Chap. 9] 



SEISMIC METHODS 



449 



When (T = J, ^ = V, and v^/Vt = \/3; that is, the longitudinal wave 

 moves faster than the transverse wave. 



The accelerations on the left side of eq. (9-15) may be set in relation to 

 the ground amplitudes. Since the inertia force, m-d^u/df, must equal the 

 restoring force ( = force per unit displacement, or spring constant c, times 

 displacement), we have m-du/dt = —cu or 



du.c „ 



— K -\ — u = 0. 



(9-17) 



Fig. 9-4. Periodic motion. 



In this equation c/m = co ; w is the angular frequency of oscillation or the 

 angular velocity of the motion of a given particle on the circumference of 

 the circle of reference (see Fig. 9-4) or 



= 27r/, 



(9-1 8a) 



where 1/T = / is the number of oscillations per second. The solution of 

 eq. (9-17) is 



w = Bo sin ut -f Co cos ut. (9-18&) 



Substituting for the two arbitrary constants, Ao = Bo + Co , sin ^ = Co/ Ao , 

 and cos ^ = Bo/Ao ; 



u = Ao sin ((at -\- \p) — Ao sin {(p -{- xj/) = Ao sin u{t + to), (9-19) 



in which <p and ^ are phase angles; ^ = uto is the starting angle corre- 

 sponding to the time to ', and tp = utis the phase angle at the time t. The 

 significance of the constants Ao , Co , and Bo may be obtained from eq. 

 (9-19) by solving for the limits ^ + ^ = 90°, <p = 0°, and <p = 90°. Then, 

 Ao is seen to be the maximum amplitude (peak amplitude) ; Co is the initial 

 amplitude; and Bo is the amplitude reached within a quarter period 

 {(p = 90°) after the start. These relations appear in Fig. 9-4, whose 

 right side shows wave motion plotted against time, while on the left ampli- 

 tudes are shown as function of phase angles on the reference circle. 



