5 approximately equal to The new aspect of the equation derived is an additional term with the interface acceleration ˙ V ( t ). As is expected, the uniform motion of the interface at V ( t ) = const does not influence the character of small interfacial oscilla- tions. For ˙ V ( t ) = 0, Eq. (8) amounts to the known re- lation determining the spectrum of crystallization waves when the melting-crystallization processes are balanced and the interface position in average is invariable, i.e., L ( t ) = const [9, 25, 26]. Undoubtedly, more realistic and complicated models of the superfluid-crystal $^{4}$He interface will improve the mag- nitudes of the effective interface density, effective growth coefficient, and surface stiffness. However, we believe that the structure of Eq. (8) is generic and holds. The values q$_{0}$ and λ$_{0}$ depend on γ , thus implying anisotropic and complicated possible surface patterns. In the opposite limit K 4 ˙ V ≪ ρ ′ $^{4}$/γρ3 ef one has roughly It is interesting that, though the spatial scale q − 1 0 is sen- sitive to the surface stiffness and its anisotropy, the time of developing the instability becomes independent of the surface stiffness and its anisotropy. On the whole, the values q$_{0}$ and λ$_{0}$ decrease as the kinetic growth coefficient reduces or dissipation with the interface enhances. From the experimental point of view this may require a crystal surface of sufficiently large sizes d > 2 π/q$_{0}$ and a large time to support the accelerated growth t > 1 /λ$_{0}$ in order to realize an interfacial insta- bility with a uniformly accelerated growth. Equation (8) can have unstable solutions depending strongly on the acceleration history of the interface. First, let us consider the stability of the plane interface with respect to small perturbations ζ$_{q}$ ∼ exp( λ ( q ) t ) for the uniformly accelerated growth of a crystal. The root with Re λ ( q ) > 0 means the interface instability, i.e., ini- tially small-amplitude perturbations of wavelength 2 π/q will grow exponentially in time. For the acceleration ex- ceeding the threshold ˙ V$_{c}$ = g ( ρ ′ − ρ ) /ρ$_{ef}$ , the interfacial perturbation will increase for the wave vectors satisfying γq 2 c < ρ$_{ef}$ ˙ V − ( ρ ′ − ρ ) g . For brevity, we put γ$_{ik}$q$_{i}$q$_{k}$ = γq 2 where γ = γ$_{ik}$n$_{i}$n$_{k}$ and n = q /q is unit vector in the direction of perturbation. Note that the threshold ac- celeration ˙ V$_{c}$ does not depend on the growth coefficient, i.e., on the dissipative properties of the interface, and is positive. The latter corresponds to the case when the interfacial acceleration ˙ V is directed to the superfluid. Thus, the Taylor instability due to non-uniform growth of the plane interface appears only for the accelerated growth of a solid. The surface stiffness stabilizes the region of long wave perturbations and establishes the most unstable wave- length having the fastest exponential growth. The value q$_{0}$ corresponding to the maximum magnitude Re λ ( q ) gives the shortest time scale for the development of the instability which will be characterized by the wavelength 2 π/q$_{0}$ . The value q$_{0}$ can be found from the equation Then, if a periodic process of melting and crystallization is realized in experiment so that the interface could oscil- late around some average position at frequency 2 ω and amplitude G/ (2 ω ) $^{2}$, equation (9) with label 2 ϵ = − G/ ˜ g transforms to a Mathieu-type equation which is identical to that analyzed in [18]. The predic- tions which follow are well known and we refer the readers where q$_{c}$ is the value related to the upper bounds of in- stability according to γq 2 c = ρ$_{ef}$ ˙ V − g ( ρ ′ − ρ ). For large magnitudes of the growth coefficient or large acceleration in the case K 4 ˙ V ≫ ρ ′ $^{4}$/γρ3 $_{ef}$, the values q$_{0}$ and λ ( q$_{0}$ ) are where we have introduced the damping coefficient Γ$_{q}$, the frequency ω$_{0}$ ( q ) of crystallization waves in the lack of damping, and the scaled accelerating amplitude ˜ g In Ref. [18] it has been shown that periodic modulation of the gravitational constant as g ( t ) = g (1 + 2˜ ϵ cos 2 ωt ) can result in parametric excitation of crystallization waves at the stationary flat interface corresponding to V ( t ) = 0 in our case. Admitting some analogy between gravity and noninertial frame, one can expect a possibil- ity of exciting crystallization waves with a periodic driv- ing, e.g., ˙ V ( t ) = G cos 2 ωt , even with the lack of gravity. To demonstrate this, let us rewrite Eq. (8) in the form INTERFACE INSTABILITIES q$_{0}$ = q$_{c}$ √$_{3}$ ( $^{1}$− ρ ′ $_{(6}$√$_{3}$$_{γρ}$$_{ef}$$_{K}$ $^{2}$q$_{c}$ ) 1 / 2 ) , λ$_{0}$ = ( 2 γ $_{3}$√$_{3}$ q 3 c ρ$_{ef}$ ) 1 / 2 − q$_{c}$ $_{2}$√$_{3}$ ρ ′ ρ$_{ef}$ K . q$_{0}$ = ( γρ$_{ef}$ ρ ′ 2 K $^{2}$q4 c ) 1 / 3 , λ$_{0}$ = K ρ ′ γq 2 c = K ρ ′ ( ρ$_{ef}$ ˙ V − ( ρ ′ − ρ ) g ) . ¨ ζ$_{q}$ + Γ$_{q}$ ˙ ζ$_{q}$ + ω 2 $_{0}$( q ) ( $^{1}$− ˙ V ( t ) / ˜ g ) ζ$_{q}$ = 0 , (9) ω 2 $_{0}$( q ) = ( γ$_{ik}$q$_{i}$q$_{k}$ + ( ρ ′ − ρ ) g ) q/ρ$_{ef}$, Γ$_{q}$ = ρ ′ ρ$_{ef}$ q K , ˜ g = g ( ρ ′ − ρ ) ρ$_{ef}$ ( 1 + γ$_{ik}$q$_{i}$q$_{k}$ ( ρ ′ − ρ ) g ) . ¨ ζ$_{q}$ + Γ$_{q}$ ˙ ζ$_{q}$ + ω 2 $_{0}$( q ) ( 1 + 2 ϵ cos 2 ωt ) ζ$_{q}$ = 0 , q 2 0 = q 2 c 3 − √$_{2}$ 3 ρ ′ √ γρ$_{ef}$ q 3 / 2 0 K ,