EXTRANEOUS FREQUENCIES 165 



where 



K = 771 r 1 !-•••. 



{la\ — UijX I 



For the tube used in these experiments a^ may be taken as l^oci which 

 gives 



R= 1 - aix/2^ + •••• 



Similar correction factors were derived for the other extraneous fre- 

 quencies measured in the experiments. 



Measurements of Plane Waves of Finite Amplitude 

 IN A Tube 



The experimental work consisted of measuring the second harmonic 

 generated along a tube. Measurements were also made of the sum 

 and difference tones when two fundamental frequencies of equal 

 pressure were simultaneously impressed on the air of the tube. For 

 the fundamental pressures and distances used in the experiments, the 

 magnitudes of the other harmonics and higher order sum and difference 

 frequencies were probably small. For high fundamental frequencies 

 and pressures, however, these other tones are important, since they 

 increase more rapidly with frequency and pressure than the second 

 harmonic; for instance, the third harmonic pressure increases as the 

 square of the fundamental frequency and as the third power of the 

 fundamental pressure. 



A sinusoidal displacement, uniform over the cross section, was 

 impressed on the air at one end of a long tube. The tube had an inside 

 diameter of 3.8 cm and was 1566 cm long. Measurements were made 

 in the first 705 cm only and the remainder of the tube was used for 

 obtaining a non-reflective termination. 



A search transmitter, comprising a small tube of 0.08 cm inside 

 diameter and 7.5 cm long, coupled to a small condenser microphone,'' 

 was used for the measurements. Attenuation in this search tube 

 was sufficiently high to prevent either overloading the microphone or 

 altering the sound wave propagated within the long tube. The search 

 transmitter was connected to a stage of amplification so operated as 

 to preclude non-linear distortion. This was followed by a band-pass 

 filter which selected the frequency desired in the measurements. The 

 filter was terminated by a measuring circuit consisting of a high-gain 

 amplifier and a vacuum tube voltmeter. A diagram of the arrange- 

 ment is shown in Fig. 2. 



' H. C. Harrison and P. B. Flanders, "An Efficient Miniature Condenser Micro- 

 phone System," Bell Sys. Tech. Jour., 11, 451 (1932). 



