The critical thickness of high temperature barrier coating is derived to avoid cycling of temperature from the finite speed heat conduction equations. When a cylinder is subject to a step change in temperature at the surface of the cylinder the transient temperature profile is obtained by the method of separation of variables. The finite speed of heat propagation is accounted for by using the modified Fourier’s law of conduction with a heat velocity of √α/τr. In order to avoid pulsations of temperature with respect to time the cylinder has to be maintained at a radius no less than 4.8096√ατr. In the asymptotic limit of infinite heat velocity the governing equation becomes parabolic diffusion equation. In the limit of zero velocity of heat and infinite relaxation time the wave equation result and solution can be obtained by a relativistic coordinate transformation. In the asymptote of zero velocity of heat and zero thermal diffusivity the solution for the dimensionless temperature is a decaying exponential in time. The average temperature of the naval warhead as indicated by UL 1709 test was estimated by using a idealized finite slab, and Leibnitz rule and an analytical expression for the average temperature was obtained using convective boundary condition. The solution is:
$For1/2>=Bi,=exp(−τ(1/2+sqrt(1/4−Bi*)))$

$ForBi>1/2,=exp(−τ/2)Cos(τsqrt(−1/4+Bi*)))$
The average temperature is damped oscillatory in time domain. Further the transient temperature profile is represented by an infinite series of decaying exponential in time and Bessel function of the first kind and 0th order. The constant can be obtained from the principle of orthogonality. The bifurcated nature of the exact solution gives rise to the lower limit on the radius to avoid cycling of temperature with respect to time. The exact solution is thus,
$u=Σ0∝cnJ0(λnX)exp(−τ(1/2−sqrt(1/4−λn2)))$
and when λn > 1/2
$u=Σ0∝cnJ0(λnX)exp(−τ/2Cos(τsqrt(−1/4+λn2))$
where,
$λn=(2.4048+(n−1)π)(√α/τr/R)$
cn is given by equation (53). The term in the infinite series onward where the contribution is oscillatory is identified.
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