6 to papers [17, 18] for details. Note only that in the free crystal growth experiments [15, 20] the pressure in the cell drops drastically down after nucleating a solid seed with the subsequent transition to the damped oscillations around the melting pressure. Crystallization waves at the superfluid-solid $^{4}$He inter- face can also be generated with the Richtmyer-Meshkov mechanism when the interface is subjected to an impul- sive acceleration, i.e., ˙ V ( t ) ∼ V (0) δ ( t ). In ordinary fluids and gases, for this purpose, the passage of a shock wave across the interface is commonly used. Unlike the Taylor case of constant acceleration when the perturbation am- plitude in the linear regime grows exponentially in time, the initial stage of interface instability in the Richtmyer- Meshkov case of shock acceleration [29] is characterized by a linear growth of the perturbation amplitude in time. The Richtmyer-Meshkov instability is independent of the direction of acceleration in contrast to the Taylor one. The late time stages of both instabilities may demon- strate a formal resemblance, showing bubble and spike morphology. Here it is worthwhile to note that the total amplification factor is proportional to the interface growth coefficient K and proves to be independent of the wavelength 2 π/q of an initially sinusoidal perturbation. On the contrary, the time for ceasing the growth of perturbation amplitude is scale-dependent. The larger the wavelength, the longer the decay time. A special feature of crystallization waves in $^{4}$He is that the growth coefficient K , and thus Γ$_{q}$, is strongly temperature-dependent. For the opposite case when the crystallization wave damping is absent Γ$_{q}$ = 0, we arrive at the persistent oscillations of the initial interface perturbation have ζ$_{q}$ ( x, t ) = qV (0) Γ$_{q}$ ( $^{1}$− e − Γ$_{q}$ t ) ζ$_{q}$ ( x , 0) = ρ$_{ef}$ ρ ′ K V (0) ( $^{1}$− e − Γ$_{q}$ t ) ζ$_{q}$ ( x , 0) , ω$_{0}$ ( q ) = 0 . ζ$_{q}$ ( x, t ) = qV (0) sin ( tω$_{0}$ ( q ) ω$_{0}$ ( q ) ζ$_{q}$ ( x , 0) , Γ$_{q}$ = 0 . ζ$_{q}$ ( x, t ) = qV (0) e − Γ$_{q}$ t/ 2 ×$^{sinh}$ ( t √ Γ 2 $_{q}$/ $^{4}$− ω 2 $_{0}$( q ) √ Γ 2 $_{q}$/ $^{4}$− ω 2 $_{0}$( q ) ζ$_{q}$ ( x , 0) , ω$_{0}$ ( q ) < Γ$_{q}$ / 2 ; (10) ζ$_{q}$ ( x, t ) = qV (0) e − Γ$_{q}$ t/ 2 ×$^{sin}$ ( t √ ω 2 $_{0}$( q ) − Γ 2 $_{q}$/ 4 ) ω 2 $_{0}$( q ) − Γ 2 $_{q}$/ 4 ζ$_{q}$ ( x , 0) , ω$_{0}$ ( q ) > Γ$_{q}$ / 2 . The maximum amplitude of oscillations is governed by the crystallization frequency ω$_{0}$ ( q ). Since roughly ω$_{0}$ ( q ) ∼ q 3 / $^{2}$, the large curvature of the initial interface perturbation can also be a factor that limits the infinite growth of perturbation amplitudes. According to Eq. (8), the growth rate of the Richtmyer- Meskov unstable interface in the linear regime can ap- proximately be described in the overdamped regime as and in the weakly damped regime as In $^{4}$He the Richtmyer-Meshkov instability can be stud- ied with a sound wave pulse hitting the interface in the normal direction. Recently [30], it has experimentally been demonstrated that the superfluid-solid $^{4}$He interface can be set in motion with a sound wave which transmits through the interface, giving rise to the processes of crys- tallization and melting. In experiment [20] a solid nucleates at the needle point and then grows free. A ratio of the crystal sizes in the different directions for the crystals grown is not drastic. The ratio of the maximum to minimum size does not exceed 2–3, see also Figs. 1 and 2. Here we consider the stability of the spherical shape of a growing solid. For simplicity, we assume the surface tension α is isotropic. In addition, we neglect the acceleration of gravity and, as above, treat the equations linearized in the interface perturbation. Here ζ$_{q}$ ( x , 0) = ζ$_{q}$ (0) cos qx is the initial perturbation amplitude at the interface immediately after the impul- sive acceleration and V (0) is an increment in the interface velocity due to acceleration of the boundary. At the lin- ear stage the amplitude of crests and troughs is approx- imately the same. The shape of the crests and troughs is similar and the interface remains approximately sinu- soidal. Staying in the linear regime, we see from (10) that the finite damping of crystallization waves is a stabiliz- ing factor and can limit the growth of the perturbation amplitude. Next, we consider two extreme cases of ω$_{0}$ ( q ) = 0 and Γ$_{q}$ = 0. For ω$_{0}$ ( q ) = 0 which corresponds mainly to the flat interface perturbations with small wave vectors q , we The equation specifying the interface motion is taken as r = R$_{s}$ ( t, Ω) = R ( t ) + ζ ( t, Ω), where R ( t ) is the ra- dius of the unperturbed spherical interface and ζ ( t, Ω) = ∑ l ζ$_{l}$ ( t ) Y$_{l}$ (Ω) is an interface perturbation expanded in the spherical harmonics of degree l = 0 , 1 , 2 . . . We look for velocity potentials of the normal v n = ∇ φ$_{n}$ and su- SPHERICAL GEOMETRY