It is shown that the criterion of incompressibility applicable to any medium, contradicts to the real meaning of this term. On the basis of expression of speed of sound in inhomogeneous medium and generalized equation of continuity of mass obtained in papers [1,2] respectively, it is proved that so called internal gravitation waves do not exist in nature. This concept appeared as a result of incorrect interpretation of incompressibility of medium. Correct understanding of criteria of compressibility or incompressibility leads to qualitatively new understanding of homogeneity or heterogeneity of medium, in particular—only strongly inhomogeneous medium can be incompressible while weakly inhomogeneous medium is always compressible. Besides, it is shown that in inhomogeneous media additional terms are added to known hydrodynamic (gas dynamic) correlations applicable to any medium which disappear at transfer to homogeneous model of medium.
It is common knowledge that all researches that have been carried out in the sphere of gas and hydrodynamics so far were based on the assumption that the speed of sound in the medium is the speed of distribution of perturbation of density which occurs as a result of change of the mass of the given volume of liquid at constant entropy and is determined by the formula
.
The medium is considered incompressible if or if, where is a characteristic speed of liquid motion. From this it follows that in the course of investigation of wave processes (except sound one), any stationary medium can be considered incompressible. Such approach leads to falling of the equation of state of medium connecting perturbed values of density and pressure from the system of hydrodynamic equations. Significance of this equation is also that it contains thermodynamic parameters of medium and in case of its absence it is unclear, which medium is considered. Besides, it is obvious that in this case perturbation of density does not occur and distribution of wave is impossible. That is why many gas and hydrodynamic tasks like the tasks of surface gravity wave and internal wave are solved incorrectly [
The above determination of the speed of sound is fair only for homogeneous medium, when and bears no relation to reality. In fact, any medium in gravitational field is inhomogeneous to a greater or lesser extent and therefore [
In the first section of the paper, based on recently published calculations [
From determination of density it follows that its change in time is calculated according to the formula [
The first summand in the right part (1.1) determines change of density at constant volume, which may occur only at constant entropy at the expense of difference of liquid flows through the surface limiting this volume. Therefore
where is speed of fluid. On the other hand
Expression, connecting perturbation of density and pressure at constant volume, has the dimension contrary to the dimension of square of speed, and as it turned out, its reciprocal value complies with the equation of sound wave in homogeneous medium when [3-5]. Therefore, the expression
is called speed of sound in medium. Curiously enough, in this connection it is not mentioned that the medium is homogeneous and it is assumed that such determination of speed of sound is fair for any medium, which leads to paradoxical results. We will demonstrate it by the example of the Earths’ atmosphere. Considering the air as ideal gas, (J/К—Boltzmann’s constant,—concentration of molecules of air,— absolute temperature) in which dependence between pressure and density in adiabatic process is defined by the following correlation
where and are initial values of pressure and density, and an adiabatic index is ratio of thermal capacities (specific heath), for air in presence of constant pressure and volume respectively, from (1.5) for speed of sound following expression is derived:
Here J/mol∙K is a gas constant, is mass of one air molecule and kg/mol—mass of one mol air. Speed of sound in the whole atmosphere is calculated according to formula (1.6) [
which succeeds to be a substitution to the mentioned in introduction formula of values from (1.6) and, physical meaning of which will be determined below. We see that sound speed in the Earth atmosphere is obviously dependent on altitude (density) and when, i.e. on the assumption of homogeneity of atmosphere, formula (1.7) transfers into formula (1.6).
The second summand in the right part of (1.1) describes change of density of the substance of constant mass as a result of change of volume which, in turn, in absence of heat source, is possible only under change of entropy, which certainly occurs at mechanic oscillation of inhomogeneous media, i.e.
Having expressed from the adiabatic equation
we will obtain
The expression has the dimension contrary to dimension of square of speed, therefore we can introduce expression
where may be called isobaric speed of sound. This value is the measure of inhomogenuity of medium and is characteristic for the speed of distribution of density perturbation related to isobaric change of volume of the given mass of substance. Applying (1.2), (1.10) and (1.11), from (1.1) we will ultimately obtain
Exactly this is the generalized equation of mass continuity which for homogeneous medium grades into the equation applied in the contemporary theory which was considered universal. Thus, compressibility of homogeneous medium cannot be ignored since from (1.12.) it follows that in such case change of density is impossible and consequently generation of wave processes is impossible.
Equation (1.12) can be written down as follows:
Integrating the Equation (1.13) on the given volume and applying Gauss-Ostrogradsky theorem on transformation of volume integral into superficial, from (1.13) we will obtain
where is the liquid mass in the given volume and is the vector in magnitude equal to area of the element of surface limiting given volume and directed along outer normal.
The first summand in the right part of the Equation (1.14) determines change of mass in stagnant volume of liquid conditioned by difference of density flows through the surface limiting this volume while the second summand determines change of mass conditioned by motion of liquid within the volume itself due to natural pressure drop in inhomogeneous medium.
As mentioned in the introduction, incorrect interpretation of significant concepts like compressibility or incompressibility leads to erroneous solution of many tasks of gas and hydro dynamics. As an example tasks of surface gravity wave and internal wave were cited. The first task was considered in paper [
The authors call gravity waves within incompressible liquid located in the Earth’s gravitational field the internal wave. With respect to incompressibility they neglect change of density related to change of pressure and assume that change of density can only be isobaric, at the expense of change of entropy under mechanic oscillation of inhomogeneous medium, i.e.
It can easily be noticed that these are the waves we were talking about in the first section and the Equation (2.1) is identical to the Equation (1.8). Thereafter they write down linearized equations of motion and mass discontinuity in the form
Having presented all perturbed values of variable values in the form, from Equations (1.9), (2.2) and (2.3) they obtain
Multiplying the Equation (2.5) by vector they receive
From Equations (2.4), (2.5) and (2.7) can easily be obtained dispersive equation in the form
where
and is angle between vector and axis. Thus, we get some kind of strange transverse wave, frequency of which depends only on direction of wave vector and it can be of any length. The reason of this paradox is that if in the equation of mass continuity the second summand related to heterogeneity of medium is not taken into consideration, then the condition means or following (2.1) and then all equations from which dispersive Equation (2.8) is obtained are nulling.
Thus, no special internal gravity wave exists. This is an ordinary sound wave in strongly inhomogeneous liquid which can be considered incompressible. Indeed, the system of linearized hydrodynamic equations in inhomogeneous medium takes the form [
Having applied medium balance equation in gravity filed and medium condition equation, where, and also applying operator to the first equation of the system (2.10) and operator to the second one, it can easily be reduced to generalized equation of gravity waves (see [
Having assumed that the medium is strongly inhomogeneous and having neglected the third summands in respect to its smallness as compared to other summands, we will get
This equation is nothing else but equation of sound wave in strongly inhomogeneous liquid which is transverse with respect to oscillations of speed and longitudinal with respect to oscillations of density. It can be obtained directly from (2.10) having laid and.
Let us consider change of density of energy of immovable volume of liquid, i.e. calculate integral
where is inner energy of unit mass of liquid. First let’s calculate the first summand of (3.1)
Having applied the generalized equation of mass continuity (1.12) and the motion equation of Euler
from (3.2) we will obtain
Here the correlation is used. From the definition of differential of thermal function of mass unit of liquid we can write
Following which from (3.5) we will obtain
Having applied correlation, and Equation (1.9) from the second summand of (3.1) we will find out
Having considered (3.6) and (3.7) for (3.1) we will obtain
Having reconstructed the first volume integral in the right part of Equation (3.8) into superficial, we will ultimately obtain
Change of density of impulse in immovable volume of liquid is described by derivative. Let us consider this derivative in components, i.e.
and write Equations (1.12) and (3.3) in the form
where indexes, moreover and—the components of unit vector directed along the axis. Having substituted (3.11) and (3.12) in (3.10) we will obtain
Integration of Equation (3.13) according to volume gives
Correlations (3.9) and (3.14) for homogeneous liquids are obtained in the monograph [
Our reasoning conclusively proves that there are no ideally homogeneous media in nature. Any medium is inhomogeneous to a greater or lesser degree depending on correlation and. Moreover, homogeneity of medium does not mean that it is not influenced by external field of force. For instance, in the Earth atmosphere isobaric speed of sound is determined by formula
[