Chapter 2 

 WAVE ACOUSTICS 



SOUND ENERGY takes the form of disturbances of 

 the pressure and density of some medium. There- 

 fore, the basic relationships between impressed forces 

 and resulting changes in pressure and density are use- 

 ful in an understanding of sound transmission. In this 

 chapter we shall derive several such relationships, 

 and shall combine them into one differential equation 

 relating the time derivatives and space derivatives of 

 the pressure changes to several physical constants of 

 the medium itself. This differential equation is the 

 foundation for the mathematical treatment of sound 

 transmission to which the rest of the chapter is 

 devoted. 



We shall see that this mathematical approach can- 

 not in itself furnish complete information on sound 

 transmission in the ocean. The physical picture must 

 necessarily be simpUfied to make mathematical de- 

 scription possible — and even this simplified scheme 

 does not yield explicit results for the sound intensity 

 in all cases. However, it is valuable to know the 

 mathematical theories even if they are partially un- 

 successful in predicting the quahties of sound trans- 

 mission. Tendencies predicted by a simplified theory 

 are often verified quaUtatively in practice. Also, 

 there is always the hope that by changes and ampli- 

 fications an incomplete theory can be made much 

 more useful. 



2.1 



BASIC EQUATIONS 



In this section we shall derive the basic equations 

 which will be put together to derive the fundamental 

 differential equation of wave propagation, the wave 

 equation. These equations are (1) the equation of 

 continuity, which is the mathematical expression of 

 the law of conservation of mass; (2) the equations of 

 motion, which are merely Newton's second law ap- 

 pUed to the small particles of a disturbed fluid; 

 (3) force equations, which relate the fluid pressure 

 inside a small volume of the fluid to the external 

 forces acting on the periphery of the volume; (4) the 

 equation of state, which relates the pressure changes 

 inside a fluid to the density changes. 



2.1.1 Equation of Continuity 



The equation of continuity is simply a mathemati- 

 cal statement of the law that no disturbance of a fluid 

 can cause mass to be either created or destroyed. In 

 particular, any difference between the amounts of 

 fluid entering and leaving a region must be accom- 

 panied by a corresponding change in the fluid density 

 in the region. 



To express this law in mathematical terms we must 

 first derive an expression for the mass of fluid which 

 passes through a certain small area of a surface in one 

 second. Let the small surface element have the area 

 4, as in Figure 1, and let the fluid move in a direction 



t«o 



tfi 



z 



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1/ 



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FiGUBE 1. Passage of fluid through area element A. 



perpendicular to A with the velocity u. In one second, 

 a rectangular fluid element of base A and height u has 

 passed through this element of area; that is, a volume 

 of Au cubic units of fluid has traversed the area. The 

 mass of fluid passing through the area per second will 

 thus be pAu, where p is the density at the point and 

 time in question. If the fluid is moving not perpen- 

 dicular to the element A, but in some other direction, 

 the mass passing A per second will still be given by 

 pAw, if M is taken to be the velocity component in the 

 direction perpendicular to A. 



Now consider a small hjrpothetical box-shaped 

 volume inside the fluid, and examine the amounts of 

 fluid entering and leaving this box (pictured in 

 Figure 2). For simplicity, we can assume that the 

 edges of the box are parallel to the coordinate axes. 

 Let the dimensions of the box be lx,lv,lz, as shown in 

 the diagram, and let the coordinates of the point H be 

 {x,y,z). Let the components of the fluid velocity at 

 the point H be Ux,Uy,Uz. 



8 



