920 MISCELLANEOUS GEOPHYSICAL METHODS [Chap. 12 



do not belong in the category of ground tests and are discussed in connec- 

 tion with geoacoustic methods on page 958. 



The reaction of the ground to the vibrator may be described by a simple 

 expression if it is assumed that the vibrator is force-coupled to a semi- 

 infinite medium (with a definite stiffness and velocity damping) whose 

 displacement x is controlled only by Hooke's law.^^ Let nio be the mass of 

 the vibrator, plus whatever portion of ground partakes in the motion, 

 coo = \/co/mo the natural frequency of the ground, w the impressed fre- 

 quency, Co a stiffness coefficient of the ground and eo a damping coefficient, 

 and X = X sin. (o}t — <p) the ground displacement having a maximum 

 amplitude X and a phase shift <p in reference to the agitator. Then the 

 centrifugal force referred to unit mass is 



F = ^, (12-9) 



ma 



where m! is the eccentric mass with a radius r on the vibrator of the mass 

 7^0 . The equation of motion of the ground is therefore given by 



X + 2€oX -f- woa; = F sin o>t. (12-10) 



This leads to a maximum ground amplitude of 



^ = W. 2 ^s2,, n> (12-lla) 



V (,wo — 0} } -T 4coco 



whose phase shift in reference to the application of the maximum force 

 is given by 



2^0 CO 

 tan ^ = "2 2. (12-116) 



Co — CO 



The power transferred to the ground is then 



P= — - — sm ^, (12-1 Ic) 



SO that, by substitution of eq. (12-9) in (12-lla) and (12-116), 



m'TitP' 

 moV (,coo — • CO ; -|- 4eoco 



and 



p = ^^.Zco'sinvp. (12-lle) 



" A. Hertwig, G. Friih, and H. Lorenz, Veroff. Deut. Ges. Boden-Mechanik 

 Heft 1, 44 pp. (Berlin, 1933). 



