A. S. Iberall 



Thus, it is not ruled out that near resonances may be excited into a turbulent 

 field. It is only by satisfying all boundary conditions that one can determine 

 what propagation modes are permitted by the field. Nonlinear excitation of 

 elastic modes cannot be dismissed in any material medium even though their 

 amplitude may not be considered to be of any importance. 



The nonlinear problem will be examined under the assumption that a turbu- 

 lent field exists with an unknown mean velocity distribution whose maximum 

 value is sufficiently removed from zero that fluctuating propagative modes per- 

 sist. Decomposed into a time-dependent (1 subscript) and time -independent 

 (0 subscript) set, they are 



P= Poo + PoC^-y.-) + L Pi(x)eH--ay-^0= p^^ + P„ + P^^, , 

 3"= 3"oo + ^o(-.y^O + L 3-i(x) ei(-^^°v^^z)^ g-^^ ^ 5-^ ^ g-^^^ 



in which the propagation constants w, a, 8 are assumed to be real. They are 

 here indexed in the primitive form of traveling waves. (The vector indices i 

 have been omitted for clarity.) 



The justification for the search in this form may be considered to be 

 Poincare's concept of characteristic exponents. As Whittaker's "Mechanics" 

 states, in discussing stability of types of motion of dynamical systems, "Hence 

 a necessary condition for stability of the periodic orbit is that all the charac- 

 teristic exponents must be purely imaginary." 



The solution technique is basically also known as the describing-function 

 technique. Even though the fluctuating components sought are trapped into 

 oscillation by the nonlinearity of the overall process, their amplitudes are as- 

 sumed to be small. Thus, they will be assumed to contribute, in the quadratic 

 terms, to the time- independent processes, but the fluctuating components aris- 

 ing from difference frequencies in the quadratic terms will be neglected. As 

 a first-order theory for the fluctuating components, it can only furnish neces- 

 sary conditions for the existence of nonlinear limit cycles. Intuitively, one 

 expects that if the fluctuating components possess small amplitude, the tech- 

 nique should be reliable. The decomposed equation sets are: 



For the mean state: 



Ro -nRo + R(i^ -oRd) --oPo + n Rq + qn(a-Ro) ■ 



Ro -nJo + R(i) •□3'(i) - (7o„-i) [Ro -DPo + R(i) -aPd)] (4) 



= ^oo[Roi,j + Roj,i]Roj,i + Eoo[R(i)i.j + R(i)j,i]R(i)j,i + ^I'o^X^ 



^oot^oCRo • °Po +R(i) • °P(i)) - (Ro •°3"o + R(i) •a3'(i))] =-a-Ro - 



708 



