Translation:On the influence of uneven temperature distribution on the propagation of sound

by Ippolit S. Gromeka

On the influence of uneven temperature distribution on the propagation of sound.

 

I. S. Ippolit

The temperature distribution in the air is known to have a significant impact under various circumstances on the propagation of sound waves. To present a theoretical explanation of this phenomenon, I outlined in an article published in the Scientific Notes of the Kazan University^[1] the derivation of the differential equations for small fluctuations in unevenly heated air mass, neglecting in these equations not only the terms proportional to the squares of the velocities, but also those that are proportional to the square of the oscillation time. Currently, I am convinced that the assumption of small oscillation times, firstly, does not seem necessary for the derivation of the mentioned differential equations, and secondly, it is in contradiction with some other assumptions. Having corrected this mistake and discarded the idea of small oscillation times, I decide in this article to present a dreive differential equations for very small oscillations in an unvenely heated air.

§ 1. Assumptions about the properties and the equation of state of the gas

Let us imagine a mass of air that is at a certain moment $t_{0}$ at rest and in equilibrium and assume that at this moment, the temperature distribution, pressure and density of the gas are known. Determining the position of each spatial points with rectangular coordinates $x,y,z$ and denoting for the moment $t_{0}$ under consideration through $p_{0}$, $\rho _{0}$ and $T$ the pressure, density and absolute temperature of the gas in any element of it at a known point $A(x,y,z)$, we will consider $p_{0}$, $\rho _{0}$ and $T$ as given functional coordinates. If at the same time $X,Y,Z$ are the projection along the axes of the external forces, then from the known equations of hydrostatics, we have the equations

(1)${\frac {1}{\rho _{0}}}{\frac {\partial p_{0}}{\partial x}}=X,\,\,{\frac {1}{\rho _{0}}}{\frac {\partial p_{0}}{\partial y}}=Y,\,\,{\frac {1}{\rho _{0}}}{\frac {\partial p_{0}}{\partial z}}=Z.$

We will assume that none of the quantities

${\frac {\partial \rho _{0}}{\partial x}},\,\,{\frac {\partial \rho _{0}}{\partial y}},\,\,{\frac {\partial \rho _{0}}{\partial z}},\,\,\,\,{\frac {\partial p_{0}}{\partial x}},\,\,{\frac {\partial p_{0}}{\partial y}},\,\,{\frac {\partial p_{0}}{\partial z}},$

${\frac {1}{\rho _{0}}}{\frac {\partial \rho _{0}}{\partial x}},\,\,{\frac {1}{\rho _{0}}}{\frac {\partial \rho _{0}}{\partial y}},\,\,{\frac {1}{\rho _{0}}}{\frac {\partial \rho _{0}}{\partial z}},\,\,\,\,{\frac {1}{p_{0}}}{\frac {\partial p_{0}}{\partial x}},\,\,{\frac {1}{p_{0}}}{\frac {\partial p_{0}}{\partial y}},\,\,{\frac {1}{p_{0}}}{\frac {\partial p_{0}}{\partial z}},$

has an infinitely greater value at any point.

Let us now imagine that, as a result of some shaking, the mass of gas began to move, and that its particles make very small fluctuations around its previous equilibrium positions. Let us denote by $u,v,w$, the velocity of the particle along the axes of the coordinates passing, at a known time $t$, through the point $A(x,y,z).$ Pressure and density at the same point for the moment $t$ are denoted by $p$ and $\rho$.

We will also assume that the changes in gas densities occurring during oscillations in each individual points of the space are very small in relation to equilibrium density. A similar assumption for pressure is also assumed. As a result, letting

$s={\frac {\rho -\rho _{0}}{\rho _{0}}},\,\,\,q={\frac {p-p_{0}}{p_{0}}},$

or in other words,

(2)$\rho =\rho _{0}(1+s),\,\,\,p=p_{0}(1+q),$

we assume $s$ and $q$ to be small quantities of the first order, on par with the values $u,v,w$ and therefore, we agree to discard in the equations any term containing the product of any pair of the these five quantities.

In addition, we will accept, as is usually done when deriving the basic equations of the theory of sound, that the oscillating gas is subject to the laws of Mariotte and Gay-Lussac, and that each particle of gas retains all its warmth. Mariotte and Gay-Lussac law for the moment $t$ is expressed by the equation

(3)$p_{0}=k\rho _{0}T,$

in which $k$ is the known constant; for the adiabatic process accompanying the vibration of particles, we will use the known equation

(4)${\frac {p}{\rho ^{\gamma }}}=h,$

in which $\gamma$ represents the constant ratio of specific heats and $h$ is a constant quantity for each individual particle. Understanding $h_{0}$ to be value of $h$ for the moment $t_{0}$ and neglecting quantities of second order, we have from equations (2) and (4)

$h=h_{0}(1+q-\gamma s).$

Let us agree to denote by the symbol $D$, the total differentiation with respect to time of any function that characterises the state of each gas particle, i.e., let us define

${\frac {D}{Dt}}={\frac {\partial }{\partial t}}+u{\frac {\partial }{\partial x}}+v{\frac {\partial }{\partial y}}+w{\frac {\partial }{\partial z}}.$

The adiabatic nature of the oscillations can then be expressed by the equation

$h_{0}\left({\frac {Dq}{Dt}}-\gamma {\frac {Ds}{Dt}}\right)+(1+q-\gamma s){\frac {Dh_{0}}{Dt}}=0.$

In here, letting

$\mathrm {ln} h_{0}=f,$

and neglecting infinitesimal terms of higher order, we find

(5)${\frac {\partial q}{\partial t}}-\gamma {\frac {\partial s}{\partial t}}+u{\frac {\partial f}{\partial x}}+v{\frac {\partial f}{\partial y}}+w{\frac {\partial f}{\partial z}}=0.$

§ 2. Differential equations of small oscillations in unevenly heated air mass

Applying the basic formulas of hydrodynamics to the large fluctuations in the air mass and discarding very small terms of the second order, we have the equations

${\frac {\partial u}{\partial t}}=X-{\frac {1}{\rho }}{\frac {\partial p}{\partial x}},\,\,{\frac {\partial v}{\partial t}}=Y-{\frac {1}{\rho }}{\frac {\partial p}{\partial y}},\,\,{\frac {\partial w}{\partial t}}=Z-{\frac {1}{\rho }}{\frac {\partial p}{\partial z}}.$

Substituting here the expressions p and p from (2) and again discarding second-order terms, we have

${\frac {\partial u}{\partial t}}=X-{\frac {1}{\rho _{0}}}{\frac {\partial p_{0}}{\partial x}}+X(s-q)-kT{\frac {\partial q}{\partial x}},$

${\frac {\partial v}{\partial t}}=Y-{\frac {1}{\rho _{0}}}{\frac {\partial p_{0}}{\partial y}}+Y(s-q)-kT{\frac {\partial q}{\partial y}},$

${\frac {\partial w}{\partial t}}=Z-{\frac {1}{\rho _{0}}}{\frac {\partial p_{0}}{\partial z}}+Z(s-q)-kT{\frac {\partial q}{\partial z}},$

which, following the relations (1) take the form:

(6)${\frac {\partial u}{\partial t}}=X(s-q)-kT{\frac {\partial q}{\partial x}},$

(6)${\frac {\partial v}{\partial t}}=Y(s-q)-kT{\frac {\partial q}{\partial y}},$

(6)${\frac {\partial w}{\partial t}}=Z(s-q)-kT{\frac {\partial q}{\partial z}}.$

This must be accompanied by the condition of continuity, which, with the accepted degree of approximation, can be represented by the equation

(7)$u{\frac {\partial \mathrm {ln} \,\rho _{0}}{\partial x}}+v{\frac {\partial \mathrm {ln} \,\rho _{0}}{\partial y}}+w{\frac {\partial \mathrm {ln} \,\rho _{0}}{\partial z}}+{\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}+{\frac {\partial w}{\partial z}}+{\frac {\partial s}{\partial t}}=0.$

Equations (5), (6) and (7) serve to determine small oscillation of gas under pressure and any external forces and under the influence of any temperature distribution.

§ 3. On the influence of gravity on vertically propagating plane sound waves.

Let us consider the vertical propagation of a plane sound wave in uniformly heated air, taking in account of the effects of gravity. Directing the $x$ axis vertically up and denoting the gravitational acceleration by $g$, we will have

for the case under consideration,

$v=0,\,\,w=0,\,\,X=-g,\,\,Y=0,\,\,Z=0,\,\,f=Const.-(\gamma -1)\mathrm {ln} \,p_{0}.$

Equations (5), (6) and (7) take the following form:

${\frac {\partial q}{\partial t}}-\gamma {\frac {\partial s}{\partial t}}+{\frac {g(\gamma -1)}{kT}}u=0,$

${\frac {\partial u}{\partial t}}=-g(s-q)-kT{\frac {\partial q}{\partial x}},$

${\frac {\partial u}{\partial x}}+{\frac {\partial s}{\partial t}}-{\frac {gu}{kT}}=0.$

Eliminating $s$ and $q$ from here, we find the equation

${\frac {\partial ^{2}u}{\partial t^{2}}}=k\gamma T{\frac {\partial ^{2}u}{\partial x^{2}}}-g\gamma {\frac {\partial u}{\partial x}}.$

This can be satisfied by the integral

(8)$u=Ae^{\varepsilon x}\cos(mx-nt-\delta ),$

in which $A$, $\varepsilon$, $m$, $n$ and $\delta$ are constants; substituting this integral into equation (8) and setting $k\gamma T=a^{2}$, we will find the relationship

$m^{2}={\frac {n^{2}}{a^{2}}}-{\frac {g^{2}\gamma ^{2}}{4a^{4}}},\,\,\varepsilon ={\frac {g\gamma }{2a^{2}}}.$

Expression (8) represents a wave propagating vertically at constant speed

$V={\frac {a}{\sqrt {1-{\frac {g^{2}\gamma ^{2}}{4a^{2}n^{2}}}}}}$

but with varying amplitude. The speed of the sound under these circumstances is increased under the influence of gravity, but it is easy to see that this increase is negligible. So for example, taking the meter as the unit of length, second as a unit of time, counting the number of oscillations per second $N=200$, and putting approximately $\gamma =1.4$, $g=9.8$ $a=330$, $\pi =3.14$, we find

$V=a(1+\omega ),$

where

$\omega <{\frac {1}{10^{8}}}.$

Slightly more significant and influential is the $e^{\varepsilon x}$ multiplier, which is with the numerical values that we took, is approximately

$\varepsilon ={\frac {1}{15875}}.$

Along the upward propagation of a sound wave, the sound strength increases slightly. But this conclusion obviously relates to the proposition that all along the way sound waves and gravity do not change, temperature is everywhere is the same, and gas particles, apart from elasticity and gravity, do not experience any other influences.

§ 4. Effect of temperature on sound propagation in the absence of external forces.

Having thus become convinced of the very weak effect gravity on the propagation of sound, let us return to the general equations (6) and, discard terms in them that depend on external forces but retain what was made earlier by assuming that the temperature is a given function of coordinates. Then instead of equation (6) we will have the following

(9)${\frac {\partial u}{\partial t}}=-kT{\frac {\partial q}{\partial x}},\,\,{\frac {\partial v}{\partial t}}=-kT{\frac {\partial q}{\partial y}},\,\,{\frac {\partial w}{\partial t}}=-kT{\frac {\partial q}{\partial z}}.$

As for equation (5), it should be noted that

in the absence of external forces, the pressure has the same value in all places of the gas and therefore

$f=Const.-\gamma \mathrm {ln} \,p_{0}=Const.+\gamma \mathrm {ln} \,T.$

As a result of this, equation (5) takes the following form for the case under consideration:

(10)${\frac {\partial q}{\partial t}}-\gamma {\frac {\partial s}{\partial t}}+{\frac {\gamma }{T}}\left(u{\frac {\partial T}{\partial x}}+v{\frac {\partial T}{\partial y}}+w{\frac {\partial T}{\partial z}}\right)=0.$

Equation (7) in this case will be

${\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}+{\frac {\partial w}{\partial z}}+{\frac {\partial s}{\partial t}}-{\frac {1}{T}}\left(u{\frac {\partial T}{\partial x}}+v{\frac {\partial T}{\partial y}}+w{\frac {\partial T}{\partial z}}\right)=0$

or othrewise using equation (10),

(10)${\frac {\partial q}{\partial t}}+\gamma \left({\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}+{\frac {\partial w}{\partial z}}\right)=0.$

The system of equations (9), (10) and (11) should serve to resolve small fluctuations in the air mass in the absence external forces. It is remarkable, however, that for any temperature distribution the functions $u,v,w,s$ can be excluded from equations (9), (10) and (11). In fact, by differentiating the first of the equations (9) by $x$, the second by $y$, the third by $z$, adding and taking into account equation (11), we find

(12)${\frac {\partial q^{2}}{\partial t^{2}}}=k\gamma \left[{\frac {\partial }{\partial x}}\left(T{\frac {\partial q}{\partial x}}\right)+{\frac {\partial }{\partial y}}\left(T{\frac {\partial q}{\partial y}}\right)+{\frac {\partial }{\partial z}}\left(T{\frac {\partial q}{\partial z}}\right)\right].$

Once the function $q$ is found as the integral of this equation, the functions $u,v,w$ can easily be determined from equation (9).

In the future, we will consider only those movements that consist of simple oscillations and have them the same period in all places of space occupied by the gas. For such movements, denoting by $n$ some constant, and by $q_{0},u_{0},v_{0},w_{0},\delta ,\delta ',\delta '',\delta '''$ some

functions of coordinates, we will have

$u=u_{0}\cos(nt+\delta '),\,\,v=v_{0}\cos(nt+\delta ''),\,\,w=w_{0}\cos(nt+\delta '''),\,\,q=q_{0}\sin(nt+\delta ).$

In this case, equations (12) and (9) will be replaced by the following:

(13)${\frac {\partial }{\partial x}}\left(T{\frac {\partial q}{\partial x}}\right)+{\frac {\partial }{\partial y}}\left(T{\frac {\partial q}{\partial y}}\right)+{\frac {\partial }{\partial z}}\left(T{\frac {\partial q}{\partial z}}\right)+{\frac {n^{2}q}{k\gamma }}=0,$

(14)$u={\frac {kT}{n^{2}}}{\frac {\partial ^{2}q}{\partial x\partial t}},\,\,v={\frac {kT}{n^{2}}}{\frac {\partial ^{2}q}{\partial y\partial t}},\,\,w={\frac {kT}{n^{2}}}{\frac {\partial ^{2}q}{\partial z\partial t}}.$

§ 5. Study of some particular cases of sound propagation in an unevenly heated medium.

1. Let us first assume that the temperature of the medium decreases uniformly along a certain straight line, and that fluctuations in the air mass occur in the same direction. Let us consider, under these assumptions, the propagation of a plane wave corresponding to the sound of a certain tone.

Since in a quiescent atmosphere the temperature usually decreases with increasing elevation above the earth's surface, and since for heights that are not too significant this decrease can be taken to be proportional to the heights,obviously the particular case under consideration is of particular interest, allowing us to indicate circumstances, which can be accompanied in nature by vertical propagation of sound waves.

Taking the direction of the mentioned straight line as the $x$-axis and denoting by $\varepsilon$ a known constant, and by $T_{0}$ the temperature of the medium in the plane passing through origin, we will have

$T=T_{0}-\varepsilon x.$

Equation (13) is accepted for the case under consideration, which once you introduce a new variable $T$, will have the form

${\frac {\partial ^{2}q}{\partial T^{2}}}+{\frac {1}{T}}{\frac {\partial q}{\partial T}}+{\frac {n^{2}}{\varepsilon ^{2}k\gamma }}{\frac {q}{T}}=0.$

Its integral can be expressed in terms of the Bessel functions.

Denoting by $A,B,A',B'$ the arbitrary constants and introducing the notation

$z={\frac {2n}{\varepsilon }}{\sqrt {\frac {T}{k\gamma }}},$

we have exactly,

$q=[AJ_{0}(z)+BY_{0}(z)]\cos nt+[A'J_{0}(z)+B'Y_{0}(z)]\sin nt.$

Hence, using equation (9), denoting by $C,D,C',D'$ some other constants, and based on the known relation

${\frac {dJ_{0}}{dz}}=-J_{1}(z),$

let us find the expression for the speed

$u=[CJ_{1}(z)+DY_{1}(z)]{\sqrt {T}}\cos nt+[C'J_{1}(z)+D'Y_{1}(z)]{\sqrt {T}}\sin nt.$

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1. Gromeka. I.S. On the Effect of Temperature on Small Variations in Air Masses; Scientific Notes of Kazan University; Kazan University: Kazan, Russia, 1888; pp. 1–40.