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37 | APPENDIX I 
1. SMALL-AMPLITUDE WAVES 
APPENDIX I 
SMALL AMPLITUDES 
For convenience of reference a summary will be given here of certain parts 
of the theory of compressive waves. For references see especially, besides books on 
sound, Lamb's Hydrodynamics and references to Riemann (10), Rayleigh (11), Becker 
(12), Bollé (13), and Epstein (14) at the end of the report. 
It is convenient to divide compressive waves arbitrarily into three types, 
which will be discussed in turn. It will be assumed, except where stated, that ef- 
fects due to heat conduction, viscosity, and thermel hysteresis are negligible. 
I. WAVES OF SMALL AMPLITUDE - THE LINEAR THEORY 
As the amplitude of compressive waves is made progressively smaller, the 
waves come to possess more perfectly certain simple properties; in the differential 
equations describing them, certain terms become negligible and the equations are then 
of the type called linear. The stock example is ordinary sound waves. The properties 
in question, predicted by theory and confirmed by experiment, are: 
1. UNIFORM VELOCITY 
"The velocity c at which the waves travel through the medium is given by the 
formula 
dp 
dp (1] 
c= 
where p is the pressure in the medium and p is its density. The value of c¢ is inde- 
pendent of wave length, and the relation between p and p follows the adiabatic law. 
In water, c increases slightly with rise of temperature or with increase of 
pressure. Some values of c at 15 degrees centigrade (59 degrees fahrenheit) and 1 
11,670 
atmosphere, expressed in feet per second, are: 
Average 
4930 16,400 
Pure Water 
4810 
1120 ft/sec 
2. UNIFORM FORM 
The Form of a plane wave does not change as the wave progresses. 
3. SUPERPOSABILITY 
Small waves can be superposed on each other, as occurs when two trains of 
waves meet. The resultant pressure is the sum of the component pressures, the re- 
sultant particle velocity is the vector sum of the particle velocities. The energy, 
however, exhibits the familiar phenomenon of interference, in exact analogy with 
light waves. 
