200, 201, 202] SOUND-WAVES. 543 



196) = 



and differentiating 192) by t and combining with this 



197) *J 



Since the motion is assumed to be irrotational, introducing the velocity 

 potential into equations 195) they become the derivatives by x, y, z 

 respectively of the equation 



198) Si''- 

 Differentiating by t, making use of equation 192), 



199) j = - - a 2 d ~ = a 2 6 = a*4<p. 



Thus J)oth the velocity potential and the compression satisfy the 

 differential equation 



200) 5 = 2 ^- 



This is known as the differential equation of wave -motion and is the 

 basis of the theories of sound and light. 



2O2. Plane Waves. Let us suppose first that the motion is 

 the same at all points in each plane perpendicular to a given direction 

 which we will take for that of the X-axis. Thus all the quantities 

 concerned become independent of y and z and equation 200) reduces to 



which is equation 109) 46 ; the equation for the motion of a con- 

 tinuous string ; or equation 182) 200, for the propagation of long 

 waves in shallow water. 



The general solution of this equation is found by introducing 

 the two new independent variables p = x at, q = x -+- at. 



We have then 



dtp dtp dp C(p dq dff dcp 



-75 = -75 QT -f- o ^TT = a -T -f- a o ) 

 dt dp dt ' d% dt dp ' dq 



dcp _ o^f 'dp iCydq _ dcp dtp 

 dx 'dp dx dq dx dp dq' 



. y , Q 

 * "*" 2 



_ 



dx* dp 

 Inserting these values in 201), we have 



