BASIC EQUATIONS 



11 



Before the equations of motion (15) can be used, 

 expressions must be derived for the force components 

 /x, /„, /. acting on the small box of Figure 2. Accord- 

 ingly, we shall calculate these forces under the as- 

 sumption that the fluid is perfect. According to this 

 assiuTiption, the box will move in the x direction if 

 and only if the pressure on face ADEH is different 

 from the pressure on face BCFG. Similarly, it will 

 move in the z direction only if the pressures on faces 

 ABCD and EFGH are unequal. Motion in the y 

 (vertical) direction is not quite so simple because of 

 the hydrostatic, or gravity-produced pressure dif- 

 ferences, which do not of themselves cause motion. 

 The box will move in the y direction if and only if 

 the pressure on face DCFE is not exactly equal to the 

 pressure on face ABGH plus the total weight of the 

 box. If the corrected pressure p(x,y,z,t) is defined as 

 the total pressure P{x,y,z,t) minus the hydrostatic 

 pressure at the point {x,y,z) when the fiuid is at 

 equilibrium, then the criterion for motion in the y 

 direction may be restated as follows. Motion will oc- 

 cur in the y direction if and only if the corrected pres- 

 sure at face ABGH differs from the corrected pressure 

 at face DCFE. We shall have little occasion to use the 

 total pressure P since the hydrostatic pressures are 

 seldom important in sound propagation. 



Figure 3. Pressure on opposite faces of infinitesimal 

 fluid element. 



Figure 3 is a duplication of the box of Figure 2 

 showing the forces acting in the x direction. If the 

 pressure at the left-hand surface is p, the total force 

 on that surface is -plylz. The pressure at the right-hand 

 surface is clearly p -|- (dp/dx)lx; and the total force 

 on that surface is therefore [p + {dp/dx)'}lxlyh. Since 

 the fluid is assumed to be perfect, these forces are 



parallel and their resultant can be obtained by simple 

 subtraction. Thus, the total force on the volume Vo in 

 the X direction is given by 



Fx = U'o 



dp 

 dx 



i'X"y''z* 



(16) 



Thus, the force per unit volume in the x direction /i 

 is given by 



dp 



/. = - 



dx 



Similarly, 



Jv — . 



f - _^ 



dz 



(16a) 



(16b) . 

 (16c) 



From equations (15) and (16a, b, c) we obtain 

 dp _ dUx dp dUy dp dUz 



dx dt dy dt dz dt 



2.1.4 Equation of State 



Our aim is to derive a differential equation which 

 will relate certain properties of the disturbed fluid 

 (pressure changes, density changes) to the independ- 

 ent variables x,y,z,t. For effective use, this differen- 

 tial equation should contain only one dependent 

 variable. The basic equations derived up to this point 

 — (4), (15), and (16) — contain the dependent vari- 

 ables (7, p, p, Mi, Uy, Uz- (7 and p are one variable since 

 they are related by equation (3). It will be seen in 

 Section 2.2.1 that the velocity components can be 

 easily eliminated by use of the equation of con- 

 tinuity (4). However, we must consider the relation- 

 ship between density and pressure before we obtain 

 a differential equation for the propagation of sound. 

 Such a relationship between density and pressure is 

 obtained from the equation of state of the fluid. 



The equation of state of any fluid" is that equation 

 which describes the pressure of the fluid as a function 

 of its density and its temperature, 



P = P{p,T). 



This function P{p,T) must be determined experi- 

 mentally for each fluid separately. In the case of sea 

 water, it depends on the percentage of dissolved salts. 

 A relation between pressure and density is obtained 

 from the equation of state by making two assump- 

 tions. First, it is assumed that a passing sound wave 



"We shall follow the common usage of physicists and use 

 the term fluids to denote both liquids and gases. 



