Contributions to a Theory of Separability of the Vector Wave Equation of Elasticity for Inhomogeneous Media

1962 ◽  
Vol 34 (7) ◽  
pp. 946-953 ◽  
Author(s):  
Joseph F. Hook
Keyword(s):  
Wave Equation ◽  
Inhomogeneous Media ◽  
Vector Wave ◽  
Vector Wave Equation

Determination of inhomogeneous media for which the vector wave equation of elasticity is separable

1965 ◽  
Vol 55 (6) ◽  
pp. 975-987
Author(s):  
Joseph F. Hook
Keyword(s):  
Wave Equation ◽  
Nonlinear Differential Equations ◽  
Inhomogeneous Media ◽  
Arbitrary Parameter ◽  
Vector Wave ◽  
Vector Wave Equation ◽  
Helmholtz Potential ◽  
Cartesian Coordinate ◽  
Constitutive Parameters

Abstract A generalization of the Helmholtz potential representation has recently been employed to effect separation of the vector wave equation of elasticity for certain types of inhomogeneous media whose properties vary with a single cartesian coordinate, z. For separability, the three constitutive parameters of the medium must satisfy a pair of simultaneous nonlinear differential equations. In this paper, integrals of these equations are derived which are explicit expressions for two of the parameters in terms of the third, whose functional form may be chosen arbitrarily. In the most general case, the arbitrary parameter must be the rigidity, μ For a special subcase, any one of the three parameters may take this role.

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