10 



WAVE ACOUSTICS 



From equations (3) and (5), 



p = po(l + <r), (6) 



V = t;„(l + co). (7) 



Since the masses at equilibrium and at time t are 

 equal, 



pv = pqVo. (8) 



By combining equations (6), (7), and (8) 



(1 + co)(l + <r) = 1. 



The product ua, a second-order term, can be neg- 

 lected, giving 



o) = -<r. (9) 



That is, imder the assumption of small displacements 

 and small density changes, the fractional volume 

 change u is the negative of the fractional density 

 change a. 



2.1.2 



Equations of Motion 



In this section, we shall apply Newton's second law 

 of motion to the mass of fluid within the volimae ele- 

 ment Vq. This law states that the product of the mass 

 of a particle by its acceleration in any direction is 

 equal to the force acting on the particle in that direc- 

 tion. 



Given the velocity distribution within the fluid as 

 a function of the position coordinates and time, 



Wx = U:,{x,y,z,t), etc.; (10) 



then the distribution of acceleration within the fluid 

 is to be calculated, 



Cx = a^{x,y,z,t), etc. (11) 



We cannot immediately say that a^ = dujdt. For, 

 in order to calculate the acceleration at a particular 

 point and a particular time, we must focus attention 

 on one particular particle. At the end of a time incre- 

 ment dt, the particle has moved to a point (x + dx, 

 y -\- dy,z -\- dz), where it has the velocity component 

 Ux{x + dx, y + dy, z + dz). The difference between 

 its new velocity and its original velocity, divided by 

 the time interval dt, gives the desired acceleration 

 component du^/dt. This value is not exactly the same 

 as the simple partial derivative dujdt, because the 

 latter does not focus attention on the change of 

 velocity of a single particle, but instead compares the 

 velocity of a particle at the point {x,y,z) and time t 

 with the velocity of the particle which occupies the 

 position {x,y,z) at the end of the time interval dt. 

 However, dujdi and du^/dt will be almost equal 



under the assumptions that second-order products 

 of displacements, particle velocities, and pressure 

 changes are negligible. To show this, we note that 



dUx dUx dUxdx dUxdy duxdz , „^ 



z — z _i_ z __ I z _^ _|_ . (12) 



dt dt dx dt dy dt dz dt ' 



which is the usual equation found in calculus texts re- 

 lating the partial and total time derivatives of a 

 function. The last three terms are second-order prod- 

 ucts, since dx/dt, dy/dt, dz/dt are merely Ux,Uy,u,. 

 Thus, the component of acceleration in the x direc- 

 tion may be approximated by dUx/dt, and similarly 

 for Oy and a^. That is. 



ax = 



dUx 

 It' 



dXly dU, 



The mass of fluid within the volume element Vo is 

 pvo. If Fx,Fy,Fz are the components of the forces acting 

 on the element, then the equations governing the 

 motion of this element are, in view of equation (13), 



Fx = pVo 



dUx 

 'dt' 



Fy = pV^ 



dUy 

 dt 



Fx = pfo— • 

 at 



(14) 



It is desirable to make the equations governing the 

 motion of the small element independent of the par- 

 ticular value of the small volume Vo. For this reason, 

 we rewrite equations (14) as 



dUx dUy dUx 



U = P— ; U = P"T7 ; h = P— I (15) 



ol at at 



where 



Ji J Jy J Ji 



Va Vo Vo 



The normalized force components may be regarded 

 as the force components per unit volume acting on the 

 small volume element. 



The next section is concerned with calculating 

 fxjyjz in terms of the pressure or density changes oc- 

 curring within the fluid. 



2.1.3 Law of Forces in a Perfect Fluid 



A fluid is called perfect if the forces in its interior 

 are solely forces of compression and expansion, in 

 other words, if the fluid is incapable of shear stress. 

 If a fluid is perfect, the force on any portion of its 

 surface is perpendicular to the surface. Fluids which 

 can exhibit shear stress in response to shear deforma- 

 tion, in addition to responding to compressive and 

 expansive forces, are called viscous. 



