multiple scales, applications of, 276-302 description of, 232,241-243 limitations of, 302-303 Generalized averaging, see Averaging, Generalized vector, 179 Generating function, 181,184, 189,190, 192,195,196,200,202,215 Generating vector, 201 Geometrical optics, 308,361, 374-377, 379 Geophysics, 110 Graetz problem, 384 Green's function, 362, 364, 380 double, 364 Group velocity, 179,219,220,267,299 Hamilton-Jacobi equation, 181, 182, 183, 186,190,199 Hamiltonian, 223,224,225 definition of, 180 for Duffi equation, 182 for Mathieu equation, 184 for swingii spring, 186 transformation of, using von Zeipel procedure, 189-200 using Lie transforms, 202, 212-216 Harmonic balance, 218 Harmonic resonance, 219-221, 234, 262-269 Harmonic wave, 76 Heat, 45,79,84,156,342,384 problem for, equation, 150-152 Helmholtz equation, 234 Hill's equation, 60
420 SUBJECTINDEX Hopf equation, 381 Hydraulic jump, 89 Hyperbolic equation, 37,42,57,99,379. See also Dispersive waves; Elastic waves; and Supersonic airfoil theory Inclined plane, see Flow, down an inclined Plane Indicial equation, 311 Induction, 13 Infiite domain, as a source, of nonuniformity, 23-31,229 of uniformity, 38,42 Inhomogeneous, 361 problems with turning points, 352-359 Initial, boundary value problem for heat equation, 150-152 layer, 23 Inner and our expansions, see Matched asymptotic expansions, Inner expansion, 110,112,114,119,145, 146,148 for bending of plates, 129-132 defition, 117-118 for earth-moon-spaceship problem, 139 for equation with variable coefficients, 122-124 for simple example, 117-118 for slider beating, 127 for thermoelastic waves, 134-135 for turning point problems, 336 Inner limit, 112,118,119,129,131,139 Inner region, 113,122,146,153 Inner solution, 112,113 Inner variables, 119,120,121,144,149, 153,154 choice of, 114-116, 122-124, 126-127, 134-135,137-138 generalized, 145 Instability, 78,269. See also Model for nonlinear wbility; Stability Integral, 10.11,12 differential equation, 18 equation, 370,380 of motion, 24,199 relations, 99 Intermediate limit, 119 Intermediate matching, see Overlapping Jacobian matrix, 202 Jacobi elliptic functions, 189 Jerky oscillations, 34 Jordan, form, 327 matrix, 327 Josephson tunnel, 221 Kamel's algorithm. 17 1 Kamel's method, 224 Kelvin-Helmholtz, 77,235 Kernel, 370 Kleincordon equation, 175,234 exercises involving, 105,226 treated, by averaging using Lagrmgh, 221-222 by Krylov-Bogoliubov-Mitropohki method, 178-179 by multiple scales, 301-302 by strained parameters, 76-77 Krylov-Bogoliubov technique, 165-168,223 Krylov-Bgoliubov-Mitropolski technique, 174-179,183,194,211,212,223, 224,246,248,303 Kruskal's technique, 168,191 Lagrange equations, 179,180 Lagrangian, 179 averaging using, 216-222,301 for Bretherton's equations, 217 for KleinGordon equation, 221 for swinging spring, 186 Lamd coefficients of elasticity, 45,90 Laminar, 39,54 Landau, equation, 382 symbols, 8,9 Langer's transformation, 308,346,384 for first-order turning point problems, 339-341 for generalization of, 341-342 successive, 350 Latta's technique, 144-154 Layer, 23,79,111; see also Boundary layer Libration points, see Stability of elliptic triangular points Lie series and transforms, 171,199,200-216,223,225,303 Lie triangle, 205,206 Lighthill's technique, 57,77-95 exercises involving, 107,108 limitations of, 79,98-100,107,109
SUBJECT INDEX 421 Limit, 7 cycle, point, or solution, 35,99, 109 distinguished, 336 Oseen s, 140 Stokes, 140 see also Inner limit; Intermediate limit; and Outer limit Limitations of, method, of composite expansions, 153-154 of matched asymptotic expansions, 144-, 145,155,156,303,339 of multiple scales, 269-270, 275-276, 302-303 of strained coordinates, 79,98-103, 107,109,110,302,303 Struble s method, 174 von Zeipel s procedure, 199 Lindstedt-Poincard technique, 56,57,58-60,95,96,185,200 Linear damped oscillator, treated by multiple scales, 228-243 Linearization,, 57,94-95 LiouviUe, equation, 236, 380, 382 problem, 314 LiouviUeGreen approximations, 49, 320, 335 higher, 315-317 successive, 324-325 LiouviUeGreen transformation, 308, 315, 340 Logarithms, 7,12,45,83,137,144,308, 311 Lommel functions, 353 Long period part, 169,191,209,214 Lunar motion, 60; see also Earth-moon- spaceship problem Mach number, 26,83,85 Matched asymptotic expansions,, 37,48,51,78,110-144,148,153, 154,235,317,346,378,384 applied to turning point problems, 336-339 limitations of, 144,155,156,303 Matching, 51,110,114,115,116 asymptotic, principle, see van Dyke s principle as guide to forms of expansions, 141 intermediate, 119 Kaplun s, principle, 119 Randtl, procedure, 112,118 refined, 118-119 Matching of inner and outer expansions, bending of plates, 130-132 caustic problem, 378 earth-moon-spaceship problem, 139 equation with variable coefficients, 124 now past a sphere, 141-144 simple example, 120-12 1 slider bearing, 127 term by term, 130 thermoelastic waves, 135-136 turning point problems, 337-339,343-344 Mathien equation, 339 exercises involving, 53,104, 223, 303, 304 treated, by averaging using canonical variables, 183-185 by Lindstedt-Poincard technique, 6062 by multiple scales, 253-257 by von Zeipei s procedure, 194-200 by Whittaker s method, 62-64 Maxwell s heat conduction law, 45 Membrane, 356 Method, of strained coordinates, see Strained coordinates, of strained parameters, see Strained parameters, of multiple scales, see Multiple scales, Missile dynamics, 233 exercises involving, 304, 305, 306 linear, 320-321 nonlinear, 291-295 Model for nonlinear instability, 50 exercises involving, 55 illustrating limitations of strained coordinates, 99-100 straightforward expansion for, 25-26 treated, by multiple scales, 264-266 by renormalization, 96-97 Momenta, 179,182,190,199,224 Moon, see Earth-moon-spaceship problem; Lunar motion Multiple scales,, 51,97, 100, 153,221,228-307,315, 317,339, 368,384 applied, to caustic, 379
422 SUBJECT INDEX to equations with slowly varying coefficients, 318-320 to Orr-Sommerfeld equation, 360 Navier-Stokes equations, 28,33, 39 Neumann, expansion, 361; see also Born expansion series, 364 Newtonian theory, 79 Nondispersive waves, 78,269-270,303. See also Dispersive waves; Elastic; Shock waves; and Supersonic airfoil theory Nonuniformity, in airfoil theory, 28,50 in asymptotic expansions, 16-18 bibending of plates, 37 in Born s expansion, 367 dependence of, on coordinates, 49-51 on size of domain, 24-31,38,42 in Duffii s equation, 24-25,49-50 in earth-moon-spaceship problem, 45 in equation with constant coefficients, 32 in flow past a sphere, 31 in geometrical optics approximation, 376-377 in interior of domain, 137 in jet instability, 99 in linear oscillator, 229 in model for nonlinear instability, 26,50 in relaxation oscillations, 35 in slider bearing, 126 in shift in singularity, 43,79 in thermoelastic waves, 4748 in turning point problems, 49,284,315, 336 variable exhibiting, 57,285 see a h Boundary layer; Region of nonuniformity; Sources of nonuniformity; and Type change of Node, 252 Normal solution, 58,60,66,311,326,331, 332 No-dip condition, 33,34 Olver s transformation, 341,384,385 Operator, 206 adjoint, 164 Faa de Bruno, 192 intensity, 372 mass, 371,381 self-adjoint. 163,164 Optics, see Geometrical optics Optimal, control, 79 coordinate, 50,51 Orbit, 64.99 Orbital mechanics, 233 Order symbols, 8,9 Orr-Sommerfeld equation, 233,360 Oscillations, 25,34,89,226,232,303 Oseen s equation, 141 Oseen s expansion, 140-144 Oseen s limit process, 140,141 Oseen s variable, 143 Outerexpansion, 110,112.114,119,146 for bending of plates, 128-129 deftnition of, 117 for earth-moon spaceship problem, 45, 138 for equation with variable coefficients, 124 for simple example, 117 for slider bearing, 126 for thermoelastic waves, 4748 Outer limit, 112,117,119,128,138,144 Outer region, 113,122 Outer solution, 111,113,115 Outer variable, 119, 121,144,145 Overlapping, 113,119 Parabolic coordinates, 79 Parabolic cylinder function, 344,378 Parabolic equation, 37,38,99,379 Paradox, 31 Parameter, 89 large, 315; see also Turning point problems perturbation, 14,16,308 see ah Small parameter multiplying highest derivative; Strained parameters, Parametric resonance, see Mathieu equation; Stability of elliptic ttiangular points Parameters, 201; see also Variation of parameters Parametrization, 88,92 Partitioning, see Asymptotic partitioning Pascal triangle, 205 Pendulum, 103,224; see Swinging spring Period, 169,191,192 Periodic, 58,60 motion, 189
SUBJECT INDEX 423 orbit, 99 solutions, 34, 77 see Mathieu equation; Stability of elliptic triangular points Perturbations, coordinate, 4-7, 308-314 parameter, 14, 16, 308 Phase, 178, 380 rapidly rotating, 168, 201, 234 speed, 76,71 Plasma, 58, 78,216,217,226,234, 235, 367 Plasticity, 90 Plates, see Unsymmetrical bending of plates Poincard-Lighthill-Kuo method, see Lighthill s technique Poisson ratio, 36,47 Potential function, 26,83,89 Prandtl s boundary layer, 113 Prandtl s technique, 111, 114 Pritulo s technique, see Renormalization, Quantum, 58, 371; see also Schrodinger equation Random, 361,362,363,365,367, 369, 372,380,381,382 Rankine-Hugonoit relation, 84 Ratio test, 6, 7, 10, 11, 16 Rayleigh-Schrodinger method, 56, 71 applied to eigenvalue problem, 68-71 Rayleigh-Taylor instability, 77, 235 Rayleigh wave speed, 47, 134 Reaction kinetics, 79 Reentry dynamics, see Missile dynamics Regionof nonuniformity, 17,18, 19,23, 79 for earth-moon-spaceship problem, 137-138 for equation with constant coefficients, 116 for flow past a sphere, 140 near a caustic, 377-378 for thermoelastic waves, 134-135 for turning point problems, 284, 336 Related equation, 341 Relaxation oscillations, 34-35 Renormalization,, 57,95-98, 308,361, 367-372 exercises involving, 103,106,107,108 Resonance, 89, 233,248 external, 225 internal, 185, 225 in linear systems, 321-324 near, 195 perfect, 189 parametric, 235 passage through, 233 see also Harmonic resonance Reynold s equation, see Slider bearing Reynolds number, 40,360 high, 33,130 small, 28,29,139 Rigidity, flexural, 36 RitzGalerkin procedure, 58 Rossby wave, 77,234 Rytov s method, 361,373 Saddle point, 252 Satellite, 233,276 Scales, 51, 110,230,231, 232,233, 242, 303; see also Multiple scales, method of Scattering, 57,95,235, 361, 364, 367, 368 Schrodinger equation, 56,71,160, 339, 342,344,346 Secular terms, 23-26 elimination of, 6548,261, 288, 302 Self-sustained oscillations, 89; see also van der Pol oscillator Series, see Asymptotic series; Lie series and transforms Shadow boundary, 378 Sharp change, 103,110,303 Shell, 233 Shift, 42,77,79,106 in singularity, 103 exercises involving, 52,53, 106, 107 straightforward expansion for, 42-43 treated, by Lighthill s technique, 79-82 by renormalization, 98 by Temple s technique, 94-95 Shock waves, 52,78,83, 84,85,99, 100, 235 Short period part, 169,191,214 Singular, 17,57,78,81,231 Singular perturbation, 17,99,340 dependence on region size, 38,42 see also Nonuniformity Singular point, definition of, 309,310
424 SUBJECT INDEX expansion near an irregular, 309-312,326-327 regular, 5,308 Singularity, 85,86 as a jump discontinuity, 32,35 branch point, 82 essential, 7 growing, 23,42,43,45,95 logarithmic, 45,83,102,137 tuning point with, 358-359 worst, 81,98 see also Nonuniformity Skin layer, 111 Slider bearing, 54,125-128 Small parameter multiplying highest derivative, 23,31-37,99, 317-318 in limitations of strained coordinates, 99,100-102 Smoothing,, 308,361,380-382 Solvability condition, 152,288,302; see Secular terms Sonic boom, 78 Sources of nonuniformity, 23-55, 140 Spaceship, see Earth-moon-spaceship prob- lem Speed, 56,57,58,76,77,83,90,91 Sphere, see Flow past a sphere Spherical, 28,78,224 Spring, 232. See also Duffii equation; swinging spring Stability, 60,77,78,99,100,162-164,235, 252,353,360. See also Mathieu equation; Model for nonlinear stability; Stability of elliptic triangular points Stability of elliptic triangular points, treated, by multiple scales, 259-262,275 by strained parameters, 64-66 by Whittaker s technique, 6668 Statistical mechanics, 236 Stochastic, 361,367 Stokes expansion, 140 Stokes limit process, 140 Stokes variable, 143 Stokes solution, 30 Strained coordinates,, 51.52, 56-109,110.114.156,235,302,. -. Strained parameters,, 56,58-77, 78,79,97,99,100,103,105,106 Strainitg, of characteristics, 87,89 dependent variable, 99,108 function, 57,78,79,81,87,99,101,102, 103,107 Stratification, 216 Stream function, 28,29,33,40,57,113, 140 Stretching transformation, for bending of plates, 128 for caustic, 377-378 for dependent and independent variables, 134 for earth-moon-spaceship problem, 137-138 for equation with variable coefficients, 122,123 for heat equation, 151 for turning point problems, 284,336 Struble s method, 171-174,176,183,194, 223 Stuart-Watson-Eckhaus technique, 162-164 Subnormal solution, 311,331-332 Supersonic airfoil theory, 50,78,93 straightforward expansion for, 26-28 treated, by Lighthill s technique, 86-89 by renormalization, 97-98 swinging spring, exercises involving, 105, 225 treated, by averaging Hamiltonian, 185-189 by Lie series and transforms, 214-216 by multiple scales, 262-264 Temple s technique, 57,94-95 Thermoelastic waves, straightforward expansion for, 45-48 treated by matched asymptotic expansions, 133-137 Thomas-Fermi model, 235 Three body problem, 199,233,276. See also Earth-moon-spaceship problem; Stability of elliptic triangular points Triangle, 205,206 Triangular points, 233; see also Stability of elliptic triangular points Transfer of boundary condition, 27 Transform, see Lie series and transforms 303 Transformation, canonical, 181,190,
SUBJECT INDEX 425 191,199 contracting, 140 Deprit, 202 Hori, 202 Lie, 201 LiouvfileGreen, 49,315,340 near identity, 51,57,168,190,201 stretching, 111,113,114,116,117,377 von Zeipel, 171,191,199,202 Transition, 380; see Turning point problems Transition curves, exercises involving, 104, 105,225,303,304 for libration points, 64-68, 259-262.275 for Mathieu s equation, 60-64, 183-185, 194-200,253-257 Transport equation, 374, 379 Tunneling effect, 342 Turning point problems, 114,308,309, 335-360 definition of, 49,122,284,335 exercises involving, 53, 305, 383, 384, 385 near caustic, 377-380 treated by multiple scales, 232, 233,284-286 Two-body problem, 201 Two-variable expansion procedure, 243, 302 applications of, 270-275 description of, 231, 240-241 limitations of, 275-276 see also Multiple scales, Type change of, 23,3742 Uniformity, see Nonuniformity Uniformization, 94, 107, 232 Unsymmetrical bending of plates, exercises involving, 155 straightforward expansion for, 35-37 treated by multiple scales, 128-133 van der Pol,, 164-165,166 van der Pol oscillator, exercises involving, 52,54,104,223,224,303,304 straightforward expansion for, 34, 34-35 treated by generalized averaging, 169-171 by Krylov-Bogoliubov technique, 167-168 by Krylov-Bogoliubov-Mitro~l~i technique, 176-177 by Lie series and transforms, 209-212 by multiple scales, 245-253, 272-275 van der Pol oscillator with delayed amplitude liiting, 104,257-259,304 van Dyke s matching principle, 114, 119 mechanics of, 120-122 Variables, see Canonical variables; Inner variables; and Outer variables Variation of parameters, 52,159-222 Variational, approach, 216-222, 227 equations, 172,183, 184, 190 Vibrations, 114,232. See also Oscillations; Waves Vlasov s equation, 78, 216 von Zeipel s procedure, 189-200, 202,231 Water waves, 56,57,58,77,216,217,234 Wave equations, 360-382 Wave number, 58,77,99,360 Waves, 58,60,77, 78,89,90,94, 216, 217, 233,234,235. See also Bretherton s equation; Dispersive waves; Elastic waves; Flow down an inclined plane; KleinGordon equation; Non dispersive waves; F lasma; Shock waves; Supersonic airfoil theory; and Thermoelastic waves Weber function, 345, 378,380 Whitham s method, 216-222 Whittaker s functions, 346 Whittaker s method, 6244,6668, 104, 105,185,200 WKBJ approximation, 49,308, 315, 320, 335,339,382 successive, 324-325 Wronskian, 160 Young s modulus, 36