ACOUSTICAL WAVE EQUATION FOR VISCOUS 

 FLUIDS 



Fundamental Equations* 



The Navier-Stokes form of the equation of motion for 

 a viscous fluid is 



p It + p ("* v > u - ^ v 2 s - (u +x ) v (vS) + vp = P i? (i) 



where p is fluid density, v is the velocity vector, P the 

 pressure, F the body force per unit mass, \i and X 1 the 

 first and second viscosity coefficients and 7 is the general 

 (three-dimensional) gradient operator. The velocity and 

 density must also satisfy the continuity equation 



^+7 • (pO) = (2 



In addition it is presumed that p and P are related by an 

 equation of state, which can be expressed in a purely for- 

 mal way as 



P=f(p) (3) 



Relation (3) in effect ignores any temperature dependence; 

 however, this is unimportant for liquids since the tempera- 

 ture variation can be considered virtually nil. This matter 

 is explored in more detail in pages 37-38. 



'"A list of symbols is given on page iv, 



26 



