Information such as energy and momentum is communicated through space and time via waves. Their study in elastic solids constitutes the subject of elastodynamics. This course presents the formulation and solution of elastodynamic problems in one, two and three dimensions. The notion of waveguides — structures that guide waves — is introduced through several examples, specially plates. Waves in anisotropic elastic media and crystals are also discussed. Experimental characterization is demonstrated.


Introductory graduate courses on (a) theory of elasticity, (b) applied mathematics covering ordinary- and partial- differential equations. Exposure to complex analysis is recommended.

Course contents:

Waves in 1-d; Method of characteristics; Three-dimensional waves; Plane and harmonic waves; Reflection and transmission; Half-space problems; Waveguides: Dispersion, 1-d, Rods, Plates; Anisotropic media; Crystals; Experimental characterization; Advanced topics.

Topics with suggested number of lectures in parenthesis

I. Review of elasticity: Navier’s equation of motion, Boundary and initial conditions. (1)

II. Longitudinal and torsional waves in 1-D. D’Alembert’s solution. (1)

III. Method of characteristics; Radiation conditions; Wave packets; Group velocity. (3)

IV. Three-dimensional waves: Helmholtz decomposition, Dilatational and shear waves. (2)

V. Plane waves. Harmonic waves. Slowness diagrams. (3)

VI. Reflection and transmission of P, SV, SH waves across interface; continuity conditions; Snell’s law; Reflection and refraction at interfaces. (5)

VII. Half-space problems: Rayleigh waves; Suddenly applied uniform normal pressure with zero body force; Cagniard de Hoop method; Buried load problem; Scattering from crack tips in mode III. (8)

VIII. Waveguides: 1-d waves; Dispersion; String on elastic foundation; Cut-off frequency; 2-d waves; Thin plates (Kirchhoff’s theory); Lamb waves; Love waves; Rods; Pochammer-Chree equation. (10)

X. Waves in anisotropic media and crystals. (3)

XI. Experimental characterization: Kolsky bar. (2)

XII. Advanced topics. One from: Plastic waves/Layered media/Visco-elastic waves/Shock waves/ Nonlinear waves/Thermal waves/Waves in discrete media/Scattering from mode I and II cracks. (4)

Textbooks and references:

  1. Wave Propagation in Solids, J.D. Achenbach, Elsevier Science Publishers,1975.

  2. Wave Motion in Elastic Solids, K. F. Graff, Dover Publications, 1991.

  3. Mechanics of Continua and Wave Dynamics, L. Brekhovskikh and V. Goncharov, Springer-Verlag, 1985.

  4. The Theory of Elastic Waves and Waveguides, J. Miklowitz, North-Holland Publishing Company, 1978.

Prepared by :

Ishan Sharma
N N Kishore
B L Sharma
S S Gupta
P M Dixit