592 Appendix A 



displacements. The operator V 2 introduced in this expression in 

 Cartesian coordinates means the following 



J 8x 2 By 2 dz 2 



Although Equation 6 can be used for many calculations useful in bio- 

 acoustics, such as diffraction patterns, its use here is restricted to one 

 observation. Any phenomenon described by an equation of the same 

 type as Equation 6 is known as a wave phenomenon, and this type of equation 

 is called a wave equation. For the mathematically initiated, this equation, 

 with its few symbols, expresses the wide variety of physical properties 

 such as interference and diffraction associated with wave motion. (For 

 plane acoustic waves in a fluid, the vector particle velocity ~v is parallel to 

 the direction of the propagation of the wave. Such waves are called 

 longitudinal (or compressional or irrotational) . These properties are not 

 expressed by Equation 6.) 



Most acoustic experiments measure not the wavelength or the particle 

 velocity but the sound pressure p (or acoustic pressure) defined by the 

 relationship 



where P is the instantaneous total pressure and P the average (or 

 equilibrium) pressure. For a plane wave traveling in the positive x 

 direction, one can show (although it is not shown here) that 



a _ ^gj2nv(t-xlc) 



and 



P+ = p™+ 



The sensation of pitch, it has been noted, is associated with frequency 

 and the sensation of loudness with the sound pressure amplitude. The 

 quality of a musical note is recognized by the number and relative 

 intensity of the harmonics present. Not only are the harmonics import- 

 ant but in a few cases the relative phases are important. Qualitatively, 

 one may think of the relative phase as the indication of the displacement 

 and velocity at the time t = 0. Symbolically, p at a given place is 

 represented by 



p = A Q e j0 e j2nvt [orp = A cos (2irvt + <p)] 



where A is a real number and 2nvt + cp is called the phase angle. If the 

 acoustic pressure wave is made up of two frequencies v x and v 2 , then 

 one may represent p at a specific place by the expression 



p = A 0i e j(p ^e i2nv ^ + Ao^e* 2 ™*' 



