(18) 



980 EXPLORATION GEOPHYSICS 



z f^ 

 where R = ., , \ — 



y/k/ca * 2 



and o = ± t7 



Equations 15 and 17 demand that the temperature increase indefinitely as z increases, 

 which is impossible. Equation 16 is excluded by the boundary conditions of Equation 12. 

 Equation 14 satisfies 12 if 



B = T. and y = w 



With these substitutions, the solution of Equation 5a is 



which yields the temperature at any time t, at any depth s from the surface. 



The range of temperature for any point below the surface may be calculated from 

 the maximum variation of the temperature at the point. Since sin d = ± 1 for a maxi- 

 mum and minimum respectively, 



- ' a/~ - ; a/IiI 



RT=:2T,e V^ "^^ =2Toe V"^ " (1^) 



where w = -— and P is the period of the variation of the sun's radiation. From Equa- 

 tion 19, it can be seen that T, is the amplitude or half range at the surface (^ = 0). 

 Consider the diurnal wave as an example. Suppose, at a certain season of the year, 



the surface temperature of the soil / — = 0.0049 ) varies from -f- 16° C. to —4" C. The 



surface half range value (7.) is then ^^ ~ ^~'^^ = 10°. P = 24 hours = 86,400 



seconds. The mean surface temperature is 6°. 



Use Equation 19 to determine the temperature range at 30 cm. and at one meter, as 

 follows : 



V^ 



V .0049 '' 86,400 



2 = 30 cm.: Rt = 2 (10) e 



= 2(10) (.07) =1.4°C. 

 5 =100 cm.: Rt = 2 (10) e-*' 



= 2 (10) (0.00016) = 0.0032° C. 



Lag, Velocity and Wave Length. — A maximum or minimum of temperature will 

 occur to a given depth at the time 



V^+(2n + l) 



t^ - V^'ca ^ ^ ^ (20) 



w 



where odd values of n give minima, and even values, maxima. 



Considering a particular (first: m = 1) minimum occurring at the surface ( ^ = 0) 



when ti = -^ , then as x and * increase, this particular minimum is propagated into 

 2w 



