164 BELL SYSTEM TECHNICAL JOURNAL 



where 



p = '^ "*" ^ P^Pfi (gj.l + (*)n)x /j^s 



P :::= T + 1 ^^^/? (cO.I — CO/<).Y , , 



2{2)-' ypo c 



and P^, Pb are the two r.m.s. fundamental pressures. 



If we extend Lamb's method of solution of equation (7) to the third 

 approximation and again consider the case ^ = a cos w/ at x = we 

 find that in the region kx"^ 1, the r.m.s. third harmonic pressure is 



In the case of the greatest r.m.s. fundamental pressure used in the 

 experiments, Pi = 8000 bars at 600 c.p.s., equation (15) indicates 

 that the third harmonic at 400 cm from the source is about 10 db 

 below the second harmonic. 



An approximate correction for the effect of attenuation in a tube 

 caused by viscosity and heat conduction can be obtained by assuming 

 that each of the extraneous frequencies and the fundamental is 

 attenuated as if it were the only wave present. Thus the r.m.s. value 

 of the fundamental at any point x is assumed to be 



Pi = Po^-"!^ or dPildx = - aiPi, 



where Po is the r.m.s. value of the fundamental at the point a; = 

 and ai is the measured attenuation factor for the fundamental. 

 If Pi is the r.m.s. value of the second harmonic at the point x and «2 is 

 the measured attenuation factor for the second harmonic in the absence 

 of the fundamental, we have by using equation (12) : 



dPi/dx = KP\ - a^P. = XPo2e-2«ix _ ^^Po, (16) 



where 



7+11 CO 



K = 



2(2)i 7^0 c 



When cti = and a^ = 0, equation (16) is equivalent to (12). The 

 solution of (16) which is consistent with the fact that the second 

 harmonic vanishes at a; = is 



P2 = [KPti'K2ai - a2)][e-«-^^ - e-'«i^]. 

 Hence 



P2^ 7 + 1 .Zi.^^.^ ,17^ 



Pi 2(2)i 7^ C "" ''• ^^^^ 



