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APPENDIX I | 38 
1. SMALL-AMPLITUDE WAVES 
4. PRESSURE AND PARTICLE VELOCITY 
Pressure and particle velocity are definitely related to each other in pro- 
gressive waves.* The form of this relationship is somewhat different, however, in 
plane and in spherical waves. 
In plane waves the "pressure" p (i.e., the excess of pressure above normal) 
and the particle velocity u are related by the equation 
Pp = peu {2] 
where p is the density of the undisturbed fluid. 
The coefficient pc may be called the acoustic or radiative impedance of the 
fluid (also called "acoustic resistivity," although no dissipation of mechanical 
energy is involved). Some values of pc are as follows, expressed for convenience in 
units suggested by the relation, pc = p/u, the pressure p being expressed in pounds 
per square inch and the velocity c in feet per second: 
eee Air (15 degrees Centigrade, 76 cm) 
0.0185 
oe ( 
In English gravitational units, the values of pc are equal to those given here multi- 
plied by 144. The value of pc is more than 3000 times as great in water as it is in 
air, because both the density and the elasticity are much greater. In a sound wave, 
where the pressure is 1 pound per square inch above normal, the particle velocity is 
less than 1/5 inch per second in water but 54 feet per second in air. 
As a spherical wave moves outward from a center, the magnitude of the excess 
pressure, positive or negative, decreases in inverse ratio to the distance r from the 
center. The particle velocity, however, does not possess a unique relationship to the 
pressure at the same point, as it does in plane waves. The reason is that the decrease 
in magnitude of the pressure as 7 increases gives rise to an additional component in 
the pressure gradient, over and above that component which is involved in the propaga- 
tion of the wave; and because of this additional pressure gradient, a compression 
accelerates the water outward while passing through it, whereas a rarefaction acceler- 
ates it inward. The additional acceleration thus produced by spherical waves is pro- 
portional to p/r. 
The particle velocity wu in a train of spherical waves spreading out from a 
center, at a point where the excess pressure is p, is given by the formula 
t 
un fet P| var + w (3] 
0 
Here u is called positive when its direction is outward. The symbol u, stands for the 
particle velocity at the point in question at a time ft, at which p= 0; and : pat’ 
0 
* The term "progressive" is meant to imply a disturbance traveling in a definite direction, as opposed 
to "standing" waves or any other admixture of progressive wave trains. 
