3 ∇ $^{2}$φ$_{n, s}$ = 0 can be represented as should acquire by the time of the overpressure release and formation of the irregular outline, we will find a rather high magnitude of several meters per second. The cor- responding acceleration which provides such increment of velocity should be about 10 $^{3}$g , g being acceleration of gravity. On the whole, large acceleration for a short time shows evidence in favor of a shock-driven character of the crystal growth. It is interesting to note that the irregu- lar patterns observed are similar in appearance to those obtained in studies [23] of a shock-accelerated boundary between two gases of different densities. The typical time of the pattern formation was of the same order of several tenths of millisecond. where r = ( x, y ) and the velocities u$_{s}$ ( t ) and u$_{n}$ ( t ) de- scribe the undisturbed motion in the superfluid. The per- turbation amplitudes A$_{s}$ ( t ) and A$_{n}$ ( t ) are assumed to be linear in ζ$_{q}$ ( t ) and will be determined later from the corre- sponding boundary conditions at the interface. The pres- sure in the superfluid is a sum of pressures P = P$_{n}$ + P$_{s}$ and In the present work we develop a linear theory on the Taylor instability of an arbitrarily accelerated boundary between the superfluid and solid $^{4}$He phases. In essence, we derive an equation which the small interfacial pertur- bations obey. We consider three typical cases of the in- terface acceleration: constant, shock, and periodic. The plane and spherical interface geometries are analyzed. $^{The index “}$∞” refers to the values taken away at infinity. Unlike previous considerations [24–27] which are also linear in the interfacial perturbations, we have to retain the quadratic terms in the superfluid and normal veloc- ities v s, n = ∇ φ$_{s, n}$ on account of nonzero value u$_{s, n}$ ( t ) and product like u$_{s}$A$_{s}$ or u$_{n}$A$_{n}$ . φ$_{s}$ = u$_{s}$ ( t ) z + A$_{s}$ ( t ) exp( i qr − qz ) , φ$_{n}$ = u$_{n}$ ( t ) z + A$_{n}$ ( t ) exp( i qr qz ) , (1) P$_{s}$ = P$_{s}$$_{∞}$ − ρ$_{s}$ ( ˙ φ$_{s}$ $^{+ (}$∇ φ$_{s}$ ) $^{2}$/ 2 ) − ρ$_{s}$gz, P$_{n}$ = P$_{n}$$_{∞}$ − ρ$_{n}$ ˙ φ$_{n}$ $^{+ (}$∇ φ$_{n}$ ) $^{2}$/ 2 − ρ$_{n}$gz. (2) j$_{ν}$ = ρ$_{n}$v$_{n ν}$ + ρ$_{s}$v$_{s ν}$ = ( ρ − ρ $^{′}$) ˙ Z , (3) v$_{n ν}$ = ˙ Z . (4) The mass continuity across the boundary gives at z = Z ( x, y, t ) Let us assume the interface is parallel to the x - y plane, with vertical position z = L ( t ) which moves at the rate V = ˙ L ( t ). The upper half-space z > L ( t ) is occupied with the liquid phase and the solid phase occupies the lower one z < L ( t ). Below we will consider the stability of the moving interface with respect to its small perturba- tions ζ ( x, y, t ) from the flat shape. Thus, the coordinate Z ( x, y, t ) = L ( t ) + ζ ( x, y, t ) gives the vertical position of the perturbed interface evolving in time. where ˙ Z is the velocity of the boundary, ν is the unit vector normal to the boundary, and j$_{ν}$ is the mass current normal to the boundary. The densities ρ$_{n}$ , ρ$_{s}$ are the normal and superfluid densities, ρ = ρ$_{n}$ + ρ$_{s}$ , and ρ ′ is the density of the solid phase. The normal components of velocities can be approximated by v$_{n ν}$ ≈ v$_{n z}$ and v$_{s ν}$ ≈ v$_{s z}$ . We also believe that there is no motion in the solid phase, i.e., v ′ = 0. To further simplifications, we suppose the normal com- ponent sticks to the interface like a viscid fluid. Also, this implies the Kapitza resistance to be infinite [25]. In ad- dition, we disregard any excitations, e.g. phonons, in the solid. So, we put at the boundary To discover the effect of nonuniform motion on the sta- bility of the interfacial shape, we make a number of sim- plifying assumptions which do not affect the main point of the phenomenon. The validity and criteria of appli- cability for the assumptions to be made below can be found in the papers on the kinetic interface coefficients and crystallization waves [24–27]. PLANE GEOMETRY AND THE INTERFACE GROWTH KINETICS Using Eqs. (1), (3), and (4), we can determine the un- known amplitudes in (1) and then the velocity fields v s and v n in the liquid. For the unperturbed motion, it is obvious So, in the superfluid we employ the usual two-fluid equations without dissipation. In addition, we assume that the growth rate of a solid V ( t ) is always small com- pared with the velocity of the first or second sound. This is an ordinary experimental situation. Thus we treat the hydrodynamics of the superfluid in the approximation of incompressible liquid and the constancy of the entropy density. In this case [28] the velocities v n and v s of the normal and superfluid motions can be described in terms of gradient of velocity potentials φ$_{n}$ and φ$_{s}$ which satisfy ∇ $^{2}$φ$_{n}$ = 0 and ∇ $^{2}$φ$_{s}$ = 0, respectively. Since finally we will discuss only the linearized equations in the perturba- tion ζ , it is convenient to consider a single Fourier mode of the perturbation ζ = ζ$_{q}$ ( t ) exp( i qr ) with wave vector q = ( q$_{x}$, q$_{y}$ ) parallel to the boundary. The solutions of The amplitudes A$_{n}$ ( t ) and A$_{s}$ ( t ) are given approximately by u$_{s}$ = − ρ ′ − ρ$_{s}$ ρ$_{s}$ V and u$_{n}$ = V . (5) A$_{s}$ ( t ) = ρ ′ − ρ$_{s}$ ρ$_{s}$ ˙ ζ$_{q}$ ( t ) q e qL ( t ) , A$_{n}$ ( t ) = − ˙ ζ$_{q}$ ( t ) q e qL ( t ) .