# Thermal-Wave Fields in One Dimension

• Andreas Mandelis
Chapter

## Abstract

Much of the Green function formalism of Chapter 1 will be used in this chapter to address several “case studies” in the form of detailed solutions to one-dimensional thermal-wave problems most frequently encountered in applications. The Green-function approach results in the use of very few constitutive formulas, which are, nevertheless, capable of covering a large spectrum of boundary-value problems. A thermal-wave field generated by a spatially arbitrary, harmonic source function Q(r,w) is the solution to the inhomogeneous equation (1.9):
$${\nabla ^2}T(r,\omega ) - {\sigma ^2}(\omega )T(r,\omega ) = - \frac{1}{k}Q(r,\omega )$$
(2.1)
where σ is the thermal wavenumber, defined by Eq. (1.5), and k is the thermal conductivity of the medium. In the Introduction chapter, it was shown that a straightforward algebraic combination of Eq. (2.1) and the equation for the Green function, Eq. (1.24′), yields the general solution for the thermal-wave field, Eq. (1.30):
$$T(r,\omega ) = \left( {\frac{\alpha }{k}} \right)G(r\left| {{r_0}} \right.;\omega )d{V_0} + \alpha \oint_{{S_0}} {\left[ {G(r\left| {r_o^s;\omega ){\nabla _0}({r^s},\omega ) - T({r^2},\omega )} \right.{\nabla _0}G(r\left| {r_o^s;\omega } \right.} \right]} \bullet d{S_0}$$
(2.2)
Here, S 0 is the surface surrounding the domain volume V 0, which includes the harmonic source.$$Q({r_0},\omega ).r_o^s$$ is a coordinate point on S 0.

## Keywords

Heat Transfer Coefficient Green Function Optical Absorption Coefficient Thermal Effusivity Dimensionless Thickness

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