The damped wave conduction and relaxation equation was derived from the free electron theory. The relaxation time is a third of the collision time between the electron and obstacle in a given material. The time to acceleration from the collapsed state to a steady drift velocity is accounted for a ballistic term in the governing equation for temperature. Six different reasons are given to seek a generalized Fourier's law of heat conduction of which the Fourier's law is a particular case. The hyperbolic PDE is solved for by four different methods for 3 different boundary conditions. The reports in the literature of a temperature overshoot is revisited. By the method of separation of variable and the use of a final time condition for the wave temperature bounded solutions given by infinite Fourier series is derived. The solution is bifurcated. For small slab, a < Pi(alpha*tr)^1/2 the temperature was shown to exhibit subcritical damped oscillations. In the case of the semi-infinite medium reports in the literature about a wave discontinuity was revisted. After dividing out the damping term the hyperbolic PDE was transformed by a substitution, eta = t^2 – X^2 into a Bessel differential equation. The transformation variable is symmetric with respect to space and time. Three regimes are reconginized in the solution – a inertial lagging zero transfer regime, a regime given by Bessel composite function of the zeroth order and first kind and a third regime given by a modified Bessel composite function of the zeroth order and first kind. The manifestation The temperature solution for a finite slab subject to constant wall temperature using the damped wave conduction and relaxation equation presented by earlier investigators showed that there can be under certain conditions a temperature overshoot and a violation of second law of thermodynamics. In this study the second order hyperbolic PDE in two variables was solved for by two methods. In the first method, the damping term was first removed from the equation by a u = exp(-nt) substitution. The resulting equation in wave temperature was solved for by the method of separation of variables. Three of the four conditions in space and time used by the earlier investigators were retained in this study. These were the boundary temperature, by symmetry the derivative of temperature with respect to space becoming zero at x = 0 and the initial temperature of the slab. The fourth time condition was the steady state condition in place of the derivative of the temperature with respect to time in the initial condition. At steady state the wave temperature W = uexp(t/2) becomes a product of zero times infinity. As a result in the time domain solution of the wave temperature the term that grows exponentially with time is neglected leaving behind the term that exponentially decays with time. The resulting solution is well bounded and within the second law of thermodynamics. No temperature overshoot can be seen in this solution. For small slabs, a < Pi(alpha*tr)^1/2 , the temperature exhibits sub critical damped oscillations. This is shown in Figure 2 for the center line temperature of the slab. Compared with some of the analytical solution given in the literature for semi-infinite slab (Baumeister and Hamill, 1969) the solution is for the complete domain from the initial condition to steady state in time. The solution presented in the literature for the semi-infinite slab showed a discontinuity at the wave front. A time taken to steady state can be read off from the graph for small slabs. This is a new feature in transient heat transfer that arises from the solution of the hyperbolic PDE. It appears as though there is another mode of heat transfer in addition to the molecular conduction, convection and radiation. This can be the wave conduction. This can be from the accumulation at the surface effects at a molecular level, or the acceleration regime in the free electron theory. When attempting to empirically fit two phenomena Fourier's law of heat conduction deviates from reality for some circumstances that are important in industrial practice. Thus the circumstances which determine whether the transfer of heat is direct exponential decay or cosinous can be derived theoretically using the damped wave conduction and relaxation equation. The Fourier infinite series solution has been found to be well bounded for the surface flux condition where there was found a “blow-up” in the parabolic solution. A aliter was presented where the direct solution of the hyperbolic PDE was undertaken. A transformation, eta = X^2 - t^2 can lump the ballistic accumulation term and the Fourier term in the damped wave conduction and relaxation equation to two terms in the similarity variable. The resulting PDE in the domain of X > t, becomes elliptic from hyperbolic. This equation can be solved for the method of separation of variables. The solution is a infinite Fourier Bessel series in the composite spatio-temporal variable. When separating the two functions after differentiation one is a function in time only and the other is a function of the spatio-temporal variable, eta. Both will be equal when the expression containing the spatio-temporal variable becomes a function of time only and this can be set equal to f. The solution is seen to obey the time and space conditions. The eigen values were solved for the boundary condition at the surface and from the roots of the Bessel function of the zeroth order and first kind. Both solutions are well bounded without any temperature overshoot and disobeyance of Clausius inequality. The periodic boundary condition is evaluated by the method of complex temperature. The system is overdamped. The damped wave conduction and relaxation equation was derived from the free electron theory. The relaxation time is seen to be a 1/3rd of the collision time between the electron and the obstacle in a given material. Free electron theory has been widely accepted in explaining the observations in the theory of electric conduction. By Lorenz analogy the ratio of the electrical conductivity and thermal conductivity and temperature was found to be universal constant. So even for metals, the relaxation time can be a non-negligible parameter. The recent work of Mitra et al, 1995 and Kaminski, 1990 has shown that for materials with a non-homogeneous inner structure the relaxation time can be measured to 10-15 seconds. The relaxation time can be related to the thermal diffusivity by re-arranging the terms obtained from the free electron theory. The electron density can be converted to a density and the Boltzmann constant to heat capacity term for ideal gases to obtain a expression for thermal diffusivity. With the advent of personal computers and continued interest in transient phenomena the damped wave conduction and relaxation need to be studied further. The analytical solution presented by Baumeister and Hamill, 1969 showed a discontinuity at the wave front. This has been improved upon by a substitution, eta = t^2 – X^2. The solution is in the form of Bessel composite function of the spatio-temporal variable and exhibits three different regimes; a inertial regime of lag and zero transfer, a rising regime described by Bessel composite spatio-temporal function and a third regime of modified Bessel composite function of the spatio-temporal variable and a falling regime in the expression for heat flux. Readily usabe solutions are obtained for the dimensionless heat flux and temperature. Some space time symmetry is seem in the solution. The wavefront solution can be obtained directly from the transformed Bessel differential equation. The manifestation of the relaxation time during a periodic boundary condition is shown in a Figure upon obtaining the solution using the method of complex temperature. It can be seen it is a ovedamped system. A storage coefficient is defined which plays a critical role in the oscillatory regimes and is given by S = rhoCp/tr and has the units of w/m3/K.

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