Equation applicable to a Plane Progressive Wave. 671 



at any moment constitute the part of the wave characterized, 

 by the density p. 

 "Suppose y =-^r(#) initially, 



then du , f/ \ . ... n 



~- =-v|r (x) initially; 

 ax r v ' 



and from what has been proved above we shall have at any 

 subsequent time 



dy 



dx 



■+(*-&$■')■ 



This equation gives us p when x, t, and the function sjr are 

 all known; but we note that it will not be possible in general 

 to exhibit p as an explicit function of x and t. 



Again, since 



du _ 1 /dp 



dp p\/ dp' 



we shall have du /dp 



U+ PT P =U+ V dp' 



so that d /dp 



Sp- u P =u+ VTp 



Imagine a plane at which the density is always p travelling 

 along with the disturbance ; then by taking account of Ihe 

 constancy of the mass within a specified small distance on 



each side of the plane, its velocity is readily seen to be -j-pn. 



Thus the value of p at time t for particles whose disturbed 

 position is given by y will be the same as the value of p at the 

 beginning of the disturbance for particles whose disturbed 



position differs by (« + \/7 ft- W e have already seen 

 above that the undisturbed position of the particles corre- 

 sponding to p at time t will differ by \/ ~r *t from the 

 initial position. Hence 



»-("V£>^K\/t-<> 

 *-H-£\/| ■<)♦(•• Vt)<- 



2Y 2 



or 



