HYDRODYNAMICAL RELATIONS 19 



(2.7) T,= -- §'■^'1 P 



Ot Po 



dp ,. 



— = -p«divv 

 dl 



From section 2.1, P is a definite function of density, and we may write 



dP ^ /dP\ dp 

 dt \dp /So di 



the subscript So indicating changes along the adiabatic having the 

 entropy of the undisturbed fluid. The total differential dP/dp, under- 

 stood to be evaluated for an adiabatic change, will be denoted by c^^, 

 and Eqs. (2.7) may be written 



(2.8) S = -^ Stad P 



ot po 



1 dP 



— — ■ = -podivv 

 Co^ dt 



A. Plane waves. It is assumed that the motion takes place only 

 along X, Eqs. (2.8) become 



dt po dx c^ dt ° dx 



Solving for P by differentiation and elimination, we obtain 



dx" ~ Co'' de 



together with a similar equation for u. 



This one-dimensional form of the wave equation is satisfied by any 

 function of the form/(^ d= x/co), the double sign choice accounting for 

 waves advancing in either positive or negative directions. 



Physically, these solutions mean that any disturbance originated 

 at some value of x travels unchanged in form with a velocity 

 Co = 'VdP/dp. For sea water at 20° C, this velocity is about 4,967 

 ft. /sec. The particle velocity u corresponding to the pressure 

 P = fit — x/co) may be found from the first of Eqs. (2.9) : 



du ^ _l^dP 



dt po dx 



PoCo \ Co/ 



