By means of the solution of the equations of ordinary and two-liquid hydrodynamics, we study the oscillatory eigenmodes in isotropic nonpolar dielectrics He I and He II in the presence of a weak alternating electric field $\textbf{E}=E_{0}\textbf{i}_{z}\sin{(k_{0}z-\omega_{0} t)}$. The electric field and oscillations of the density become ``coupled,'' since the density gradient causes a spontaneous polarization $\textbf{P}_{s}$, and the electric force contains the term $(\textbf{P}_{s}\nabla)\textbf{E}$. The analysis indicates that the field $\textbf{E}$ changes the velocities of first and second sounds by the formula $u_{j}\approx c_{j}+\chi_{j} E_{0}^{2}$ (where $j=1, 2$, $c_{j}$ is the velocity of the $j$-th sound for $E_{0}=0$, and $\chi_{j}$ is a constant). We have found that the field $\textbf{E}$ jointly with a wave of the first (second) sound $(\omega,k)$ should create in He II hybrid acousto-electric (thermo-electric) density waves $(\omega + l \omega_{0},k + lk_{0})$, where $l=\pm 1, \pm 2, \ldots$. The amplitudes of acousto-electric waves and a change in the velocity of the first sound should resonantly increase at definite frequencies $\omega$ and $\omega_{0}$. These solutions can be verified experimentally.

中文翻译：

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存在交变电场时 He I 和 He II 中的声模式

通过求解普通流体动力学方程和双液体流体动力学方程，我们研究了弱交变电场存在下各向同性非极性电介质 He I 和 He II 的振荡本征模式 $\textbf{E}=E_{0} \textbf{i}_{z}\sin{(k_{0}z-\omega_{0} t)}$。电场和密度振荡变得“耦合”，因为密度梯度导致自发极化 $\textbf{P}_{s}$，并且电力包含项 $(\textbf{P}_ {s}\nabla)\textbf{E}$. 分析表明场 $\textbf{E}$ 通过公式 $u_{j}\approx c_{j}+\chi_{j} E_{0}^{2}$ 改变第一和第二声音的速度（其中 $j=1, 2$, $c_{j}$ 是 $E_{0}=0$ 时第 $j$ 个声音的速度，$\chi_{j}$ 是一个常数）。我们发现场 $\textbf{E}$ 与第一（第二）声波 $(\omega,k)$ 应该在 He II 混合声电（热电）密度波中产生 $( \omega + l \omega_{0},k + lk_{0})$，其中 $l=\pm 1, \pm 2, \ldots$。声电波的振幅和第一声速的变化应该在确定的频率 $\omega$ 和 $\omega_{0}$ 共振增加。这些解决方案可以通过实验验证。