788 EXPLORATION GEOPHYSICS 



After completing the study of free vibrations of the oscillating particle, Equation 

 126, which describes the theoretical seismometer when disturbed by certain types of 

 ground motion, may be studied in more detail. This equation may be written in a more 

 famihar form 



y -{- 2hny + ni'y = — Z, 



and now is similar to Equation 133 except for the acceleration term ( — Z) which 

 represents an outside force. A vibrating particle which is under the influence of an 

 outside force (in addition to the force of attraction toward the rest position) is said 

 to be executing a forced vibration. Assume that the particle is subjected to the external 

 force H ; then the equation of motion will have a term on the right-hand side such that 



TT 



Z =. ~. When Z is a constant or is a mathematical function of time (0 only, Equa- 

 tion 126 may be solved by the differential treatment for equations with constant co- 

 efficients. However, if Z is a function of both (0 and (y), the motion cannot be that 

 of a simple forced vibration. From experience gained through working with problems 

 of this nature, it has been found that if 



C 



— Z z= a cos wt where a = r-r 



M 



an approximation is obtained to the real motion of the ground caused by a buried ex- 

 plosion or an earthquake. It will be noted that the force, C cos wt, is a simple harmonic 

 force impressed on the system. Equation 126 now becomes 



y + 2hny + n^y = a cos wt (139) 



A solution of this equation is given by 



y = A cos {wt — <p) (140) 



Substituting in Equation 139 and expanding the sine and cosine of the difference, 



[a — A { (n^ — zif) cos ^ -f 2nhw sin <p}] cos wt 



-{■ [—A (n^ — w^) sin (j) -\- ZAnhzv cos <p] sin wt = ( 141 ) 



This equation is satisfied for all values of (0 only if the coefficients of sin wt and 

 cos wt are zero separately. Equating the coefficients of sin zvt to zero, 



2nhzv 

 sm <p 



y/ in" — zsfy + {2wnhy ' 2nhw 



tan ((> = 

 n" — zi^ 

 cos = ; 



n^ — zs/^' 



V {rv" — ziiry-\- {2zmhy ' (142) 



Placing the coefficients of cos zut equal to zero, 



J a a 



(u' — z«/) cos <p + 2nhw sin ~~ -^ (^n"" — zii'y + (2zmhy 



Now y= > ■ , Z , ro r ., '^os (wt - <p) , tm<p = -p^ (144) 

 y/ (nr — zif)" + (2nhwy nr — zir 



This solution of Equation 139 contains no arbitrary constants ; therefore it is a so-called 

 particular solution. To obtain a complete solution to Equation 139, proceed in the fol- 

 lowing manner : 



