EXTRANEOUS FREQUENCIES 163 



This approximation restricts the dilatation di,jdx to values small com- 

 pared with unity or the excess pressure to values small compared with 

 7/>o, but the restriction need not be as severe as would be required for 

 the linear approximation: 



By a method of successive approximations, carried to the second 

 approximation, Lamb ^ derives the solution of (7) : 



I -I 2 



^ — a cos aj(/ — x/c) + ■ — ^ a^x\^l — cos 2co(/ — x/c)] (8) 



corresponding to a motion ^ — a cos cot imposed on the air at x = 0, 

 and assuming complete absence of reflection. By virtue of (4) and (1) 

 we have for the pressure: 



p = Ml - yd^/dx + •••). 



Neglecting the terms of small amplitude, in the region where 47rjc is 

 large compared with the wave-length X we have 



where 



or 



where 



P= Pdc + Pl-\- P2, (9) 



Pdc = Po- ypo- ((t + l)/8) • (a;Vc2)a2, 

 pi = — jpo-((joalc) sin co(/ — x/c), 

 pi = ypo-((7 + l)/4)-(a;Vc^)-a2xsin 2aj(/ - x/c), 



pi = 2'Pi COS [co(/ - x/c) + 7r/2], 



p2 = 2iP2 COS l2o}{t - x/c) - 7r/2], (10) 



Pi = ypocoa/2k, (11) 



P,= y^.^.^, (12) 



2(2)^ ypo c 



P\ and Pi are thus the r.m.s. fundamental and second harmonic 

 pressures, respectively. 



Lamb ' also gives a solution of (7) for the case when the forced 

 motion at :;c = is: ^ = ^a cos oo^^ + $b cos cob/. In addition to the 

 two fundamentals and two second harmonics, the pressure now in- 

 cludes components whose frequencies are, respectively, the sum and 

 difiference of the two primary frequencies: 



p^ = 2«Pg cos [(co^ -\- coB){t' — x/c) — x/2], 

 pd — 2^Pd cos [_{oiA — wb)(/ — x/c) + x/2], 



