Wave Motion in Elastic Solids

by Graff, Karl F.

• ISBN13: 9780486667454
• ISBN10: 0486667456
• Format: Paperback
• Copyright: 1991-06-01
• Publisher: Dover Publications

Summary

Comprehensive textbook for students and research workers offers self-contained coverage of a variety of topics ranging from the elementary theory of waves and vibrations in strings to the three-dimensional theory of waves in thick plates. Emphasis is on analytical and experimental results, in addition to theoretical development. Appendices contain introductory material on elasticity, transforms and experimental techniques. Over 100 problems.

Table of Contents

Introduction	1	(1)
General aspects of wave propagation	1	(2)
Applications of wave phenomena	3	(2)
Historical background	5	(4)
Waves and Vibrations in Strings	9	(66)
Waves in long strings	9	(20)
The governing equations	9	(1)
Harmonic waves	10	(3)
The D'Alembert solution	13	(1)
The initial-value problem	14	(3)
The initial-value problem by Fourier analysis	17	(2)
Energy in a string	19	(2)
Forced motion of a semi-infinite string	21	(2)
Forced motion of an infinite string	23	(6)
Reflection and transmission at boundaries	29	(7)
Types of boundaries	29	(1)
Reflection from a fixed boundary	30	(1)
Reflection from an elastic boundary	31	(3)
Reflection of harmonic waves	34	(1)
Reflection and transmission at discontinuities	35	(1)
Free vibration of a finite string	36	(6)
Waves in a finite string	36	(1)
Vibrations of a fixed-fixed string	37	(3)
The general normal mode solution	40	(2)
Forced vibrations of a string	42	(9)
Solution by Green's function	43	(1)
Solution by transform techniques	44	(5)
Solution by normal modes	49	(2)
The string on an elastic base-dispersion	51	(7)
The governing equation	51	(1)
Propagation of harmonic waves	51	(3)
Frequency spectrum and the dispersion curve	54	(2)
Harmonic and pulse excitation of a semi-infinite string	56	(2)
Pulses in a dispersive media---group velocity	58	(10)
The concept of group velocity	59	(3)
Propagation of narrow-band pulses	62	(3)
Wide-band pulses---The method of stationary phase	65	(3)
The string on a viscous subgrade	68	(3)
The governing equation	68	(1)
Harmonic wave propagation	69	(1)
Forced motion of a string	69	(2)
References	71	(1)
Problems	72	(3)
Longitudinal Waves in Thin Rods	75	(65)
Waves in long rods	76	(4)
The governing equation	76	(1)
Basic propagation characteristics	77	(3)
Reflection and transmission at boundaries	80	(4)
Reflection from free and fixed ends	80	(2)
Reflection from other end conditions	82	(1)
Transmission into another rod	83	(1)
Waves and vibrations in a finite rod	84	(11)
Waves in a finite rod---history of a stress pulse	84	(2)
Free vibrations of a finite rod	86	(4)
Forced vibrations of rods	90	(1)
Impulse loading of a rod---two approaches	91	(4)
Longitudinal impact	95	(13)
Longitudinal collinear impact of two rods	95	(5)
Rigid-mass impact against a rod	100	(5)
Impact of an elastic sphere against a rod	105	(3)
Dispersive effects in rods	108	(17)
Rods of variable cross section---impedance	108	(6)
Rods of variable section---horn resonance	114	(2)
Effects of lateral inertia---dispersion	116	(5)
Effects of lateral inertia---pulse propagation	121	(4)
Torsional vibrations	125	(3)
The governing equation	126	(2)
Experimental studies in longitudinal waves	128	(6)
Longitudinal impact of spheres on rods	128	(1)
Longitudinal waves across discontinuities	128	(2)
The split Hopkinson pressure bar	130	(3)
Lateral inertia effects	133	(1)
Some other results of longitudinal wave experiments	133	(1)
References	134	(2)
Problems	136	(4)
Flexural Waves in Thin Rods	140	(73)
Propagation and reflection characteristics	140	(15)
The governing equation	140	(3)
Propagation of harmonic waves	143	(1)
The initial-value problem	144	(4)
Forced motion of a beam	148	(5)
Reflection of harmonic waves	153	(2)
Free and forced vibrations of finite beams	155	(15)
Natural frequencies of finite beams	155	(4)
Orthogonality	159	(1)
The initial-value problem	160	(1)
Forced vibrations of beams---methods of analysis	160	(7)
Some problems in forced vibrations of beams	167	(3)
Foundation and prestress effects	170	(10)
The governing equation	171	(2)
The beam on an elastic foundation	173	(1)
The moving load on an elastically supported beam	174	(1)
The effects of prestress	175	(2)
Impulse loading of a finite, prestressed, visco-elastically supported beam	177	(3)
Effects of shear and rotary inertia	180	(15)
The governing equations	181	(3)
Harmonic waves	184	(3)
Pulse propagation in a Timoshenko beam	187	(8)
Wave propagation in rings	195	(7)
The governing equations	195	(4)
Wave propagation	199	(3)
Experimental studies on beams	202	(7)
Propagation of transients in straight beams	202	(4)
Beam vibration experiments	206	(2)
Waves in curved rings	208	(1)
References	209	(2)
Problems	211	(2)
Waves in Membranes, Thin Plates, and Shells	213	(60)
Transverse motion in membranes	213	(16)
The governing equation	214	(1)
Plane waves	215	(2)
The initial-value problem	217	(3)
Forced vibration of a membrane	220	(2)
Reflection of waves from membrane boundaries	222	(2)
Waves in a membrane strip	224	(1)
Vibrations of finite membranes	225	(4)
Flexural waves in thin plates	229	(29)
The governing equations	229	(5)
Boundary conditions for a plate	234	(1)
Plane waves in an infinite plate	235	(1)
An initial-value problem	236	(3)
Forced motion of an infinite plate	239	(6)
Reflection of plane waves from boundaries	245	(1)
Free vibrations of finite plates	246	(9)
Experimental results on waves in plates	255	(3)
Waves in thin cylindrical shells	258	(12)
Governing equations for a cylindrical membrane shell	259	(4)
Wave propagation in the shell	263	(3)
Longitudinal impact of a membrane shell	266	(4)
References	270	(1)
Problems	271	(2)
Waves in Infinite Media	273	(38)
Wave types	273	(8)
The governing equations	274	(2)
Dilatational and distortional waves	276	(2)
Plane waves	278	(3)
Waves generated by body forces	281	(13)
Certain classical solutions	282	(1)
The simple SH wave source	283	(5)
More general body forces	288	(6)
Cavity source problems	294	(15)
Harmonic dilatational waves from a spherical cavity	294	(3)
Dilatational waves from a step pulse	297	(3)
General case of dilatational waves with spherical symmetry	300	(3)
Harmonic waves from a cylindrical cavity	303	(3)
Transient dilatational waves from a cylindrical cavity	306	(3)
References	309	(1)
Problems	310	(1)
Waves in Semi-Infinite Media	311	(83)
Propagation and reflection of plane waves in a half-space	311	(18)
Governing equations	312	(5)
Waves at oblique incidence	317	(5)
Waves at grazing incidence	322	(1)
Surface waves	323	(5)
Wave reflection under mixed boundary conditions	328	(1)
SH wave source---method of steepest descent	329	(14)
The SH wave source---formal solution	330	(2)
Direct evaluation of the integral	332	(4)
Method of steepest descent	336	(5)
SH wave source solution by steepest descent	341	(2)
Surface source problems	343	(32)
Waves from a harmonic, normal line force	344	(9)
Other results for harmonic sources	353	(3)
Transient normal loading on a half-space	356	(12)
Results for transient loads on a half-space	368	(7)
Waves in layered media	375	(7)
Two semi-infinite media in contact---plane waves	377	(3)
Waves in layered media---Love waves	380	(2)
Experimental studies on waves in semi-infinite media	382	(7)
Waves into a half-space from a surface source	382	(2)
Surface waves on a half-space	384	(2)
Other studies on surface waves	386	(3)
References	389	(2)
Problems	391	(3)
Scattering and Diffraction of Elastic Waves	394	(37)
Scattering of waves by cavities	395	(14)
Scattering of SH waves by a cylindrical cavity	395	(5)
Scattering of compressional waves by a spherical obstacle	400	(9)
Diffraction of plane waves	409	(19)
Discussion of the Green's function approach	409	(1)
The Sommerfeld diffraction problem	410	(17)
Geometric acoustics	427	(1)
References	428	(2)
Problems	430	(1)
Wave Propagation in Plates and Rods	431	(151)
Continuous waves in a plate	431	(33)
SH waves in a plate	432	(3)
Waves in a plate with mixed boundary conditions	435	(5)
The Rayleigh-Lamb frequency equation for the plate	440	(2)
The general frequency equation for a plate	442	(4)
Analysis of the Rayleigh-Lamb equation	446	(12)
Circular crested waves in a plate	458	(2)
Bounded plates---SH and Lame modes	460	(4)
Waves in circular rods and cylindrical shells	464	(16)
The frequency equation for the solid rod	464	(4)
Torsional, longitudinal, and flexural modes in a rod	468	(7)
Waves in cylindrical shells	475	(5)
Approximate theories for waves in plates, rods, and shells	480	(52)
An approximate theory for plate flexural modes	482	(12)
An approximate theory for extensional waves in plates	494	(16)
Approximate theories for longitudinal waves in rods	510	(11)
Approximate theories for waves in shells	521	(11)
Forced motion of plates and rods	532	(29)
SH waves in a plate	533	(5)
Pulse propagation in an infinite rod	538	(5)
Transient compressional waves in semi-infinite rods and plates	543	(18)
Experimental studies on waves in rods and plates	561	(14)
Multiple reflections within a waveguide	561	(2)
Dispersion of a sharp pulse in a cylindrical rod	563	(2)
Experimental results for step pulses	565	(6)
Other studies of waves in cylindrical rods and shells	571	(4)
References	575	(4)
Problems	579	(3)
APPENDIX A. THE ELASTICITY EQUATIONS	582	(21)
A.1. Notation	582	(1)
A.2. Strain	583	(2)
A.3. Stress	585	(1)
A.4. Conservation equations	585	(3)
A.4.1. Conservation of mass	586	(1)
A.4.2. Conservation of momentum	587	(1)
A.4.3. Conservation of moment of momentum	587	(1)
A.4.4. Conservation of energy	588	(2)
A.5. Constitutive equations	590	(2)
A.5.1. Green's method	590	(1)
A.5.2. Cauchy's method	590	(1)
A.5.3. Isotropic elastic solid	591	(1)
A.6. Solution uniqueness and boundary conditions	592	(2)
A.6.1. Uniqueness	592	(2)
A.6.2. Boundary conditions	594	(1)
A.7. Other continua	594	(2)
A.8. Additional energy considerations	596	(4)
A.9. Elasticity equations in curvilinear coordinates	600	(3)
A.9.1. Cylindrical coordinates	600	(1)
A.9.2. Spherical coordinates	601	(2)
APPENDIX B. INTEGRAL TRANSFORMS	603	(26)
B.1. General	603	(1)
B.2. Laplace transforms	604	(10)
B.2.1. Definition	604	(1)
B.2.2. Transforms of derivatives	605	(1)
B.2.3. The inverse transform	605	(1)
B.2.4. Partial fractions	606	(1)
B.2.5. Solutions of ordinary differential equations	607	(1)
B.2.6. Convolution	608	(2)
B.2.7. The inversion integral	610	(4)
B.3. Fourier transforms	614	(6)
B.3.1. Definition	614	(1)
B.3.2. Transforms of derivatives	614	(2)
B.3.3. The inverse transform	616	(1)
B.3.4. Convolution	617	(1)
B.3.5. Finite Fourier transforms	617	(2)
B.3.6. The Fourier integral	619	(1)
B.4. Hankel transforms	620	(1)
B.4.1. Definitions	620	(1)
B.4.2. Transforms of derivatives and Parseval's theorem	621	(1)
B.5. Tables of transforms	621	(7)
B.6. Fourier spectra of pulses	628	(1)
APPENDIX C. EXPERIMENTAL METHODS IN STRESS WAVES	629	(8)
C.1. Methods for producing stress waves	629	(3)
C.2. Methods for detecting stress waves	632	(5)
References to Appendices A, B, C.	634	(3)
Author Index	637	(4)
Subject Index	641