4 SOUND WAVES 



Octave. — An octave is the interval between two frequencies having a 

 ratio of two to one. 



Transducer. — A transducer is a device by means of which energy may 

 flow from one or more transmission systems to one or more other trans- 

 mission systems. The energy transmitted by these systems may be of 

 any form (for example, it may be electrical, mechanical or acoustical) and 

 it may be the same form or different forms in the various input and out- 

 put systems. 



The example of Fig. 1.1 has shown graphically some of the properties 

 of wave motion. It is the purpose of the next section to derive the fun- 

 damental wave equation. It is not necessary that the reader digest all 

 the assumptions and processes involved in order to obtain valuable infor- 

 mation concerning the properties of a sound wave. 



1.3. Acoustic Wave Equation. — The general case of sound propagation 

 involves three dimensions. The general relation for sound propagation 

 of small amplitudes in three dimensions will be derived and then these 

 relations will be applied to special problems. 



A. Equation of Continuity. — The fundamental equation of hydrokinetics 

 is the equation of continuity. This equation is merely a mathematical 

 statement of an otherwise obvious fact that matter is neither created nor 

 destroyed in the interior of the medium. That is, the amount of matter 

 v.'hich enters the boundaries of a small volume equals the increase of matter 

 inside. Consider the influx and efflux through each pair of faces of the 

 cube of dimensions hx, Ajy and Az, the difference between the latter and 

 the former for the whole cube is 



d{p'u) ^ djp'v) ^ d(p 

 dx By dz 



z J 



+ ^^— ^ + -^ — - Ax Ay Az 1.1 



where x, y, z = coordinates of a particle in the medium, 



u, V, w = component velocities of a particle in the medium, and 

 p' = density of the medium. 



The rate of growth of mass — Ax Ay Az in the cube must be equal to 



dt 



the expression 1.1 This may be written as 



dp' d(p'u) d(p'v) d(p'w) 



— -f ^ ^ -\- -^^— ^ + = 1.2 



dt ^ dx dy dz 



where / = time. 



This is the equation of continuity which signifies the conservation of mat- 

 ter and the three dimensionality of space. 



