ABSTRACT
Plane waves emerge as very special solutions of homogeneous wave equations122 for homogeneous materials, that is to say, we look for solutions of the homogeneous equation (7.20)
µ∆v(R, t) + (λ + µ)∇∇ · v(R, t) − ρ ∂ 2v(R, t) ∂t2
= 0 (8.1)
for isotropic nondissipative materials; applying one time integration, we can equally write this equation in terms of the particle displacement (Equation 7.24):
µ∆u(R, t) + (λ + µ)∇∇ · u(R, t) − ρ ∂ 2u(R, t) ∂t2
= 0. (8.2)
“One-dimensional” means that all field quantities should only depend upon one (Cartesian) coordinate. We choose the z-coordinate, that is to say, we postulate independence of x and y putting all derivatives with regard to x and y to zero:
∂
∂x ≡ 0, ∂
∂y ≡ 0. (8.3)
With (2.182), (2.180), and (2.186), respectively, the requirements (8.3) yield as
µ ∂2u(z, t)
∂z2 + (λ + µ)
∂2uz(z, t) ∂z2
ez − ρ ∂u(z, t)
∂t2 = 0. (8.4)
K12611 Chapter: 8 page: 219 date: January 13, 2012
K12611 Chapter: 8 page: 220 date: January 13, 2012
We take the three Cartesian components of this one-dimensional vector wave equation:
µ ∂2ux(z, t)
∂z2 − ρ∂
2ux(z, t) ∂t2
= 0, (8.5)
µ ∂2uy(z, t)
∂z2 − ρ∂
2uy(z, t) ∂t2
= 0, (8.6)
(λ + 2µ) ∂2uz(z, t)
∂z2 − ρ∂
2uz(z, t) ∂t2
= 0. (8.7)
We obtain three mutual independent (decoupled) equations for the respective components of u(R, t) with a similar mathematical structure that can be solved independently. For example, from the outset, we can choose trivial solutions for two equations, e.g., ux(z, t) = uy(z, t) ≡ 0 or ux(z, t) = uz(z, t) ≡ 0 or uy(z, t) = uz(z, t) ≡ 0.
