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Strongly attenuated symmetrical waves in superfluid helium
Volume 43A, number 4
26 March 1973
PHYSICS LETTERS
STRONGLY ATTENUATED SYMMETRICAL WAVES IN SUPERFLUID HELIUM R.P. WEHRUM and L. MEINHOLD-HEERLEIN Institut fiir Physik, Universitiit Mainz, Germany
Received 30 January 1973 We prove the existence of a series of strongly damped symmetrical wave propagating in a He II layer of arbitrary width and calculate some of their properties (phase velocities, attenuation coefficients).
In several previous works the propagation of waves in a layer of superfluid helium has been studied [e.g., l-51. Two different waves have been found to exist; in [S] they were called mode l-4 and mode 2-5 because, depending on the width of the layer and the frequency of the wave, they can represent a wave process between first sound and fourth sound, and a wave process between second sound and fifth wave mode [4] (“no sound” [2], thermal wave [3]), respectively. The aim of this paper is to prove the existence of further waves propagating in a He II layer of arbitrary width and to calculate their dispersive behaviour. The solution to this problem has not only some formal significance, but may also be of importance in treating the excitation of layer waves. In dealing with waves of small amplitude we may start from the linearized Landau-Khalatnikov equations [6]. Being mainly interested in questions of existence, we shall only take into account the first viscosity and neglect the thermal expansion of He II. Let us consider a layer of He II bounded by two infinitely extended parallel plates. A coordinate system is introduced so that one plate coincides with the plane y = + d and the other with the plane y = -d. (The x and z coordinates can be fixed at will.) We are looking for special integrals which describe harmonic waves propagating in the He II layer with an (arbitrary) angular frequency w in the positive x direction. As was shown in [5], the consideration of boundary conditions leads to tuo transcendental equations which implicitly comprise the dispersion relations of all possible layer modes. Here we confine ourselves to investigating the symmetrical layer modes?, which means to find the solutions of: tit is characteristic of a symmetrical wave that the x@ components of the velocity fields are even (odd) functions of y.
FcS)(k; w, d; P, T) = (@I+ [(u’-u;)Z+s+c_ = 0,
u’) l+ I_ lvs+ s_ cv
- (+u~)Z_c+s_]
k2sv}lv (1)
where k is the wave number (source problem: k E Cc, w E IR, w > 0); k gives us the phase velocity u = o/Re k and the coefficients of attenuation Im k and fl= Im k/Re k. We have introduced the abbreviations (j EJ = {t, -, v}), $ = 4-i sj = sinh (lid), ci = cos (+I); Uj and kj denote the complex phase velocities and complex wave numbers of first sound, second sound, and the viscous wave, respectively; u4 is the real phase velocity of fourth sound; up kj, u4 must be read as (known) functions of (w, P, T); P, T stand for pressure and temperature in the global thermodynamic equilibrium of the fluid. In the following we describe the methods we employ to find new solutions of eq. (1): 1. We calculate numerically the distribution of zeros of F@) in a subset of the complex wave number plane by means of the “principle of the argument” [e.g., 71. Thus we succeed in proving explicitly the existence of a series of (more than 50) symmetrical waves. (The application of this principle is restricted to bounded domains where Fe) can have at most a finite number of zeros.) 2. Moreover, we calculate numerically the dispersion relations of some of these new modes, i.e., phase velocities and attenuation coefficients as functions of (0, d). This is done by a procedure which is based on a complex Newton-Raphson iteration as was described in [5]. 3. The knowledge of the dispersion relation of a certain wave allows one to determine its different fields. Here we calculate us and u,, , the velocity fields 391
Volume 43A, number 4
PHYSICS LETTERS
26 March 1973
PLV‘ “LV,CcM b
\\ -105 1, Jm(k) [--I
j=26
3.10’
100
'j=56
Fig. 1. The positions of the wave numbers k(LV.) (j=2,26, 56) for different values of d which are indicate d on the curves (w = 2x lo3 s-l , P= saturated vapour pressure, T= 1.5 K).
of the super and normal fluid flow. We arrive at the following conclusions: i) As might be expected, the waves l-4 and 2-5 are the only soundlike waves, in the sense that there exist intervals of w and d for which the attenuation is weak. All the other modes are strongly damped, /I > 1, and characterized by limd,, k = k, (w constant). This means the dispersive behaviour of each one of these new modes approaches that of the viscous wave by increasing d (cf. fig. 1). In this sense these new modes appear as layer modifications of the viscous wave. Therefore we call them (symmetrical) LV (layer viscous) waves. Different LV modes show a different attenuation coefficient for constant values of w, d, P, T, so that they may be classified according to their attenuation coefficients. ii) As a function of 6 = d/penetration depth of the viscous wave, /3 proves to be independent of the frequency, see fig. 2; this figure also shows the interesting behaviour of the phase velocity. iii) The propagation of an LV wave involves in the main a normal fluid flow. The profile of the u,, field (u, as a function ofy at constant X, t) of each LV mode displays a characteristic number of nodes, this number being an even integer (for the symmetrical mode). A classification of LV modes according to the number of nodes of the profile of the velocity field
392
,uLVL
h
10’
:,6
102
Fig. 2. Phase velocity and attenuation coefficient for the wave LV4 as functions of b (P= saturated vapour pressure, T= 1.5K). unx turns out to be equivalent
to a classification with respect to the attenuation: the higher the number of nodes, the larger the attenuation coefficient. A physical explanation for this is obvious from the fact that it is the normal fluid (viscous) part that primarily supports the wave propagation. Surely, at least for small d, it will be difficult to prove the existence of a single LV wave experimentally. On the other hand, the number of LV modes is large, and depending on the kind of wave excitation, a measurable amount of energy may go into them. (This could be of importance for, e.g., sound absorption measurements.) Further details of our calculations including an investigation of antisymmetrical waves will be given elsewhere.
References [l] K.R. Atkins, Phys. Rev. 113 (1959) 962. [2] G.L. Pollack and J.R. Pellam, Phys. Rev. 137 (1965) A1676. (31 I.N. Adamenko and MI. Kaganov, Soviet Phys. JETP 26 (1968) 394. [4] L. Meinhold-Heerlein, Z. Physik 213 (1968) 152. [S] R.P. Wehrum and L. Meinhold-Heerlein, Z. Physik, to be published. [6] I.M. Khalatnikov, Introduction to the theory of superfluidity (W.A. Benjamin, New York, 1965). [7] L.V. Ahlfors, Complex analysis (McGraw-Hill, New York, 1953).
26 March 1973
PHYSICS LETTERS
STRONGLY ATTENUATED SYMMETRICAL WAVES IN SUPERFLUID HELIUM R.P. WEHRUM and L. MEINHOLD-HEERLEIN Institut fiir Physik, Universitiit Mainz, Germany
Received 30 January 1973 We prove the existence of a series of strongly damped symmetrical wave propagating in a He II layer of arbitrary width and calculate some of their properties (phase velocities, attenuation coefficients).
In several previous works the propagation of waves in a layer of superfluid helium has been studied [e.g., l-51. Two different waves have been found to exist; in [S] they were called mode l-4 and mode 2-5 because, depending on the width of the layer and the frequency of the wave, they can represent a wave process between first sound and fourth sound, and a wave process between second sound and fifth wave mode [4] (“no sound” [2], thermal wave [3]), respectively. The aim of this paper is to prove the existence of further waves propagating in a He II layer of arbitrary width and to calculate their dispersive behaviour. The solution to this problem has not only some formal significance, but may also be of importance in treating the excitation of layer waves. In dealing with waves of small amplitude we may start from the linearized Landau-Khalatnikov equations [6]. Being mainly interested in questions of existence, we shall only take into account the first viscosity and neglect the thermal expansion of He II. Let us consider a layer of He II bounded by two infinitely extended parallel plates. A coordinate system is introduced so that one plate coincides with the plane y = + d and the other with the plane y = -d. (The x and z coordinates can be fixed at will.) We are looking for special integrals which describe harmonic waves propagating in the He II layer with an (arbitrary) angular frequency w in the positive x direction. As was shown in [5], the consideration of boundary conditions leads to tuo transcendental equations which implicitly comprise the dispersion relations of all possible layer modes. Here we confine ourselves to investigating the symmetrical layer modes?, which means to find the solutions of: tit is characteristic of a symmetrical wave that the x@ components of the velocity fields are even (odd) functions of y.
FcS)(k; w, d; P, T) = (@I+ [(u’-u;)Z+s+c_ = 0,
u’) l+ I_ lvs+ s_ cv
- (+u~)Z_c+s_]
k2sv}lv (1)
where k is the wave number (source problem: k E Cc, w E IR, w > 0); k gives us the phase velocity u = o/Re k and the coefficients of attenuation Im k and fl= Im k/Re k. We have introduced the abbreviations (j EJ = {t, -, v}), $ = 4-i sj = sinh (lid), ci = cos (+I); Uj and kj denote the complex phase velocities and complex wave numbers of first sound, second sound, and the viscous wave, respectively; u4 is the real phase velocity of fourth sound; up kj, u4 must be read as (known) functions of (w, P, T); P, T stand for pressure and temperature in the global thermodynamic equilibrium of the fluid. In the following we describe the methods we employ to find new solutions of eq. (1): 1. We calculate numerically the distribution of zeros of F@) in a subset of the complex wave number plane by means of the “principle of the argument” [e.g., 71. Thus we succeed in proving explicitly the existence of a series of (more than 50) symmetrical waves. (The application of this principle is restricted to bounded domains where Fe) can have at most a finite number of zeros.) 2. Moreover, we calculate numerically the dispersion relations of some of these new modes, i.e., phase velocities and attenuation coefficients as functions of (0, d). This is done by a procedure which is based on a complex Newton-Raphson iteration as was described in [5]. 3. The knowledge of the dispersion relation of a certain wave allows one to determine its different fields. Here we calculate us and u,, , the velocity fields 391
Volume 43A, number 4
PHYSICS LETTERS
26 March 1973
PLV‘ “LV,CcM b
\\ -105 1, Jm(k) [--I
j=26
3.10’
100
'j=56
Fig. 1. The positions of the wave numbers k(LV.) (j=2,26, 56) for different values of d which are indicate d on the curves (w = 2x lo3 s-l , P= saturated vapour pressure, T= 1.5 K).
of the super and normal fluid flow. We arrive at the following conclusions: i) As might be expected, the waves l-4 and 2-5 are the only soundlike waves, in the sense that there exist intervals of w and d for which the attenuation is weak. All the other modes are strongly damped, /I > 1, and characterized by limd,, k = k, (w constant). This means the dispersive behaviour of each one of these new modes approaches that of the viscous wave by increasing d (cf. fig. 1). In this sense these new modes appear as layer modifications of the viscous wave. Therefore we call them (symmetrical) LV (layer viscous) waves. Different LV modes show a different attenuation coefficient for constant values of w, d, P, T, so that they may be classified according to their attenuation coefficients. ii) As a function of 6 = d/penetration depth of the viscous wave, /3 proves to be independent of the frequency, see fig. 2; this figure also shows the interesting behaviour of the phase velocity. iii) The propagation of an LV wave involves in the main a normal fluid flow. The profile of the u,, field (u, as a function ofy at constant X, t) of each LV mode displays a characteristic number of nodes, this number being an even integer (for the symmetrical mode). A classification of LV modes according to the number of nodes of the profile of the velocity field
392
,uLVL
h
10’
:,6
102
Fig. 2. Phase velocity and attenuation coefficient for the wave LV4 as functions of b (P= saturated vapour pressure, T= 1.5K). unx turns out to be equivalent
to a classification with respect to the attenuation: the higher the number of nodes, the larger the attenuation coefficient. A physical explanation for this is obvious from the fact that it is the normal fluid (viscous) part that primarily supports the wave propagation. Surely, at least for small d, it will be difficult to prove the existence of a single LV wave experimentally. On the other hand, the number of LV modes is large, and depending on the kind of wave excitation, a measurable amount of energy may go into them. (This could be of importance for, e.g., sound absorption measurements.) Further details of our calculations including an investigation of antisymmetrical waves will be given elsewhere.
References [l] K.R. Atkins, Phys. Rev. 113 (1959) 962. [2] G.L. Pollack and J.R. Pellam, Phys. Rev. 137 (1965) A1676. (31 I.N. Adamenko and MI. Kaganov, Soviet Phys. JETP 26 (1968) 394. [4] L. Meinhold-Heerlein, Z. Physik 213 (1968) 152. [S] R.P. Wehrum and L. Meinhold-Heerlein, Z. Physik, to be published. [6] I.M. Khalatnikov, Introduction to the theory of superfluidity (W.A. Benjamin, New York, 1965). [7] L.V. Ahlfors, Complex analysis (McGraw-Hill, New York, 1953).