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Basic Theoretical Physics: A Concise Overview P31. This concise treatment embraces, in four parts, all the main aspects of theoretical physics (I . Mechanics and Basic Relativity, II. Electrodynamics and Aspects of Optics, III. Non-relativistic Quantum Mechanics, IV. Thermodynamics and Statistical Physics). It summarizes the material that every graduate student, physicist working in industry, or physics teacher should master during his or her degree course. It thus serves both as an excellent revision and preparation tool, and as a convenient reference source, covering the whole of theoretical physics. It may also be successfully employed to deepen its readers’ insight and. | 310 40 Phenomenological Thermodynamics Temperature and Heat and postpone the evaluation of the real part to the very end i.e. in effect we omit it . Below the surface z 0 we assume that T z t G. b e k b.-e k with complex wavenumbers kj j 1 2. The heat diffusion equation then leads for a given real frequency w-1 or x2 365 1 to the following formula for calculating the wavenumbers i j Dw kj giving D G j i. With F 1 i Vi v 2 one obtains for the real part kj1 and imaginary part kj2 of the wavenumber kj in each case the equation The real part kj1 gives a phase shift of the temperature rise in the ground relative to the surface as follows whereas for z 0 the maximum daytime temperature average over the year occurs on the 21st June below the ground for z 0 the temperature maximum may occur much later. The imaginary part kj2 determines the temperature variation below ground it is much smaller than at the surface e.g. for the seasonal variation of the daytime average we get instead of T-x_ b1 T z . bi however the average value over the year TTO does not depend on z. The ground therefore only melts at the surface whereas below a certain depth z c it remains frozen throughout the year provided that If e-fc 2 lzlc bi lies below 0 C. It turns out that the seasonal rhythm w-1 influences the penetration depth not the daily time period w2. From the measured penetration depth typically a few decimetres one can determine the diffusion constant Dw. 2 With regard to Green s functions it can be shown by direct differentiation that the function x2 e 4Dwt G x t 4nDw t is a solution to the heat diffusion equation 40.3 . It represents a special solution of general importance for this equation since on the one hand 40.4 Solutions of the Diffusion Equation 311 for t to G x t propagates itself more and more becoming flatter and broader. On the other hand for t 0 with positive infinitesimal 6 G x t becomes increasingly larger and narrower. In fact for t 0 G x t tends towards the Dirac delta .
