D,6 • APPLICATIONS 



of this strength located at a: = 2d, i.e. a distance x = {2 — ^)d from the 

 plane at a: = ^d. It can be shown that, with two such sources located sym- 

 metrically about X = d, the conduction of heat in the region ^ x ^ dis 

 governed by the same differential equation and initial and boundary con- 

 ditions as in the slab during a time t* < t*^, when the temperature rise 



1.0 





0.5 



CD 



1.0 



2.0 



^=hV 



yj^ 



kpc 



Fig. D,5. Temperature transients in semi-infinite solid (cf. Eq. 5-1). 



at rc = is not yet influenced by the image source at a; = 2d. The tem- 

 perature in the slab given by Eq. 4-7 is therefore accurately represented 

 at short times r* < rf^ by the superposed effect of the two sources: 



QiX. r) = 0(iV.=J,, T*) + &(A^.=(2-i)., T*) 



(5-3) 



-ViVi 



where the conversion from r* to t for the slab is r =- t / ly^^d- 



In practice, Eq. 5-3 is accurate within 1 per cent of 6 up to times 

 when the Fourier representation (Eq. 4-7) requires only one term for 1 per 

 cent accuracy. Computations of d{Nx, r*) in Eq. 5-1 and 5-3 are facilitated 

 by tables of e'^ erfc z and appropriate asymptotic expansions given in [6], 



D,6. Applications. The temperature distribution in the slab given 

 by Eq. 4-7 may be regarded as a superposition of damped spatial modes, 

 cos iiin(l — ^), with decreasing amplitudes A„ and increasing exponential 



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