20 HYDRODYNAMICAL RELATIONS 



the prime indicating differentiation with respect to the argument. 

 Integrating, we have 



1 r' 



U — Uo 



^^o J . \ Co/ 



lo 



_ P- Po 



PoCo 



If the constants of integration are chosen to make u = when P = P, 

 the pressure in the undisturbed fluid, we have 



P - Po 



U = 



PoCo 



The relation for the wave f{t + x/co) travelHng toward negative x is 

 found in the same way to be 2^ = — (P — Po)/poCo. As a result the 

 point value of either P or u suffices to determine the other if the wave 

 motion is one-dimensional. 



The relative magnitudes of pressure and particle velocity are greatly 

 different in liquids and gases initially at atmospheric pressure. For ex- 

 ample, an excess pressure of 0.15 Ib./in.^ (one hundredth atmospheric 

 pressure) corresponds to a particle velocity of 0.056 ft. /sec. in sea water 

 at 20° C, but for air at 20° C. the particle velocity for the same pressure 

 is 3,700 times as great. 



B. Spherical waves. The simplest form of spherical wave is one in 

 which the disturbance is a function of radial distance from a source and 

 not of the angular position. If the radial component of particle velocity 

 is Ur, other components being zero, and P = P{r), Eqs. (2.8) become 



(2.10) 



(■■•?) 



If the second equation is differentiated with respect to f, and the particle 

 velocity eliminated by the first equation, we obtain 



r2 dr \ dr ) ~ Co^ ^dt 



It is easily verified that any function of the form P{r, t) — Po 

 ^ {l/r)f{t — r/co) is a solution, the negative sign corresponding to an 



