Maria Bayard Dühring - Solid Mechanics

2.2 Modeling of wave problems 7
in space and time and can be described in a similar way by second order differential
equations. In the case of elastic waves in a solid material, the displacement ui
of a point with the coordinates xk varies with time t, such that ui = ui(xk, t). By
combining Newton’s second law with Hooke’s law for a solid inhomogeneous material
the wave equation is obtained [3]
∂ ∂ul
cijkl − ρ
∂xj ∂xk
∂2ui = 0, (2.1)
∂t2 where cijkl are the stiffness constants and ρ is the mass density, which all depend on
the position xk. The expression in (2.1) is a system of three second order differential
equations for an anisotropic, inhomogeneous material in 3D. In the case of a fluid
the wave equation has the scalar form
∂ ∂u ∂
α1 − α2
∂xi ∂xi
2u = 0, (2.2)
∂t2 where u is the pressure, α1 is the inverse of the mass density and α2 is the inverse of
the bulk modulus. If u describes the out of plane displacements in a solid material,
α1 is the shear modulus and α2 denotes the mass density, equation (2.2) models
the behavior of elastic shear waves. In a similar way, the electromagnetic waves
can be described by a system of three second order differential equations in the
general case, which can be reduced to the scalar equation (2.2) for the special cases
of in-plane modes. In the case of transverse electric modes u describes the electric
field, α1 is the inverse of the dielectric constant and α2 is the product of the vacuum
permittivity and the vacuum permeability.
In many situations it can be assumed that the forces, and thus the waves, are
varying harmonically in time with the angular frequency ω. It is therefore convenient
to eliminate the time-dependency in the equations. In the case of for instance a fluid,
the field variable is described on the complex exponential form u(xk, t) = û(xk)eiωt ,
where û describes the complex amplitude and the phase indirectly. The exponential
represents the time variation. This means that the operator ∂2
∂t2 can be written as
−ω2 and when ûeiωt is introduced in the wave equation (2.3) and the hat is omitted
it reduces to the simpler form
∂ ∂u
α1 + ω
∂xi ∂xi
2 α2u = 0. (2.3)
This equation is known as the Helmholtz equation and is valid for the field oscillating
at the single frequency ω. The real physical value of u is found by taking the real
part of the complex solution to the problem. The conversion from a time dependent
to a time-harmonic problem can be done in an equivalent way for the other wave
problems and only these types of problems are considered in this work.
A solution to a wave problem can be found by applying a set of boundary conditions
that give information about free and clamped surfaces and interfaces between