Appendix A 591 



the latter represents the experimental value. To illustrate this, one may 

 write the following 



Real part Complex notation 



£ = A 1 cos 2-nvt + A 2 sin 2-nvt £ = Ce j2nvt C = A - jB (2) 



di 



v = — = 2ttv[ — A 1 sin 2-nvt + A 2 cos 2-nvt] 



v = j t = 27TvjCe j27lvt = 2-nvj£ (3) 



d 2 £ 

 a = -jj = - {2ttv) 2 [A x cos 2-nvt + A 2 sin 2-nvt] 



= -(2rco) 2 £ 



a = ^= -(2rrv) 2 Ce^ 



= -(2w) 2 | (4) 



No matter what type of object is vibrating, be it a piano-wire, an organ 

 pipe, or a part of the ear, these same relationships are valid. 



In addition to a local particle velocity, the wave velocity c is often used 

 in acoustics. When a displacement is transmitted, the rate at which a 

 wave front moves through the medium is called the wave velocity. For 

 media such as air, water, and most tissues, c is independent of frequency. 

 For anv non-viscous fluid 



c = VB/ Po (5) 



where B is the adiabatic bulk modulus and p is the average (or equi- 

 librium) density. For gases, B is related to the average pressure p 

 by the equation 



B = yp and hence c — Vyp /p 



In this expression, y is the ratio of the specific heat at constant pressure 

 to the specific heat at constant volume. For air under normal pressure 

 and temperature c= 3.4 x 10 4 cm/sec. For ideal gases, the ratio 

 polpo is proportional to the absolute temperature. Hence, their wave 

 velocity c will increase as the square root of the absolute temperature. 

 It is shown in physics and math texts that the equation 



will describe the motion of a nonviscous medium subject to infinitesimal 



- c 2 V 2 v (6) 



