Cartesian Thermal-Wave Fields in Three and Two Dimensions

Abstract

Thermal-wave fields must be expressed in Cartesian coordinates when the relevant geometries are rectangular, even if the sources used for the creation of these fields possess circular symmetry (such as Gaussian laser beams). As a subset of rectangular geometries, laterally infinite structures can also be treated in the Cartesian coordinate system with relatively simple analytical integral solutions within the Green function formalism developed in Chapter 3. Other equivalent representations using the cylindrical coordinate system are investigated in Chapter 6. The starting point for various boundary-value problems worked out in this chapter is the thermal-wave field equation (2.2):

$$ T(r,\omega ) = \left( {\frac{\alpha }{k}} \right)G(r\left| {{r_0}} \right.;\omega )d{V_0} + \alpha \oint_{{S_0}} {\left[ {G(r\left| {r_o^s;\omega ){\nabla _0}({r^s},\omega ) - T({r^2},\omega )} \right.{\nabla _0}G(r\left| {r_o^s;\omega } \right.} \right]} \bullet d{S_0} $$

((4.1))

where S ₀ is the surface surrounding the domain volume V ₀, which includes the harmonic source Q(r ₀,w). The thermal diffusivity, a, and conductivity, k, are assumed to be independent of the coordinate in V ₀. dS ₀ indicates an infinitesimal vector in the outward direction normal to the boundary surface S ₀: dS ₀ = ñ ₀ dS ₀, with ñ ₀ being the outward unit vector, r ^s₀ + c is a source-coordinate point on S ₀. This expression has been shown to offer a unique solution for the thermal-wave field (see Theorems I.1.a and I.1.b) under any inhomogeneous boundary condition of the Dirichlet, Neumann, or third-kind type.