ACOUSTIC WAVE EQUATION 5 



B. Equation of Motion. — Referring again to the space Aat Ajy Az the 



acceleration of momentum parallel to x is p' A.v Ajy Az — . The mean pres- 



dt 



sures on the faces perpendicular to x are 



where p^ — pressure in the medium. 



The difference is a force — • Lx Ay Az in the direction of increasing x. 



dx -^ ^ 



Equating this to the acceleration of momentum, the result is the equation 



of motion, 



1.3 



1.4 



C. Compressibility of a Gas. — The next property of a gas which is used 

 to derive the wave equation depends upon the thermodynamic properties 

 of gases. The expansions and contractions in a sound wave are too rapid 

 for the temperature of the gas to remain constant. The changes in pres- 

 sure and density are so rapid that practically no heat energy has time to 

 flow away from the compressed part of the gas before this part is no longer 

 compressed. When the gas temperature changes, but its heat energy 

 does not change, the compression is termed adiabatic. 



In the case of an adiabatic process, 



po \p) 



1.5 



po 



where pQ — static pressure. The static pressure is the pressure that 

 would exist in the medium with no sound waves pres- 

 ent. The unit is the dyne per square centimeter, 

 p = static or original density, 

 po = total pressure (static -f- excess), 

 p' = instantaneous density (static -t- change), and 

 7 = ratio of specific heat at constant pressure to that at con- 

 stant volume and has a value of 1.4 for air. 



