Wave motions in rotating helium II

ANNALS

OF

PHYSICS:

43, 158-194

Wave

(1967)

Motions

in Rotating

Helium

II

H. A. SNYDER AND P. J. WESTERVELT Department

of Physics,

Brown

University,

Providence,

Rhode

Island

02912

A general account is given of wave propagation in uniformly rotating He II. A formal solution is presented for the four types of waves which occur in the interior of the fluid: first sound, second sound, inertial waves, and vortex waves. Boundary-layer solutions are not considered in detail. Several special cases are solved to illustrate the mechanics and to gauge the magnitude of small effects. The results are applicable to ordinary fluids as well as to He II. I INTRODUCTION

The Navier-Stokes equations cannot be used to describe the motion of liquid He II because quantum properties of the liquid are observed on a macroscopic level. Several atOempts have been made to derive a consistent set of equations which describe t’he motion of He II but each treatment of the problem is based on an ad hoc premise. Consequently, the only way to select an appropriate set of equations is to test each proposal experimentally. One of the most commonly used experimental techniques in the study of He II hydrodynamics is the investigation of wave propagation [Hall (I)]. It is well known (as will be shown explicitly in this paper) that there are four kinds of waves which propagate in rotating helium II: first sound, second sound, inertial waves, and vortex waves. A great deal of information on the structure of the motional equations can be found by observing the dispersion relations of these waves. In fact, a comparison between the theoretical and empirical dispersion provides sufficient information to reject certain sets of equations which have been proposed. It is necessary, then, to solve the equations to be tested with the boundary conditions appropriate to the experimental situation. A search of the literature will show that these solutions are not available even for the most widely used geometries of the boundaries. It is the purpose of this paper to remedy that deficiency to some extent by presenting solutions to the most widely accepted sets of equations for the most frequently encountered experimental conditions. The immediate motivation for this work is the need for an analytic method to reduce the experimental resuts of Snyder and Linekin (2). The present paper is the second in a series of three on the propagation of second sound in rotating 158

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MOTIONS

IN

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11

159

He II; t.he first of t’he series is by Snyder (3) and the t,hird is Ref. (2). [See also Snyder and Westervelt (4)]. Accordingly the first problem to be treated is that of a rot’ating resonator of rectangular cross section and driven by a uniform sound source on one wall of the cavity. The methods employed are sufficient11 general so that the results may be extended to other geometries. The analysis has been carried beyond the needs of this particular experiment and represents a general account of wave motions in rotating helium. The most general solution for wave motion in a rotating container of He II is the sum of several oscillatory solutions. In the interior of the fluid there is a superposition of the four types of waves listed previously. Each type of wave has associated with it one or more damped solutions which have appreciable amplitude only in the boundary layer. The boundary-layer effects will not be treated here but a subsequent paper is anticipated in which they will be treated in detail. a4l1other effects associated with the linearized equations will be dealt with. The result’s reported here apply to the response of a rotating resonator to: (i) first sound in an ordinary fluid; (ii) first sound in He II; (iii) second sound in He II; and (iv) inertial and vortex waves in He II. The method of attack used below was first developed by Westervelt (5) to study first sound in a rotmating system. In the first section which follows, the equations of motion are stated and are linearized. The resulting equations are transformed to a rotating frame. Kext a simple case, that of second sound in a square cavity, is solved with several small effects neglected in order to illustrate the method. ,4 general and consistent, solution is set down formally in the following section. Various special cases are considered in Section V. This is followed by a discussion of some physical prohlems relat,ed to the solutions of Section V. The final part is deveoted t,o proving n general theorem on the frequency shift of normal modes by rotation. II.

THE

EQUATIONS

OF

MOTIO?r’

The equations for He II developed by Landau (6) agree wilh experiment when the liquid is not in rotation and when the analogue in He II of turbulence is not present. Landau’s equations have been accepted generally under the above restrictions. In fact, the problems that, remain in He II dynamics are largely concerned with turbulence and the effects of rotation. Only t,wo theories of rotating He II hydrodynamics have been discussed widely in the lit’erature; the one is by Lin (?‘), and the ot,her is based on work hi by Hall and Vinen (I), (8), and by Bekarevich and Khalatnikov (9) (henceforth called HVBK theory). BothLin’s equations and those of HVBK include Landau’s equations as special caseswhen there is no rotation. Accordingly, there is no need to treat Landau’s equations separately.

160

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Some predictions of Lin’s theory are at variance with experimental results, while the predictions of the HVBK equations appear to be in good agreement with the known data when there is no turbulence in the flow. It is appropriate, therefore, to use the HVBK equations to illustrate the method presented here. The corresponding results for Lin’s equations will be stated for the simplest case. The HVBK equations have been derived by two distinct methods. Hall and Vinen (8) and Hall (1) starting with Landau’s equations added terms which would appeal’ if macroscopic vortex lines were present in the superfluid. The approach is based on the model of quantized vortex lines envisaged by Onsager (10) and by Feyman (11). Bekarevich and Khalatnikov, in deriving their equations, used only the laws for the conservation of energy, mass, and momentum, together with one additional assumption. The postulate which in the latter derivation leads to the terms associated with vortex lines is that, as a result of rotation, the internal energy of the liquid is increased locally by an additional term which is proportional to the curl of the superfluid velocity: 6e = s,& where & = / v x v, 1 and Q is a constant. The latter derivation does not depend upon a particular internal motion but is consistent with the model used by Hall and Vinen. Bothmethods lead to the same set of equations. The HVBK equations may be found in correct form in a recent monograph by Khalatnikov (1%‘) ; (the original published form of both Hall and of Bekarevich and Khalatnikov contain typographical errors). The constant q, may be related to the microscopic properties of the vortices by using Hall and Vinen’s model: 7, = (5/2nz)p, In (b/so). Here 71~is the mass of the helium atom, ps the density of the superfluid and b/a0 is the ratio of the distance between vortices to the effective core radius of the vortex line. It is appropriate for the calculations which follow to have the governing equations set down in a frame rotating at a constant angular velocity. Also we shall restrict our discussion to the case of small acoustic velocity amplitudes so that nonlinear terms in the equations of motion may be dropped. In making the transformation we will use v=V-tiOxr,

(1)

where v is the velocity in the rotating frame, V the velocity in the nonrotating frame and o,, is the constant angular velocity vector. When Eq. ( 1) is applied to the HVBK equations, the terms not due to vortex lines are unaffected but the Coriolis and centrifugal forces are added to the momentum equations; also, all terms in Q = v x V transform into 20~ + o. Here o = v x v. The only transformations which are not simple arise from the expression v x (O/I w I). Assuming 1 w j = ) v x v / << wowe expand as follows: #3,/lQ j = CL,- cz,(&.,)/2W~

+ 0/2wo.

(2)

We shall use the convention throughout this paper that a vector with a caret

WAVE

MOTIONS

IN

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HELIUM

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161

will mark is of unit length: thus, & = W/I o. /. A little algebraic manipulation show that the two expressions associated with tension in the vortex lines transform under Eqs. ( 1) and (2) and in Cartesian coordinates as

Co/l Gi I) x [ti x V x (o/l 4 1) + ii

[-- w,i + w, j].

1-i)

The z axis will always be assumed to be in the direction of o0 . Notice that Eqs. ( 3) and (3) do not approach zero explicitly as w0 + 0. The linearized equations of motion in the rotating frame are 9s follows : dV, _-.

zz

at

-~vp+sv2’+

P

+ By [wo(.v,- v,) + woBr,, -

l3’ e o. x P

dV,

- = -‘VP at P

- !? SvT

+ 2 ([I - ps c3j v(v*(v*

$2 - ps t1 + $J! +;

-

vIl,>

Pn

(

”>

v(v*v,)

- ;

(v

x)2vrl

- B ; [W(,(V,- vs) - oo*(v, - v,) &I - “00 x vn

( li i

162

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AND

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In these equations the commonly accepted symbols for the density, pressure, entropy per unit mass, temperature, time, and velocity have been used, i.e., p, p, S, T, t, v, respectively. The subscripts s and n refer to the superfluid and t,he normal fluid respectively. Thus, the radial coordinate of displacement from equilibrium is designated r, for the normal fluid and rs for the superfluid. The coefficients associated with dissipation arc: T,, for the first viscosity of the normal fluid; and x for the thermal conductivity. The t1 , tz , and & for t,he second viscosity; mutual friction parameters of Hall and Vinen are written as B and B’. Here, pn and q3 are used for the source terms. The pressure terms in the above equations are renormalized and include the addition caused by rotation: p = p. + Q( 2~0 + w,) , where po is the pressure without rotation. The momentum flux tensor to first order in v and transformed according to Eq. (1) is TIik

=

[PO

+

~(2~0

+

c+)l&k

+ji(a0

x

r)k

+

jk(m0

x

r), (9)

+

~(00

x

r)i(o0

x

rJk

-

rls(6i,w,k

+

&~zwsi)

+

7ik,

where 7;lCis the viscous stress tensor and j = pBvs + pnvn is the momentum density. In the transformed frame Tik has the same form as in the space frame. Under the same restrictions the vector of energy flux density is Q = PST v,, + pj - xVT i- 2~0 % [h -

+ w. ,,s B’;

( oo.vs) ;,,I + woqs B ; [ho x (vn - vs)l

[v, - vs) - cjo*(v,, - vs) ho]

(10)

where the chemical potential is designated as p and the t,hermal conductivity by x. Equations (9) and (10) are useful in specifying boundary conditions. The terms in Eq. (5) which are due to tension in the vortex lines make the equation of second degree and consequently it is necessary to specify two boundary conditions on v, ; the equation for v,, is also of second degree. If it can be assumed that the mechanical and thermal impedances of the sides of the resonator are infinite, then j/p generally equals zero unlessthe surface is in forced motion in which caseit equals the velocity of the surface. Also, the flux of heat W through the surface must be carried away by the normal fluid: vn = W/ST. The component of vn parallel to the surface must equal the tangential velocity of the surface. On the surfaces where the vortex lines terminate the tangential component of v, must satisfy the following condition to first order in v and in a rotating

WAVE

MOTIONS

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163

WC have used v, for the velocity of the surface and both {’ and { are adjustable parameters which specify vortex slippage at the surface. For the present we will assume that, the boundaries have infinite mechanical and thermal impedance. At, this point it is appropriate to consider under what circumstances the above equations and conditions apply. A number of approximations have been necessary to bring the formulation intro the present state. First, the mutual friction coefficient B, in the direction of rotation has been assumed to vanish. This is in accordance wit,h the experimental work of Snyder (3) and the theoretical work of Bekarevich and Khalatnikov (,12). Secondly, the omission of terms qundrnt,ic in 21has further consequences for liquid helium than for ordinary liquids. In the lat’ter case, t’he Reynolds stress is removed by neglecting such terms, but in the former case not only the Reynolds stress but also the scalars p, p, and p depend upon (Un - ~1~j’. Thirdly, the approximation stated in Eq. (21 is good to order (o/woj3 for velocity fields for which &,*ti = 1o j. We will see that this condition holds for the sound fields to be considered below. Fourt’h, the density p has been assumed to be constant. However, the centrifugal force term wir, sets up a radial pressure gradient, which in turn makes p a fun&ion of ro . We have neglected this effect of rot’ation. Finally, it must be kept in mind that the microscopic interpretation of the mutual friction terms ceases t,o hold near the X-point where the exc!t:ltions can no longer exist individually. Whether it is appropriate in this region tc) apply the equat’ions st’a ted above is questionable ( 13). Also, since some of the effects to be considered are quite small, it is always necessary t,o determine whether t.he approximations listed above change the result to :I considerably lesser extent than the effect in question. III.

A SIMPLE

SOLUTION

Equations (5)-( 11) are rather complicated and it appears pedagogically :~lvantageous t,o solve a highly simplified case first and then to proceed to the more complex situations. It is fortunate that the solution needed to reduce the data of Ref. (2) is simple and accordingly this case will be treated in this section. The effect,s to be considered later do not modify the response curve derived in the

164

SNYDER

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present section by more than 5 % except very close to the X-point. Therefore this “simple” solution is nearly a complete solution Consider a resonator of square cross section in the plane perpendicular to the axis of rotation and filled with He II. Assume that the walls have infinite impedance; neglect boundary-layer effects, the coupling of first to second sound, and the nonlinear generation of higher harmonics. Also, let the viscosities and the thermal conductivity of the liquid approach zero. The resonator is drivenwith a plane and uniform transmitter of second sound which coincides with one wall. Under these circumstances, as will be shown in Sec. IV, the vortex lines do not execute torsional oscillations. The problem is to calculate the response of the cavit,y as a function of t.he driving frequency. For second soundvp z 0 and pnvn w - psv, . Thus, it is necessary to solve for only one velocity field G . Let v, = vOeeint and substitute into Eqs. (6) and (7). If we take into account the simplifications which result from the approximations listed in the preceding paragraph, iQ(1 + iBw&)vo

=

(ps/pn)SVT'

+

(2

-

B')oox

vo,

(12)

and V-vo

=

(iC,IST)T'

+

qn,o,

(13)

where a prime indicates the time varying part of the quantity and C, is the specific heat. It is also assumed that there is a simple source strength density, viz., -at qn

=

qn.Oe

.

The wave equation for T’ is formed by solving Eq. ( 12) for V*VO and by substituting the result for V=VO in Eq. (13). In order to find V*VO from Eq. (12) it is necessary to apply V. andv x and solve the resulting pair of simultaneous equat,ions. The wave equation for T’ obtained in this manner contains (w&) tosecond order only and the lowest order of (we/Q) in the velocity field derived from T’ likewise is quadratic, Yet, it is evident from Eq. ( 12) that the velocity field should be first order in (w&). The difficulty is due to our implicit assumption that solutions of the wave equation for T’ so derived are general solutions. Recall that for any vector vim: vo

= VP + V x a,

(14)

where p is a scalar potential and a is a vector potential. It follows that V-v0 = V'p and that V x vo = -0'a. The scalar wave equat.ion can only produce solutions derivable from the scalar potential; solutions arising from V x a are absent. It is possible to include the effect of V x a in the scalar wave equation for T’ by selecting a suitable source q,,.o ; and this is the next step in the solution. First, solve Eq. ( 12) for Q and a by applying V and V x successively and by l

WAVE

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MOTIONS

IN

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HELIUM

1%

II

between the pair of equations that are formed. Then cp =

-iPsSIPnQ

(1:)

T’(f/d2,

and a _

PSS

p

(2

-

B’&. p

Pn fl

(16)

II ’

where

gz = f -

(2 - B’)20~/i22.

The wave equation takes the form (f/g)‘(u2/fl))2V2T’

+ T’ = (iST/M’,)q,,o

,

(17)

where 2 u2

We will seek solutions

of standing

= PsTX’l,‘pnCp . waves with

T’ = (& cos as/l

the form

+ 3:: cos ?rylf)e?’

ilS!

where E is the length of one side of the resonator. Note that Eq. ( 15%)represents eigensolutions of Eq. (17) when gn,o = 0. Substituting Eq. (IS) into Eqs. (13, 16) and thence into (14) we find the velocity field at the surfaces x: = 0, z = P. y = 0, y = e is L~~,~(~x= 0) = tio,J.r = 8) = i-v

x a =

cIwo(aje)32

sin

“y/e,

(19)

and uo,,(y = 0) = ~~.~(y =

e)

= j-V

x a=

-cIwo(T/e)31

sin

TX/.F.

(a )

Here C1 = p&‘( 2 - B’)/( p&g’). Accordingly, the velocity field does not meet t.he boundary condition that its normal component must vanish at each surface. It is not possible to remedy the situation by invoking viscouseffects. The assumed solution, Eq. (IS), is, therefore, not satisfactory. A correct expression for 7” consists of an infinite Fourier expansion in the unperturbed normal modes of the cavity.Thenormalmodeswithoutrotationarenlodifiedbyrotatiorlandnolongersatisfy Eq. (17) with yn,o = 0. In other words, the set of Eqs.( 14)--( 17) is inconsistent for any particular normal mode if qn,O equals zero. The set may be made consistent, however, by choosing a volume source which makes ZI~,~(X = 0, 8) and ao,,( 9 = 0, e) identically zero; and this is a sufficiently strong condition t’o determine Q~,~. Then, the eigensolutions of Eq. (17), using the correct value of CJ~.~

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are the eigenfunctions with rotation and the effect of rotation is included in T’ to all orders of approximation; however, the effect of the centrifugal force on the density, which is second order in :c’ohas been neglected. The source ~,,,o can be expanded in unperturbed normal modes of the cavity along with T’, so that qn = -vo,z(x

= 0)6(x)

- vo,z(x = e)s(x) - V”,zl(Y = 0)6(y) =

q0e 4Qt

=

( Q1 cos

- Vo,a(W = 4)6(y)

T2qe +

Q2 cos

TTY/P

(21) +

e. .

)evint.

A similar expansion for T’ must replace Eq. ( 1s). Then, the sameprocedure used to get Eqs. ( 19)) (20) can be applied with the complete normal expansion of T’ to find Q,, via Eq. (21) in terms of 3, . In this manner the wave equation becomes an infinite series expansion in 3, ; but, following standard procedure, the coefficients of normal modes in the resulting expansion of Eq. (17) may be equated to zero, thus yielding an infinite series of linear simultaneous equations in 3, . By truncating the series we arrive at the approximate solution. Using only the first two terms, & and & , the method imphes that 31 and & are to be chosen so t,hat the spatial average of vo over a surface is zero. The neglected terms are necessary to make v. identically zero along each boundary. In a later section the approximation will be carried to higher order and t,he neglected terms will be shown to be small. Applying the procedure out’line here, we find (E?l= SC1w$,/!!” and (3* = -SC&&/!“. Also we have assumedthat the cavity is driven by an external source at .r = 0. It is necessary to include the external source in the wave equation in order that t,he frequency response of the resonator may be calculated. Using qefor the external source we have int = W/( pST)G(x)e-i”t E SW/( pDXTl)e-“2t cos n-x/l, qe = qe,Oe (22) where W represents the heat flux from the transmitter. Only the first term in the series expansion of 6(x) is retained. Returning to the wave equation (17)and substituting for qn,oits equivalent (qo + qe,o),the result is an equation for s1and s2which is homogeneousin cos TX/~ and cos ?ry/4LThe coefficient, of the latter two terms must vanish separately. Thus, there is a pair of simultaneous equations which determine 31and s2. Whereupon it can be shown that 2i[Q2 - (~20f/g)21~~o w/7rpcp u2) 31 = [a2 - (!&f/g)“]” - [8(2 - B’)w,O/?r2g2]”

ww - cws)‘l = (Q2- 3+*)(0’ - Q-2)-

(23)

Here, a0 is the resonant frequency when wo = 0 and W equals 2iflQ,W/( ?~pCgu~).

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Also Q+ M f&f/g + 4( 2 - B’)uo/?r’g’,

(24a)

CL z !&f/g - 4(2 - B’)uo/?r2$.

(24b)

Note that the validity of Eq. (23) is not dependent upon the magnit’ude of (we/Q). Accordingly, the single resonance peak of the stationary resonator is split by rotation into two peaks with a separat!ion AQ = (S/7r’)(2

- B’)oo,/g”.

(2.j)

The splitting and t,he response curve are symmetrical about t,he unperturbed frequency. Recall that, to first order in we/O, 1f / = 1g 1 = 1, so that we may write Eq. (23) as ~(9’ - n,’ + iBwo%) ‘l = (~2 - Q+” + iBwoQo)(Q2 - L” + iBwoQo)

( L'(i

)

when terms in ( w~/Q)~ are neglected. To the same accuracy we have -;w[8(2 - B’)Qouo/,r’l ” = (n2 - Q+2 + iBwoQo)(Qz - L2+-iBwj~)

’

(51

If the resonator is not square the same procedure may be used with trial func*Cons of the form cos (r.r/&) and COY(Q/C,). Then

wzD2 - &/f/Y)“1 (L’S) J1= [!? - (Q,f/g)2][Q2- (Q2,f/g)2]- [S(2 - B’)wofLQt,/x”g”]” and W, differs from W in that Q2,replaces Qo. The t’wo unpert’urbed resonant frcquencies are 12,and 3, . To first order in we/Q, Eq. ( 28) has the form 31

=

w,(Q” - q2 + dB woi&,) (al- - n;,+ + iBwofio) (St” - Q:,- + iBw,,iZo) - a:( A@,)” ’

7-i

( “‘I ’

where fro = ( Q, + Qy)/2, AR,, = (12, - Q,) and

We have assumedas before that the x mode is externally driven. The results for the case when 8%f &, differs in a fundamental way from t,he responseof a square. In the former the unperturbed resonancesare separated in frequency and the effect of rotation is to increase the spread bet’ween the peaks. The additional separation is not linear in wobut is a complicated function of AtIn and w. . Note t,hat when there is no rotation there is only one peak at R, while t’here are two peaks when wg# 0. The frequency-response curve is no longer syn-

168

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metrical about the central minimum when 4, # t8 but Eq. (29) reduces to Eq. (26) as 4& 4 4, . The above analysis could be generalized by including all of the dissipative terms in Eq. (17) and by extending the series of Eqs. (IS), (21) and (22) to higher order. However, it is not appropriate to consider these refinements without first investigating some other effects which may be quantitatively as large, Therefore, the next section is devoted to deriving a solution of very general applicability. Equations (26) and (29) are actually good approximations to the general case in which all the second-order effects are included in a consistent manner; the corrections are all less than 5 % except for conditions not usually encountered experimentally. IV.

A.

DISPERSION

THE

GENERAL

SOLUTION

RELATTONS

The various types of wave motion that may be excited in rotating He II may be exhibited by considering the dispersion relations of Eqs. (5)-(S). If the equations of state p’ = --paT’

+ (y/z&~’

(31)

and 25” = (C,/T)T’

- (a/&’

(32)

are combined with Eqs. (5)-(8) it is possible to eliminate T’ and p’ in the motional equations and to write them entirely in terms of the six velocity components. In Eqs. (31) and (32)) o( is the volume thermal expansion coefficient, y is the ratio of specific heats, and u1 is the adiabatic velocity of first sound. By considering each velocity component proportional to exp i( az + 6y + cz - Q2t) and substituting in the mot,ional equations, the resulting set of homogeneous equations defines a 6 X 6 determinant which must vanish. The resulting equation is the dispersion relation. It is evident that the determinant has symmetry in the (x, y) plane and that there is no loss of generality in choosing 6 = 0. When b = 0 the determinant is as shown in Table I. The resulting equation is too complicated to permit easy display of all mechanisms of importance but several subcases will be worked out which illustrate that there are four types of waves which propagate and that each type is, in most cases, modified by the others. To simplify the presentation we will set all dissipative terms equal to zero. This has the effect of suppressing the boundary layer solutions. Unless dissipation is large it has a negligible effect on the real part of the wave number; if the dissipation is not small, then experiments are not practical since the Q’s of the various modes are small.

WAVE

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TABLE THE

DISPERSION RELATION WITH THE DEFINITION

HELIUlVI

169

II

I

FOR PLANE WAVES NEGLECTING DISSIPATION, OF SOME SYMBOLS USED IN THE TABLE AND IN

TOGETHEI~

SEC. IV

D, = w&n/~ Ds’ = woB’pn/p D, = --&P./P D,’ = wo(2 - B’P,lP) Es = -D. - (k,%sD,)/2w,~s E,’ = 2w,, - Da’ + (k.%[l - (D,‘/2wo)ll~,l

En = -Dn + (~z2d.M/2w,p. En’,’ = 3wo - D,’ + (k,%~,[l v,

=

TslPs

qsn

=

El

v =

-

)InlPn

Psi3

qns

(Dn’/2oa)?/p.l

llss

=

P&/P,,

=

P&3

-

P.%IPrt

= 52/h - 2p&/p, f p&/pn + 4v/3 F, = in + E, F,, = iR + D,,

q on IL,

=

p’/p

-

ST’

1~.

=

p’lp

$

psST’/p,,

$.

=

mQ.v

x

-

qssV*v, -

amv-vs

~anVv,, -

snnr~vo

vn

Consider firA, the case of propagation in the direction of the rotational As: a = b = 0. The dispersion relation is ( iI2 - 4c.d:) [cl” - (2wo + v, c2)] [!a4 - ( 63 + e )n2 c” + ( 6x - @lo) c4] = 0

aud has several solutmions : Q = 2WO) c = [(a -

2wo)/v.y2,

i x1 1

170

SNYDER

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and c = ~

63 + e f

I.

l/(63 + e)4 - 4( CBC?- cm> 1’2 2(CBe - cm)

(36)

Equation (34) describes pure inertial waves; Eq. (35) is the expression for pure vortex waves; the two roots of Eq. (36) represent first sound modified by second sound and second sound modified by first sound. Equation (36) also occurs when the fluid is not in rotation and has been studied in that context by Dingle (14). Note that if, in Eqs. (31) and (32), CY= 0 so that there is no coupling between the pressure and temperature fields, then (a + e) = u12 + ~22and (M? - a~) = (ulu2)“. In that case c = Q/u1 or fi/ue . In general, a/p << C,/T and LY<< y/u12 and t’he coupling between the pressure and temperature fields is small. See Dingle (14) for a more detailed discussion. For the particular case when both the sound wave and the vortex waves travel in the axial direction there is no coupling between them. This is why the dispersion relation, Eq. (33), factors as it does. The next degree of complexity occurs when b = c = 0. Then the dispersion formula is: Q = ( cl2 - 4w02)1’2

(63 + e> f [

I.

&3 + e>2 - 4(@C + @a>) l/3 2(&K? - aa>)

(37)

The inertial and vortex waves do not appear since they cannot propagate perpendicular to the axis of rotation. However, the inertial wave frequency is a lowfrequency cutoff for both first and second sound. The coupling of first and second sound is the same as in the nonrotating problem. The general case of arbitrary a and c is too complicated to consider, but if a = c = t all the effects are present and the expression is not too formidable: [(cl2 - 4&) +

[2Q2 -

-

(2Q2 - 4wl)

c?f2/fi2][Q2 -

(2wo - d”)“]

(2wo + V,f2)( 2wo + 2v,f2)][( &XL? -

cm)( 2Q2 -

If we set vs = 0 in Eq. (3s)

so that vortex

4002)f4/Q4

(38)

( !a2 - 4w&BJf2/522] = 0.

waves are excluded we find:

and the first and second sound branches are found similar to the nonrotating case but with a cutoff frequency of 0 = 2~0. There is appreciable dispersion near fi = wo . It can be shown that in the general case of a sound wave propagating at an arbitrary angle 0 to the x axis, the first bracket in Eq. (39) is ( Q2 - 4w0’)/ (a” - 4w02 cos2 0). When vortex waves are allowed, it is not possible to factor Eq. (3s) or to

WAVE

MOTIONS

IN

ROTATING

HELIUM

171

II

writ#e down the solution in a simple form. Since Eq. ( 3s) is of fourth degree in f’, me expect four roots for ?. However, Descart’es’ law of signs indicat,es that t,here are at, most three positive roots; the negat’ive root represents a highly damped wave. The three roots correspond to ( a) first sound, (b) second sound, and (a) vortex waves. Each is modified by the others and each has a low frcquency cut off at Q = 2~0 . For special values of v, , ~1 and ~2 it is possible to have peculiar t)ypes of waves for Q < 2~00but it does not appear that these have any practical importance. In summary, the dispersion has four branches corresponding to inert,ial waves. vortex waves, first sound, and second sound. The branches have a functional dependence which differs only slightly from the relations obtained by making the simplifying approximations which lead t’o pure waves. There is a low-frequency cut.off in the vicinity of o. for all four types of waves except in exceptional circumst’anres. In what, follows we will be interested in wave motions when the const,raint l!, = 0 is applied; therefore, it is necessary to study the dispersion relation whcll Pz = 0. For t.he case a = 6 = 0 we find (52” - 4w02)[QZ - (~2WO+ v, c”,] = 0.

(40 1

Comparing E:qs. (33) with (40) we note that pure inertial waves and pure vortcs waves are possible but that the sound field is supressed. This is t,o be espetted since sound is primarily a longit,udinal wave motion and the condition P; = 0 prevents motion in t,he z direction. When b = c = 0 the constraint does not effect the relation and Eq. ( 37 ) still holds. The third case studied above, b = 0, a - c = t leads to:

C&f6 + [iwov$+ (w2- atfqf - (Q”- 4woy(az + e’)f + (Q” - 4woZ~p2 -

i2wo + Yf’,“]

= 0

(11,

and this expression differs considerably from Eq. ( 3X 1. Recalling that pure sound waves are longitudinal and that both pure inertial waves and pure vortex waves arc transverse? it is not surprising that the unphysical requirement t,hat 11~= 0, changes the nature of these waves drastically. If we consider waves for which v, = c’, = 0, me find for a = b = 0 that c is given by Eq. (36), so that sound waves are allowed and have the same dispersion relation as in the non-rotating case. Since our equations are all linear n-e can combine the sound waves found when c, = v, = 0 with those allowed by zsi = 0 and get all the sound waves allowed by Eqs. (5)-(S) which propagate either parallel or transverse to the axis of rotation. Moreover, these waves will have the same properties as the solutions to the full set of equations. The situation for inertial and vortex waves is similar to that for sound waves: the propagation of t,hese waves parallel or normal to the axis of rotation is t,hc

172

SNYDER

AND

WESTERVELT

same whether v, is constrained or free. We conclude then, that, so long as we confine our interest to those waves whose wave vector is either parallel or perpendicular to the rotational axis, there is no loss in generality in setting v, = 0 in the approximation that the coupling of waves due to dissipative effects can be neglected. Dissipative coupling is small, even within boundary layers, and will be considered in future reports. By neglecting disipation we have reduced the degree of the equations of motion and thus the order of the dispersion relation. The omitted roots refer to the damped waves necessary to fit the four types of waves discussed above to the boundary conditions; they are the boundary-layer solutions. Note that the major source of dissipation in the interior of a resonator is attributable to the terms in Bwo ; thus, neglecting this effect neither changes the degree of the dispersion relation nor reduces the number of roots. Having found the relation between D and the wave vector, it is possible to derive the ratios of the velocity components for the different types of waves. Only wave propagation along, or perpendicular to, the rotational axis will be considered. For either first or second sound waves traveling in the direction of 00, there is no effect of rotation; they are purely longitudinal and the ratio of velocities is vn/v,

=

1 +

(~P/P~s)(u1”u*“)/(u1”

-

u2”)

=

(42)

a1

for first sound and 21,/v, =

-Ps/Pn

+

(aP,‘SPn)(U12U22)/(U12

-

d>

=

a2

(43)

for second sound. Plane sound waves traveling along the x-axis are not purely longitudinal but have a transverse component : v,, = -i( 2w&)vsi: ; vny = -i(2wo/n)vm ; VII, = v,, = 0. This is why v x a is not zero in this case and is also the reason why these waves violate the normal boundary condition, as we saw in Eqs. ( 19, 20). Notice that the solution, Eqs. (23, 2S), are such that at every point v, = -i( 2w&)v, , The ratio v,,/v,~ is not effected by rotation and is given by Eqs. (42, 43). For both inertial and vortex waves propagating along wo the results are: ; and v,~ = &iv,, . These waves are purely transverse. V nz = vsz = 0; v,, = ~iv,, If we can set V-V = 0, then, inertial and vortex waves are transverse for any direction of propagation-the condition V-v = 0 suppresses first and second sound. Recall that the dispersion relation was simplified by neglecting dissipative terms and the results of this section are valid only when this procedure is valid. The largest source of dissipation is due to B thus, it is not generally possible to neglect this term. However, the inclusion of B in the dispersion equation increases the number of algebraic manipulations to such an extent that recourse

JVAVE

MOTIONS

IN

ROTATIKG

HELIUM

must be made to a computer solution. We will not attempt such as Eqs. (42, 43) for the inclusion of R. B.

SOUND

11

173

to correct equations

WAVES

B solution of Eqs. (5-S) includes all effects except those associated with finite-amplitude waves. Under certain restrict,ions we cm use the same approuc&h as that of Sec. III to solve this set of equations. The first, st,ep is to tind the scalnr sncl vector potentials for each velocity field: v, = VIA + V x a, 1

I14

11, = Van + C x a,, .

( 4.7 I

i

Again it, is assumed that all variables vary sinusoidally in time? i.e., as e- ‘!!I. Throughout, the remainder of the paper, unless stated, this exponential timck dependence will be suppressed. The result’ of taking the divergence and curl 11t Eqs. (5) and (6) is a set of eight, scalar equat8ions. The propagation of waves :IS predicted by these equations is anisotropic since t’he axis of rotation is :L pr(‘-ferred direction. Terms in the spatial derivatives of 11~couple the eight equations. (In this section if a subscript is omitted, it is implied that the result holds for :mJ subscript under consideration.) The equations can be divided naturally into two groups of four, and if avJ& = 0 the two groups are independent. In facat it’ aI1 spaCal derivatives of v, do not vanish the method of Sec. III cannot~ be appli(~(l unless these derivatives are known explicitly. When it is not desirable to restrict the problem as in the last paragraph, it is still possible to solve Eqs. (5)-(S) directly by reducing t’he set’ to six c~oul~lccl ordinary differential equations. It must be assumed, however, t’hat each of thcb variables (in this case, the velocity components’) are separable and that tllcb dependence of t,hree out of the four fact’ors making up the variable are 1mJwtr. This “straightforward” method requires a great deal more algebraic computntic,n t’han the previous method. There is only one effect, that can he dealt, with by I hcl conventional scheme that, is intractable by the procedure of Sec. III : that is tilts coupling of inertial and vortex waves to sound w:~vcs-~~a small second-ortl(br effect. In the previous subsection it was shown that, for waves traveling parallel 01’ at right angles to the axis of & , the dispersion relation of :I particsular wave i.s t IIP same whether or not, v2 = 0. The ratios of t,he velocity components in the various waves also indicated that the coupling of inertial srld vortex w:lves t,o sountl waves does not’ occur for this particular direction of propagation. Considering the complexity of the problem it, is desirable to design cspcrimerits so that v; = 0. In all the experiments known to the aut’hors the armtlgc~merits are suc*h that 21, = 0 except in boundary layers. Therefore, our nttdysis

174

SNYDER AND WESTERVELT

will be confined primarily to caseswhere all spatial derivatives of v, are zero or where v, is a function of x only. Let us return, I~OW, to the t’wo groups of four scalar equations which result from taking the divergence and curl of the equations of motion. Replacing ~‘/d,~.~ by -k,‘, we can write down the solution of the first group as follows:

[F, Ds En F,, E,’

D,’

& E,,'

0s') f

v-v,

D,‘l 1

V-V,,

' h0.V x v, j i 6o.V x v,

-D,

-F,

1E,,’ D,,’ - E,

-F,

‘I

'"'"I v2*n

=

!46)

O I I V2h J

where the symbols have been defined in Table I. Equation (46) may be solved by successive approximation 011 the term & . First set v = 0, (v occurs only in &,) . Then use&*v x vn to find a first approximation to & and proceed to higher order if necessary. In most applications even the first-order term in & is negligible. If we define the 4 X 4 matrix in Eq. (46) asA and its determinant by 1A ( and also let Arnnstand for the cofactor of the element A,, of the matrix, we can write the solution as follows: CF~= (&k

+ tenon + A”I&)//

cpn= &~s

+

~5 o*a, = -(dads

A”22h

A I;

(47) (45)

+ ~z,~v)lI A 1;

+

&2+n

+ &i~,)lj

A 1;

(49)

+

&zl(/n

+ &Wl

A 1.

(50)

and &,*a, = -(hh

It will be shown below t#hat under the restrictions imposed here that the sound

field is determined completely by Eqs. (47)-(50) once the pressure and temperature fields are specified. C. INERTIAL

WAVES

AND VORTEX

WAVES

In the remaining set of four equations only four variables appear &u,/dz and &J,/&. In matrix notation the equations have the form:

ah a2 (E,’

D,’

1F,

D,

E’,’

D,’

I E,,

F,

-Es Es’ -En E,’

-Da Ds -F, Dn’

ah, ax ah, a2

a&l, -. a2

0 I 0 2 ah vv -

a2

vv 2 -ah, a2 I

.

(51)

WAVE

MOTIONS

IN

ROTATING

HELILW

II

175

It is not appropriate in Eq. (51) to set v = 0 since the terms retained are of the same order of magnitude as the terms which would be omitted. It is clear that the variables treated in Eq. (51) require exponential solutions and that r’? becomes -k,‘. It was shown above that the inertinl and vortex waves which arise from Eq. (51) cannot travel in the X-Y plane: therefore, k~ = 0. Upon replacing the Laplacian by t,he square of the wave number and rearranging terms, we (‘a11 write Eq. (51) with no terms on the right. The new matrix will be denoted b> C and it differs from the matrix of Eq. (37) in that CaA = -F, + vX.,’ :mcl c,, = -c,, . Set,tiug the determinant 1 C 1 equal to zero yields the dispersion relation for k, vs. 12.Sinre 1C 1 = 0 has four root s, there :Lre f( )ur possible types ()f waves. As a first approximation we can set B and R’ equal to zero in the dispel,sion relation to obtain the wave numbers k and t!he penetration dept’hs X (of t#hc waves. In the limit B, B’ + 0

The numerical value of v, is of order 1O-3 cgs so that, k has a large imaginnry part. except for the special case keg and 0 > 2~0. The penetration depths for all but the special case are less than 1 mm at all temperatures for commonly used values of Q and w. . The value of X:,’ for Q < %wois t,he same as Xii’ but for 12> kc, it is infinite (but only because we neglected B j. The waves I<$, are examples of boundary layer waves; they are inertial waves with damping. The waves k’:’ art also boundary layer waves except for t’he case k$’ for D > 2w0 which is :L pure vortex wave. Both types of waves were first studied by Kelvin (1~7). The inertial waves occur when a fluid in solid body rot,ation is disturbed; while the vortex waves occur in a velocity field of the form ual/~ when it’ is disturbed from cquilibrium. For all of these waves it can be shown t’hat ~1~= &z~, ; they are circul:ulp polarized and transverse. The penetration depth of pure vort,es wave?: X, is limit,ed by the dissipation inherent in B and it can be shown that A, = p[(n - 2c&J1r2(

Bp,,ll)

and X, is of the order 1 cm in many experiments. The four dispersion relations resulting from / C j = 0 may be found in a revi,\, :u%icle by Andronikashvili et al. (16’). This article also includes a discussiori ot solutions applicable to bodies oscillating in rotating He II. It, is shown in Ref. ( II;) that t,he effect of B and B’ is to decrease the penet,ration depths X:,5’ :rntl Xl:’ and also to couple the four different types of waves. Computer calculations are presented in Ref. (16) to show that t’his coupling is nearly negligible ill ~11 the experimental work reported in the literature. III the work reviewed hy

176

SNYDER

AND

WESTERVELT

Andronikashvili et al., it has been assumed that the liquid is incompressible. It has been shown above that for wave vectors along & this is a satisfactory but not a necessary premise. The components of a, and a, , which remain to be determined, follow from Eq. (51). Signify the 4 X 4 matrix of Eq. (51) by B, the determinant by 1B 1 and the cofactors by &, , and notice that i-v x v = -@/ax. Then we have !*a, = Y(&a2hy/a2

- Bs4atinz/a2)// B 1,

i*a, = --v(Bloav,,/a2 i-a n =

- B14ah/aX)/(

043av~yIa~ - mvnzia4il

B 1, B 1,

(55) (56) (57)

and i-a, = - V(BZ3ahy/aZ - Bz,av,,/ax)/j

B (.

(55)

The terms dzJ,,/dx and dv,.,./dz which are necessary to complete Eqs. (55)-( 58) can be determined from the matrix C and t,he boundary conditions. The matrix C defines a homogeneous set of equations and the ratio av,,/dz/dv,,/& can be written as a ratio of cofactors of C. The boundary condition can be used to complete the determination. D. THE TRIAL SOLUTION The most general solution is a sum over all the different types of waves discussed in the previous subsections. For each type we must also sum over all values of the wave vector. The boundary conditions will specify the correct amount of each wave in the summation. Equations (46) and (51) and also the equations for the pressure and temperature that can be derived from the former are of the Helmholtz type. It can be shown that all the equations which occur in the analysis are separable in all their independent variables. The sound field will be of the form

and V ZI =

C

EjVSj(Z).

(59c)

X)ZJ2j(y)V4j(Z),

(6Oa)

Likewise, for the vortex waves we have 2’211

=

V yap =

C

djVlj(

iv,11 and oZlr = 0.

(Gob)

WAVE MOTIONS

IN ROTATISG

HELIUM

II

17-i

The inertial waves have a structure similar to vrr and will be designated by vrJl . Boundary-layer solutions must also be added in the form

The complete solution, then, is the sum v = vr + vI1 + vrzr + vIr- . Under our assumptions vZI and vUI are determined completely by ICqs. (-Ii I-(50) and t’he condition at the wall t’hat the normal component of the velocity equal t,hat of the wall. (A more complicated condition pertains if there is a net flux of heat through the wall.) There are no effectIs of rotat,ion on uZJother t’han the compressive effect, of the centrifugal force and it,s solution may be found direct lp from Eqs. ( 5j-(S) by assuming all variables to be function:: of x only and that 00 = 0. Likewise, vlr and vrrr are determined complet’ely by ICqs. (.55jH 5X) with the condition of vanishing normal component, of velocit,y relative to t,he w:tlls of the cavity. Since we are considering propagation of vortex waves and inerti waves along t#herotational axis only and since for these waves P, = ill, , t~hon 2’, = I’, = ~,,.~,11 at’ all walls parallel to the rotational axis. There are no conditions on vrr and vlrr at the boundaries normal to ov . Because ~VCare considering oscillations with wave vect’ors that are parallel ()1 perpendicular to w. only, the compressibility of t’he liquid does not8enter intO the analysis of vortex and inertial waves; the results of AndroIlikashvili et al. apply. If, however, the wave progressesat a nonzero angle with respect of 00, the fact that Y-V f 0 modifies the results. Also, upon reflection at, a surface inclined to the rot’ational axis, the wave number and amplitude of both vortex waves and inrrti:ll waves may change. [See Philips (17).] We shall not consider in this paper how v IV may be found but a future art ic*lo will be devoted to this problem. E. SOLUTIONS

FOR THE SOUND FIELD

The sound field can be specified by t.wo scalars: the pressure and the ttbnlperature. For a geometry which is rectilinear, the normal mode expansion will be of the following form:

17s

SNYDER

AND

WESTERVELT

and p’ = C

6,,,

m,n

cos (mrx)/&

cos (n7ry)/tu

Pt.

(63)

If the normal impedance of the walls is not sufficiently large, then it is necessary to generalize Eq. (59) to T’ = c 3,,, cash (2rgz:n)/& cash (2ag,y)/& rn,ll

eWsr

(64)

and a corresponding expression for p’. To a first approximation, gz = (k/2) + (K - iu)2&/nn1X and there is a similar expression for gI with VL replaced by n. Here (K - iu) is t’he acoustic admittance ratio of the surface: pulv/p’ or puev/T’. Also, X is the wave length of the disturbance. The expressions for T’ and p’ determine & , tin , and #y , and these in turn are used to find v, and v, via Eqs. (46)-( 50). First, eliminate tiy from Eqs.( 33)-( 36) by successive approximation. Thus,

so that Gv h” v(A”~IV*~, + A”d’2&,/1

A I.

(66)

A consideration of the relative magnitude of the dissipative parameters shows that it is not necessary to go to higher order than the first in evaluating J/” except for the boundary layer solutions vrv . Next, it is necessary to relate $+ and tin to T’ and p’ and this may be done by combining Eqs. (31) and (32) with Eqs. (7) and (8)) so t,hat v=v~=~~Q[(--cY+&)T’+(~-~)~~]+cw~T’

(67)

and V-v,

= iQ[(-a-$)Tr+(~+?$-)p’]-~zW2T’,

(6s)

where 8 stands for x/pXT. Since C,/ST >> cxand y/pu1* >> t~/pS in the t’emperature range where secondsound is well defined, it is usually possible to drop terms in arl to the degree of approximation dealt with here. Also, it is appropriate and consistent to neglect terms in 87. Thus,

and

WAVE

MOTIONS

IN

ROTATING

HELIUM

II

179

The wave equation for the sound field is derived by equating V-V,, and V-V, from Eqs. (67) and (GS) to their equivalents in the form of Eq. ( 46 1. However, Eqs. (67) and (68) must be complemented with the source terms. The driving force is best introduced as a source at the surface of t’he transmit,ter. If we select t!he plane s = 0 for the transmitter, then t,he esternal sour(*c is the normal mcdr expansion of the delta function ate .r = 0. Thus,

=

?(I’

an tl YC,S= ?(,I’ -

WjpST

7.2 I

where L’ is the velocity amplitude of the transmitter, and W,lpST has been defined above. When 1; or TV are functions of x or 1~t,he series expansion in Eqs. ( 71 ) and (72 ) will be more complicated. If ye were the only source in the wave equation, the only modes excit,ed would be t,hose which have no 1/ dependence. It’ follows t*hat the velocity field, in that case, would be a function of z only. Equations (5) and (6) clearly show that it is not possible to have U, = 0 everywhere when wg # 0 and u,< # 0. It, is nccessar) to add sources to t.he wave equations as explained in Sec. III. To determine the sources, first observe that, while dp-r.ld.t vanishes at, .r = 0 and s = C, , and that, +/a~ is zero at !/ = 0 and !/ = &, due t#oour choice of normal modes, the velocit’y contribution of v x (D, . a )k violates the boundary ~OIIditions. By direct, calculat8ion, when Eqs. (62) and (63) are substituted into 151s. 117’)-( 50) and the result applied to Eqs. (44) and (45 i, we find u,(:r = 0, y) = --coo C (3,,,‘2 m,I,

+ (P,,,,5$)(nn!lJ,)

sin (ring) ‘fV

( i:i i

+ @m,n~)(7r~7r/C.c) sin (7~7r.r:) /tL .

( 71 I

and W .r, !I = 0) = b412

( 3,,,T

Also ! z&(2 = l,, 2/) = (-l)me’J.r

= 0, !/I

( it5 1

an d Q,(s, y = e,, = i -lJnL’,J.C,

j/ = 01,

( 7ti 1

where 2 and 9 are defined in Table II. A distribut,ion of sources cm bc chowr which makes t,he right hand sidesof Eqs. (73) and (71) equal to zero and t,his u PII.

180

SNYDER

AND

WESTERVELT

TABLE DEFINITIONS

OF SYMBOLS

USED

in/we+ -3

D,’

&I

A=

@ =

@ =

@ =

@ =

@ =

@ =

in/w,

+ D,

E’.’a

D,’

CL’

D,’

F”

EL,’

D,’

D,’

-F.

-D,

D,,’

-E,

En

E’ n

Eh’

-F,

WAVE

EQUATION

ES’

DS’

-32’

D,’

-&l/w0

-

B,

--DS

-E,,

-&2/q

-

D,

DE?

E,’

DS’

D,’

-F,

-D,

-F,

D,

--IX,

-F,

D,’

F8

@ =

-D,

E’ n

-E

En

Fn

D,’

E,’

D,’

-D,

E,’

D,’

-F,

(En

Fn

E,’

D,’

-F,

El’

D,’

---En

n

II

IN THE GENERAL

@ =

-F,

@ =

En’

D,

Es’

D,’

F,

E,’

D,’

D,’

-F,

F,

D,

D,’

E,

F,

Dll’

E,’

D,’

-D,

8

=

1

@ =

ES’

Es

-F,

-LA

E,’

-h’,, Fs

D,

-F* D,’

E,’

D,’

-DS

Is,’

D,’

1 Fs

D,

-F, ES’

E,’

D,’

-F,

E’ n

D,’

-E,

1 Fs

E,’

D,’

1 E,

E,’

Dll’

1E.’ F, F, Ds

-D,

@ =

DS’

-D, E,’

E,

F,

E’ n

E,’

D,’

-FS

OF SEC. IV

WAVE

MOTIONS

TABLE

IN

ROTATING

HELItJaI

II

IS1

II-Conhwxi

dition is sufficiently strong to specify the source cwlnplet,ely. Accordingly, t\l(> velocity produced by the source at R: = 0 must be -v,( :r = 0, !I ) and siulilal],v :~t, the other three boundaries of the resonator; \vhen(ae

- 1$( !/ = 0)6(y); hut8 q can he expanded in the normal modes of the system q = t: Qm CO8 (prs)/& P4 Combining

Eqs. i 70)-( 75) we see that

cos (q7ry)/-II,, .

( iS 1

182

SNYDER AND WE~TERVELT

and also

(81) where S,,, = 3,,,X + @,,,\l-‘. All values of Q,,, not accounted for by Eqs. (79)(51) vanish. It is now possible to write down the wave equations: P~vsp’+ TI,,T’ + P2,sV2p’ i- Td’T’

-I- P4J4p’ + TQ4T

= iQdqe,s + qd;

(82)

and P~
-I- Pz,,V’p’ + TQ2T’

+

Pd4p’

+ T4,nV4T’ = iQp(qe,n + qn);

(53)

where the coefficient,s PI,, through T 4,” are defined in Table II. The final step in determining the general solut.ion for the sound field is to replace p’, T’, qe, and q by their expanded forms given by Eqs. ( 62)) (63)) (71)) (72)) and (78)-( 81). In the resulting equations terms of each normal mode may be equated, and this leads to an infinite set of linear equat’ions in J,,, and @,,, which are to be solved simultaneously. This completes the formulation of the general solution of the sound field. F. SOLUTIONS

FOR THE TRANSVERSE

WAVES

The velocity field for the normal modes of vortex waves and inertial waves must go to zero on those rigid boundaries which are parallel to the rotational axis. Under the restrictions we have imposed, the transverse waves must be generated by motions of the walls which are perpendicular to oo. We will consider that the driving force is due to motions of the wall z = 0 and that all walls have infinite mechanical impedance both t,o normal forces and shear. Then, in rectangular coordinates the velocity field has the form 21, = e--in2lsin [(PZz)k,] 2

W,,, sin (naz)/&

sin (nzny)/& ,

(84)

where B, is a root of the dispersion relation ) C 1 = 0. Also, recall that v, = iv, . Since we are not considering viscous and thermal boundary layers in this paper, only one of the four root’s of ] C ] = 0 can be used here for k, , i.e., k:-‘. In this mode of oscillat,ion Eq. (37) indicates that ( v, 1>> 1vn 1.The coupling of v, to V, is small and depends on D, and D,‘, both of which approach zero as T approaches zero. The coupling is linear and we may set V, = CV,where Eis a constant, possibly

WAVE

MOTIONS

IN

ROTATING

HELIUM

II

1 s3

complex, for each value of L?,and for a given mode. The value of e may he found by substituting Eq. (84) into Eq. (51). It remains to find v,,, in terms of the motion of the boundary. The boundary condition on v, is given by Eq. (11). If we assume v, = EV, we cannot use t,he usual condition v, = v, since the latter condition would be imonsistent with Eq. (11). The trouble lies in neglecting the boundary solutions which arc the threeroots of / C 1 = 0 other than kb-‘. In the boundary layer all four solutions foi k, must be used-the non-zero value of v, at the surface due to Eq. (84) k rancelled by t’he other additive terms in v,, . Outside the surface layer Eq. (81) together with vn = ev, gives the correct solution. Proceeding now t’o find v,,, it is possible to show that l
+ b’)[(a%

-

~‘,$/a= l/(a’

+ b’)[(bll

+ a%)u,,, + (b% - a%)/~,];

(85,1

b%)v,,, + (0% + b9L)z*,];

and ( 8li 1

where the new symbols are defined in Table III. At 2 = f then, z’,~ = ~1,~= z’,,> = hl, = 0; but, at z = 0 we must have sin e$, C ‘u,,, sin (n7rz)!B, m,,I

sin (7~y),:f~

= t’,,, ,

1871

and

Obviously Eqs. (87) and (88) cannot be satisfied for arbitrary ~b? . Again the difliculty arises from neglecting the viscous solutions and in this case the solutions of the sound field may to complete t)he set’. It’ is possible to decompose z’,,, and z*,, at, sum of ckularly polarized motions: P,,, I2 = izlUdVand P,,,‘L = TABLE I~EFINITIONS

OF STMBOLI;

USED

III

TO SPECIFY THE BOTND.\RT VORTEX LINES

Ct = -1 + B’p,/“p 6s

=

BP~/~P

u = 63 + 6.b,k,2/2wo 1: = a + @,&2/2WO

values of z’,~, :ultl boundary layer also be necessary each point into 3 - z’z’.?+~ and aimi-

- iyk,/2w, - ir’k;/2wo

err = v,*i at surface [‘.I, = v,=j at surface l’,,r = v,.i f’,., = v,-j

C~NI)ITIONX

ON

SNYDER AND WESTERVELT

1%

larly for 2& . Only t’hat part of v,, with the correct helicity generates vortex waves; the remainder generates damped waves. This procedure when applied with Eqs. (87) and (88) completes the determination of the velocity field for transverse waves. V. SPECIFIC

A. SOUKD WAVES

IN A NEWTONIAN

EXAMPLES FLUID

As a first example of the methods developed above we will work out the details of the velocity field for a resonator of square cross section in the plane perpendicular to o. and driven at the x: = 0 plane. It is agreed that the walls all have infinit’e impedance. The fluid is an ordinary fluid like air or water. The velocity of the transmitt’er located in the x = 0 plane is taken to be v,

=

qe

-i$tt

zz 0

J/,/s < y < 3q4

@a)

P/4 > y > 3-e/4,

(89b)

where v. is constant. Since there is no motion of the boundaries which are normal to o. no inertial or vortex waves are generated. Equations (.5)-(8) can be reduced to the correct form for sound in an ordinary fluid by setting v, = B = B’ = ps = ,t = 0 and P,, = p. Then, f” = 1 and g2 = f” - 4w02/fi2,and to first order in v we have

and

(91) where *=

[

4 ivQ 1 -;+y&P’

1

As an assumedsolution use: p’ = 6+ cosIrx/e then,

and

+ p2 cosTy/e;

(92)

WAVE

MOTIONS

IN

ROTAlYXG

HELKM

II

Q, = no.f/g rt 8wo ‘a2g2

1s.s

( 97 1

k. = &+fto~12+ (7 - 1j~n~,!(~~C,~~’), ( 9X

A-* = ko f

i

( l12voo/37r2poul”).

Terms in wov have been neglected in the final expression Eci. ( !Ni) . Notice that the only effect of using a nonuniform transmitter is to decrease t,ltc response in proportionate to the area of the transmitter. j4 uniform transmit tcr generat’es terms of the form Q”,,~ cos pm/tZ only, but t,he uneven source can generate terms e”,,, cm p~x/4Z cos pry/t. However, the response of ipl contains :L Rionnnce denominator of t’he form l/(Q” - 1202j2,SO that unless the driving frc>quency is close to the resonance frequency of one of the Cp, (/ ) modes the eff(lct of unevenness is not, appreciable. B.

FIRST

POVXI)

IN

HELIUM

II

To find the proper motional equation for a wave driven by :L surface mnilltsinrtl at) constant temperature but vibrating normal to its plane, mult,iplying IGl. ( J I by ps and Eq. (6’) by p,, and add the result,ing equations. To find t,he c~ontinuit> ecluat’ion mult’iply Eq. (7) by pn and Eq. (S) by p. and add. This proc*cdurc elknntes 7” from the equations. The equation so obtained are identical to those for sound in an ordinary liquid if we replace +$v 11p i a:;q,,,'p + &lp ). The result s ot subset. VA apply. C.

SECOND

Soum

IN

HELIUM

II

If t,he transmitter is impervious to the superfluid, rigid, and heated by a11 oscillating elect’ric current, the surface varies in temperature at a frequency twicse that of theheating current and half the power input goes iuto second souud. \Yv assume the walls of the container are perfect insulators. The motional equations are found by subtracting Eq. (5) from (6) and setting P,,P~ = -PIP; in the equ:ttion obtained. Also, apply the same procedure to Eqs. ( 7‘) and ( S). The potentials arc

186

SNYDER

AND

WESTERVELT

an d

(100) to first order in V. Here, J/ = E ST’ - & ‘?&& T’ - -f!?

V’T’,

(101)

PST

where 417 --”

y2~ps Pn

(

-

31

+

3P

52

+

(102)

PE3

P

>

and the last term in Eq. (101) may be dropped to our approximation. out the standard procedure,

Carrying

Ql = f ( 8~02/n2u22)00~a~z 2 1

(103)

for a square cavit’y and, if the z = 0 surface is driven uniformly, In the final expression

Q$

2imo W( si” - Qo2f2/g2+ iOo2ko} ” = TU? PC, ( fF - fLz + islo2k.} ( !A2 - fl+2 +

iQ202k+)

’

BW/XT.

(104)

we have used &* = flof/g f 4( 2 - B’)wo/T2g2, k-0

=

j-2~ov2/g2u22

+

fioxlPcpu22,

(105) (106)

and k, = ko f

S( 2

-

B’)[vz

+

psv//$h&2g2.

(107)

The largest correct’ion to Eq. (104), due to neglecting terms in vp,wo,/pQo , is the loss of a term in Eq. (107) of the same order of magnitude as t,he last term. Observe that if we set 8 = v? = 0 we retrieve Eq. (23). To justify Eq. (23) we must, show that Bwo,& >> liu , k, . Since B ,P 1 and wo/Qo c lo-’ to 1O-3 in most experito assume ments, and ko is never larger than lop6 we see that it is satisfactory k. = k+ = 0 when evaluating experiments. Inclusion of the effects of boundary layers can be shown to modify Eq. (104) only through changing 1~0. If a constant R (weakly dependent on Qo and ~0) corresponding to the dissipation in the boundary layers is added to Eq. (106) the analysis is essentially complete. The calculation of R is complicated but it can be determined experimentally rather easily by measuring the Q. of the nonrotating cavity. Then R z L?fi,“/Q,“. Figs. l-4, sh ow the response curves of & for various values of the parameters.

WAVE MOTIONS

IN ROTATING

HELIUM

IS7

II

FIG. 1. Response curves of a square resonator to second sound [Eq. (104)] with Cl0 = 270. B = 1.0, B' = 0, and R = 7,000. The curves for three values of the rotation rate 00 art’ indicated and the arrows show the resonant frequencies in each case.

5

I

I

FIG.

2. Response

cklrves

I

269

268

of a square

270

resonator

G to second

271

sound

272

[&I.

(104)]

with

QO = 270.

B = 1.0, B' = 0.2, and R = 7,oiu D.

COUPLING

OF FIRST AND SECOND SOUSD

In a linearized treatment there is no coupling of the t’wo kinds of sound waves; the so called coupling is due to the generation of both types of waves by the driving source. When the dispersion relations were derived it was shown thut if dissipative effects may be neglected, rotation does not change the ratio P,in,,, in either wave. The ratios of ZJ,/V,of Eqs. (42) and (,43) are used t.o determine the coupling. Khalatnikov (1.2) illustrates the coupling when a plane oscillates irl :I plane normal to itself while its surface temperature varies periodically.

188

SNYDER

AND

WESTEBVELT

3. Response curves of a square resonator to second sound [Eq. (104)] with 1.0, B' = 0.2, wo = 1.0 fo r t ‘wo values of R corresponding to Q0 = 370 and amplitude scale for the former case has been increased by a factor of 109. FIG.

B =

FIG.

layer and

4. The dissipation

response added].

of a rectangular cavity The z-mode is driven

to second sound [Eq. (28) and 0, = 269.8, Bzr = 270.2,

with

B =

no = 270, 1200. The

boundary1.0, B’ = 0,

R = 7,000.

Cvnsider a surface which is porous t’o the superfluid but impervious to the normal fluid. If the temperature of the surface varies periodically, then the magnitude of u, is W/PST. The superfluid velocity in this case could be other than -PnVnlPs ; assume, say, v, = -I*v, . Let A, be the amplitude of first sound and A2 that of second sound. Then, Al + A, = -r(aJl + czJ2) and A? = --[(I + ral)/( 1 + RJ~)]A~ . This ratio in the amplitudes of second to first sound is strongly dependent on 1’. If 1’ 2: pn/ps , we have mostly second sound; if 1’ de-

WAVE

MOTIONS

IN

ROTATING

HELIUM

II

189

viates appreciably from pn/ps , the ratio is strongly dependent upon the temperature and may range from an extreme of all first sound to all second sound. E.

LIN’S

EQUATIONS

The equations proposed by Lin (7) for rotating He II can be transformed to a rotat,ing frame and linearized as follows:

dvn _ -- 1 vp - fi XvT

at

P

> x iv, - vs) ; - 20~ x v,, + W” r. + ps ~ :W

Pn

iiosb

P

and dV, -= at

--1p+SvT+h

xv,+w,&-P”“wo

P

P

x (v,,-vv,).

iio9)

The continuity equations are the same as Eqs. (7) and (S). For second sound waves subtract Eq. (109) from Eq. (108) to eliminat,e p:

&VII - v,) at

-

=

2&1’. pn

ill01

Thus P = - ~ipq’p&“f)

T’,

(111)

where ( vn - v, j = Vq and a = 0. Since a = 0, a complete solution is of t’he form ‘I” = 3 cos ar/‘P and there is only one resonance peak. The experiments of Snyder (;j’), (4) ; Snyder and Linekin (2) ; and of Lucas (18) show a double peaked resonance curve with the separation of the peaks proportional to oc in agreement wit,h the HVBK equations and at variance with Lin’s equations. Icor first sound, remove VT from Eqs. (108) and ( 109) ; thus,

ant1

and finally, a-w = (2/Q)

iwo;ti)p':

i 113b 1

so that the mode degeneracy of first sound is remoced by rot,ntion. E‘.

HIGHER-C)HDER

DIODES

I~YA

CIRCULARRESONATOR

The method which has been demonstrated above has two orders of approximution: first the potentials are expanded in powers of ~(w~lti)~; and second, tht>

SNYDER AND WESTERVELT

190

infinite set of linear equation obtained by equating the coefficients of the various modes has been truncated with the two degenerate modes whose resonance is close to the driving frequency retained. W:: will consider the effect of omitting all but two modeshere by treating the caseof a cylindrical resonator. The potentials Eqs. (15, 16) and the wave equation Eq. (17) apply. The trial function has the form :

and the sourcehas a similar expansion in Q>,, and Q”,,, . The radius of the cylindrical resonator is a and (Y,,, is a constant such that clJ,(?ra)/da = 0. Upon calculating the radial velocity at the wall of the cavity it is found that: 2, (,. r

=

u)

=

(2

-

B’)P*SUO

g2p, QZ

Then, since q = --6( I’ - a)v,(r

(115)

= a) it follows that

A similar expression results for QL,, in which $,, of Eq. (116) is replaced by -3z7.n * Observe that it is not necessary to know qe in order to find the splitting of the degenerate frequencies. If, in Eq. (116), we let CJ= n and retain only the term 3,,, in the sum on the right, we get for the splitting iI* = f&j/g f

(2 - B’)w1w0/[g”(u”7r2a1,, - m’)],

(117)

a result which agrees with t’he calculation by Lucas (18) using perturbation theory. Equation (116) shows explicitly the effect of higher order terms and how they will modify the first order perturbation; terms in n other than n = p should be included in the set of linear equations. Note that the induced source terms for any CC, p depend only on 3: ,,- there is a separate set of equations for each 1)~. Thus, modes of different WLdo not mix. It is somewhat easier to seethe resu1t.sof including more terms in the determinant for the case of a square resonator. G. HIGHER-ORDERMODESINASQUARERESONATOR The formal procedure for including more than two modes in t’he analysis was described in Set 1V.E. Using the formulas of Eq. (78)-(81) with & = r; and

WAVE

MOTIONS

IN

ROTATING

HELIUM

1!)1

II

6 = 0, we set up the infinite set of linear equations and solve various portions of the set. We have considered the pair 30,1 and $,o above and the splitting is given in Eq. (24). If various pairs of degenerate modes such as so,2 , &,@ are examined, il- is found that modes of the form ?/0,29and s2q,o produce no splitting of the resonant’ frequency. However, for modes with one zero and one odd subscript like &,2q+l , &.+1,0, the splitting is 1/(2q + 1)” times that for so,l , sl,o . This can he explained physically by observing that the induced source q changes sign at nodal lines of the wave function as shown in I?ig. 5. By symmetry the strength of the source is the same in each area bounded by nodal lines. The net source for modes with even subscript is obviously zero, while that for odd subscript is 1 ,v’. The same argument can be used for modes like CC,,, to show that the larger n and m, the smaller the frequency splitting. The closestresonance to (01) and ( 10) is ( 11) ; accordingly, consider the t,hrer modes (01) , ( lo), and ( 11). The solution is 311= 0 and the mode ( 11) does not affect the result; the splithing is the samefor (01) and (10) if ill) is included 01 neglected. Now t#ry t’he four modes (01)) ( lo), (03)) and (30). If we abbreviat’e Eqs. ( ~3 :t and (27’) thus, &,o = xW/(r” + y”) and sl,o = -yNTf’(.r + y’). Then in the present case we can write Lila = .r(3.r)2W/[.r2(3x)2 301

=

-(x

+

+ y*(x

+ ~.zT)~],

3r)(3x)yjW~[s2(3x)2

Eho = (3x)a2W/[xz(3x)2

+ y?,r

+ %.rJ2],

+ y2(a + 3x)7,

and 303 =

-(x

+

3x)xgv[z2(3x)”

+ y’(.r

+ 3x)“].

+ + + + -El+ El +-+

Now each of these expressions reduces near the resonant frequency to t,he result, expected when the degenerate modes are treated in pairs. When the same procedure is applied to the four modes (Ol), (lo), (02), and

(a)

FIG.

and

(h)

5. The (03)(30).

sign

of the source

(b)

in

various

areas

of the

container

for

the modes:

(a) (W)(20)

192

SNYDER AND XVESTERVELT

(20) we find that near resonance for (Ol)(lO) the effect of (02)(20) is small. However, near the (02)(20) resonance, it is no longer true that there is no splitting of the resonant frequency-but the splitting is much smaller than that for the (Ol)(lO) modes. H. VORTEX WAVES IN A SQUARE RESONATOR Let us suppose the bottom of the resonator oscillates along the x-axis. It is necessary for the driving motion to be expressed in the form v, + ivU . The assumed motion can be written as x( v, + iv,) + >$( vZ - izb) . The latter term generates only evanescent waves. Turning to Eq. (84) we must choose w,,, so that sin Ck, cm,, ‘u,, sin rnr.r/e cos nay/C is a constant equal to half the velocity amplitude of the driving surface. Since this motion is independent of y we see that the only solution is ‘u,, = 0 and it is impossible to generate vortex waves in this way. Now consider a motion of the bottom of the form v2 = V cos fit sin TX/~ sin ay/C and v, = V sin fit sin ax/e sin ry/e. The, 2)11= V/sin &lc, and all other terms in II m,nare zero. VI.ON

THE PHYSICAL INTERPRETATION

OF THE RESULTS

There is a problem of interest regarding the results presented here. It has to do with whether there is a uniform density of vortex lines in the rotating resonator. The HVBK equations hold only if this is the situation. Bendt and Oliphant (19) first pointed out that the free energy of rotating helium is not always a minimum for a uniform dist’ribution of vortices. A method of finding t)he distribution of vortex lines by minimizing the free energy is discussed by Khalatnikov (1.2) where the case of cylindrical symmetry is treated. The situation is somewhat different in a rectangular container and Snyder (20) has shown how to treat. the new complications which arise in this case. For the shapesof resonators used in the experiments presently reported in the literature, it appears that the HVBK equations hold to better than 1% [Snyder (20)]. It is possible, however, by making a container long and narrow t’o create regions of appreciable extent where there are no vortices according to the principle of minimum energy. Recently Fetter (21) has criticized Khalat,nikov’s method. VII. A THEOREM ON ROTATIONAL SPLITTING When dissipative effects can be neglected and the equation can be reduced to an Euler type equation such as dV

-=

at

1 --VP P

- 20~ x v + wo2r0

(11s)

and

(119)

RAVE

MOTIONS

IN

ROTATING

HELIUM

II

I!):-5

it can be shown that a simple relation exists between the splitting of the resonant frequency of degenerate modes and the averaged properties of the waves [W estcrvelt (511. The theorem is (1”O)

fb - Qo = =F(oo*4)Qo/Eo,

where Lo is the angular momentum of the wave in question averaged over the container and E. is the energy of t’he wave averaged in the same way. The UIIperturbed values of L and h’ are to be used in the evaluation. The unperturbed wave is to be considered to be the superposition of the two degenerate waves which are split by rot#ation. For example, a plane wave is to be viewed as being made up of two circularly polarized waves with opposite helicity. Thus, for a first sound wave in a square resonator, t,he unperturbed wave in the x-direction i:: p = [cm nax;‘! sin

Sit + cos n?ry/e cos tit] + [cos n7rx,lt sin

Then the displacement do defined as the equilibrium position is: do = ~p/p&~’ and vc can be found in terms of p. Rut Lo = (do x vo), and the average energy calculat,ions on each part, of Eq. (121 j we

Qt - cos nay/8 cos fit].

(1’1)

excursion of a fluid particle from its the velocity is v. = ad,/at. Thus dc and the average angular momentum is is B. = Q,“(dl). Carrying out thf>se find:

- 1)3wo Qf - fit0= f 3- icesn7r &” in agreement with Eq. 197) and other work

in Sec. V.

.h!KSOWLEDGMEiXTS The research of one of us (P. J. W.) was supported in part Command, the other (H. A. S.) benefited from a grant from tion No. (:P2857. Support by the Advanced Research Projects RECEIVED:

July

by the U. S. Air Force Systems the National Science FoundaAgency is also acknowledged.

19, 1966 REFERENCES

1. 8. 5. 4. 5.

HI. II. H. II. P.

E. H-ILL, Phil. ililag. Suppl. 9, 89 (1960). A. SNYDER, AND D.M. LINEKIN, Phys. Rev. 147,131 (1966). A. SKYDEK, Phys. Fluids 6, 755 (1963). A. SNYDER, AND P. J. WESTERVELT, Phys. Rev. Letters 16, 748 (1965). J. WESTERVELT, Combustion Inst,ability Quarterly Progress Report, No. 3, Bolt, Beranek, and Newman Inc., Cambridge, Massachusetts, August 1963 (rmpublished). 6. L. L.\PiD.iU, J. Phys. USSR 6,71 (1941). 7. C. C. LIN, Liquid Helium, in. “Proceedings of the Enrico Fermi International School of Physics, Course XXI.” Academic Press, New York, 1963. 8. II. E. HALL, AND W. F. VINEN, Proc. Roy. Sot. (London) A238, 204 (1956); ibicl. A288, 215 (1956).

194

SNYDER

AND

WESTERVELT

9. I. L. BEKAREVICH, END I. M. KHALATNIEOV, Zh. Eksperim. i Teor. Fiz. 40, 920 (1960) [English transl.: Soviet Physics-JETP 13, 643 (1961)]. 10. L. ONSAGER, Nuovo Cimento 6 [Suppl. 21 249 (1949). 11. R. P. FEYNM-IN, “Progress in Low-Temperature Physics,” Vol. 1, Chap. II. North Holland, Amsterdam, 1955. 12. I. M. KHAL.ITNIKOV, “Introduction to the Theory of Superfluidity.” Benjamin, New York, 1965. 13. L. P. PIT~EVSKII, Zh. Eksperim. i. Teor. Fiz. 35, 408 (1958) [English transl.: soviet Phys.-JETP 8, 282 (1959)]. 1.4. R. B. DINGLE, Proc. Phys. Sot. (London) A63,638 (1950). 15. SIR W. THOMSON, Phil. Mug. 10, 155 (1880). 16. E. L. ANDRONIKASHVILI, Y. G. MAMAL~DZE, S. G. MSTINYAN, AND D. S. TSAKADZE, /Tsp. Fiz. Nauk 73, 3 (1961) [English transl. : Soviet Phys. Uspekhi 4, 1 (1961)]. 17. 0. M. PHILLIPS, Phys. Fluids 6, 513 (1963). 18. P. LUCAS, Phys. Rev. Letters 16, 750 (1965). 19. P. J. BENDT, AND T. A. OLIPHANT, Phys. Rev. Letters 6, 213 (1961). 20. II. A. SNYDER, AND Z. PUTNEY, Phys. Rev. 160, 110 (1966). 21. A. L. FETTER, Phys. Rev. 162,183 (1966).