The processes of coronal plasma heating and cooling were previously shown to significantly affect the dynamics of slow magnetoacoustic (MA) waves, causing amplification or attenuation, and also dispersion. However, the entropy mode is also excited in such a thermodynamically active plasma and is affected by the heating/cooling misbalance too. This mode is usually associated with the phenomenon of coronal rain and formation of prominences. Unlike adiabatic plasmas, the properties and evolution of slow MA and entropy waves in continuously heated and cooling plasmas get mixed. Different regimes of the misbalance lead to a variety of scenarios for the initial perturbation to evolve. In order to describe properties and evolution of slow MA and entropy waves in various regimes of the misbalance, we obtained an exact analytical solution of the linear evolutionary equation. Using the characteristic timescales and the obtained exact solution, we identified regimes with qualitatively different behaviour of slow MA and entropy modes. For some of those regimes, the spatio-temporal evolution of the initial Gaussian pulse is shown. In particular, it is shown that slow MA modes may have a range of non-propagating harmonics. In this regime, perturbations caused by slow MA and entropy modes in a low-\(\beta \) plasma would look identical in observations, as non-propagating disturbances of the plasma density (and temperature) either growing or decaying with time. We also showed that the partition of the initial energy between slow MA and entropy modes depends on the properties of the heating and cooling processes involved. The exact analytical solution obtained could be further applied to the interpretation of observations and results of numerical modelling of slow MA waves in the corona and the formation and evolution of coronal rain.

先前已证明日冕等离子体加热和冷却过程会显着影响慢磁声 (MA) 波的动力学，导致放大或衰减，以及色散。然而，熵模也在这种热力学活跃的等离子体中被激发，并且也受到加热/冷却失衡的影响。这种模式通常与日冕雨和日珥的形成有关。与绝热等离子体不同，连续加热和冷却等离子体中慢 MA 和熵波的特性和演化是混合的。不同的失衡机制导致初始扰动演化的各种场景。为了描述各种失衡状态下慢 MA 和熵波的性质和演化，我们得到了线性进化方程的精确解析解。使用特征时间尺度和获得的精确解，我们确定了慢 MA 和熵模式的性质不同的制度。对于其中一些状态，显示了初始高斯脉冲的时空演变。特别是，它表明慢 MA 模式可能具有一定范围的非传播谐波。在这种情况下，由慢 MA 和低熵模式引起的扰动 结果表明，慢速 MA 模式可能具有一系列非传播谐波。在这种情况下，由慢 MA 和低熵模式引起的扰动 结果表明，慢速 MA 模式可能具有一系列非传播谐波。在这种情况下，由慢 MA 和低熵模式引起的扰动\(\beta \)等离子体在观察中看起来是相同的，因为等离子体密度（和温度）的非传播扰动随时间增长或衰减。我们还表明，慢 MA 和熵模式之间初始能量的分配取决于所涉及的加热和冷却过程的特性。获得的精确解析解可进一步应用于日冕中慢MA波的观测和数值模拟结果的解释以及日冕雨的形成和演化。