Earthquake waves through, the earth. f>57 



We may assume the change of pressure corresponding to a 

 change in x to be small, otherwise the equation will not be linear. 



Hence we neglect the product -4- a. 



P J) dqj 



Also e being a large pressure, we neglect — -~ -~ , and our 



B (LOO 



equation becomes, 



(_r _]\dp m o^ 

 V P e) dx dx*' 



Since p is the pressure at M ', and is a function of £, and f is 

 a function of x, therefore p is a function of x, and when £ becomes 



£ 4- j^ Bx, p becomes p + -J- Sx. 



Hence p + ~ Bx is the pressure at N', and -J- Bx is the 



difference of the pressures on the two faces of M'N', and acts 

 towards the origin, and is therefore negative. 



The mass of M'N' on which it acts is the same as that of the 

 original mass MN, viz. D8x. 



Hence by the equation of motion 



ax dt 2 



frD t D\d 2 %_d 2 % 



\ P e J dt" dx 



If there was no extruded gas, or the disturbance due to the 

 expansion of the liquid only, we should have 



e d 2 % _ d 2 g 

 Bdx 2 ~ dt? ' 

 This is the differential equation to a wave of which the 



velocity is \f j^ . It indicates a wave which will be propagated 



within the substance of the liquid, independent of any gas. 



If on the other hand the liquid was inexpansible, or the 

 disturbance due to the extruded gas alone, then we should have 



rD dx 2 dP ' 



This would be the differential equation to a wave whose 



/p 

 velocity would be a / — ^ , and would probably be less than that 



of the wave due to the expansion of the liquid alone. 



