**K. D. Cole**

In this section, steady-periodic heat conduction is treated. Also called
time-harmonic or thermal-wave behavior, this special case is important
whenever the causal effect is harmonic in time and has continued long
enough for any start-up transients to die out.

**Rectangular Coordinates. Steady-periodic 1-D.**

Consider a one-dimensional region in which the temperature is sought.
The transient temperature distribution satisfies

(1) | |||

Here is the thermal diffusivity (m s), is the thermal conductivity (WmK), is the volume heating (W m), and is a specified boundary condition. Index represents the boundaries at the limiting values of coordinate . The boundary condition may be one of three types at each boundary: boundary type 1 is specified temperature ( and ); boundary type 2 is specified heat flux (); and, boundary type 3 is specified convection where is a constant-with-time heat transfer coefficient (or contact conductance).

Since in this section the applications of interest involve steady-periodic
heating, the solution is sought in Fourier-transform space,
and the solution is interpreted as the steady-periodic response at
a single frequency .
For further discussion of this point see Mandelis (2001, page 2-3).
Consider the Fourier transform of the above temperature equations:

(2) | |||

Here is the steady-periodic temperature, is the steady-periodic volume heating, is the steady-periodic specified boundary condition, and .

The temperature will be found with the Fourier-space
Green's function, defined by the following equations:

(3) | |||

(4) |

Here and is the Dirac delta function. The coefficient preceding the delta function in Eq. (3) provides the 1-D frequency-domain Green's function with units of sm. This is consistent with earlier work with time-domain Green's functions.

If the steady-periodic Green's function is known (given below), then the
steady-periodic temperature is given by the following integral
equation:

(5) |

For a derivation of this equation see Beck et al. (1992, pp. 40-43). Next the steady-periodic Green's functions are given for 1-D bodies for cases XIJ.

- Infinite Body, 1-D, Steady-Periodic GF
- Semi-infinite body, 1-D, Steady-Periodic GF
- Plate, 1-D, Steady-Periodic GF

2004-08-10