Sound Wave as a Particular Case of the Gravitational Wave

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1 Oen Journal of Acoutic, 1,, htt://dx.doi.org/1.436/oja Publihed Online Seteber 1 (htt:// Sound Wave a a Particular ae of the Gravitational Wave Vladiir G. Kirtkhalia Ilia Vekua Sukhui Intitute of Phyic Technology, Tbilii, Georgia Eail: it@it org.ge, v.kirtkhalia@gail.co Received June 7, 1; revied July 18, 1; acceted Augut 5, 1 ABSTRAT It i deontrated that the univerally acceted yte of ga-dynaic (hydrodynaic) equation i alicable only to hoogeneou (ientroic) edia require advanceent to get alicable to non-hoogeneou edia. A generalied equation of gravitational wave for adiabatic ideal edia i obtained fro advanced yte. Fro thi equation, in turn, i obtained an equation of acoutic wave, which i lane different for the known equation in that the hae eed of the wave in the Earth atohere obviouly deend on altitude, i.e., T intead of acceted T. Thu, acoutic wave i a hort-eriod gravitational wave in which gravitational effect are revealed at 3 altitude.3 1, which lead to alification of refraction of ound. The here of alicability of the equation i deterined it i deontrated that it i true only u to the uer boundary of the troohere ( 11-1 k.) above which anoalou rocee develo in the atohere. Keyword: Seed of Sound; Atohere; oreibility; Incoreibility 1. Introduction In the aer [1] it wa hown that the exreion of adiabatic eed of ound, 1 kt with the hel of which the eed of ound i currently deterined in lower a well a uer layer of the atohere, i alicable only u to the altitude 1 3. Above thi altitude obviou deendence of the ound eed on coordinate i revealed along which the atohere i non-hoogeneou a a reult of influence of gravitational field of the Earth. Revelation of the factor of gravitational filed becae oible due to the fact that in the equation of the tate of the atohere, conidered to be an ideal ga, entroy i taken into conideration i written in the for, intead of generally acceted, which i true only for iotroic edia i not alied to the Earth atohere. Such aroach enabled to deterine that along with adiabatic echani of generation of ound wave, there exit an iobaric one exactly thi echani lead to deendence of the eed of ound on altitude or on denity, which i the ae. The true value of the quare of ound eed which i deduced fro the quare of adiabatic iobaric eed of ound i obtained fro equation of the tate of the ediu i defined a the coefficient connecting erturbation reure denity. The reent aer deontrate that the known yte of ga-dynaic (hydrodynaic) equation i alo atifactory only for hoogeneou (iotroic) edia require advanceent to get feaible for non-hoogeneou edia. Thereafter, the advanced yte of equation i reduced to the generalied equation of ga-dynaic (hydrodynaic) wave in non-hoogeneou ediu. Alication of thi equation to erturbation of the Earth atohere with frequency bwidth correonding to the ound frequencie of range (1-14) H, how that the equation of ound wave in the Earth atohere i in the for of lane wave with the only difference that the quare of ound eed in it coincide with the value deterined in the aer [1], i.e.,, T in con- trat to currently acceted T.. Advanced Syte of Ga-Dynaic (Hydrodynaic) Equation Motion of frictionle liquid (or ga) in the gravitational field of the Earth i decribed by the equation of Euler: v vv t g (1.1) which i olved in conjunction with the equation of continuity of a divv (1.) t oyright 1 SciRe.

2 116 V. G. KIRTSKHALIA adiabatic equation v (1.3) t The Equation (1.) deterine change of denity liquid article in given volue V ay that it equal to difference of a flow incoing outgoing through the urface liiting uch volue []. However, change of denity ay occur at the cot of either change of ubtance a in contant volue or change of volue of contant a of ubtance. Indeed: d dv V d d dt dt dt dtv V (1.4) d d ln V dt dt V Auing in econd u of the right art of the Equation (1.4) = 1 i.e. V 1, we obtain: d d d lnv ln dt dt dt finally fro (1.4) we find out d d d dt dt dt V (1.5) Hence, colete change of denity conit of two art, the firt of which i deterined by the Equation (1.), i.e. d divv (1.6) dt t V The econd art decribe change of denity of the ubtance of contant a a a reult of change of volue which ay occur only due to change of teerature which, in turn, in the abence of heat ource, i oible only under change of entroy. d v (1.7) dt t Adiabatic Equation (1.3) i ued here. For hoogeneou (iotroic) ediu v (1.5)-(1.7) ake obviou that change of denity of liquid article i indeed deterined under the Equation (1.). For non-hoogeneou ediu it can be written v v (1.8) then, roceeding fro (1.5)-(1.8) the equation of continuity of a in non-hoogeneou edia will have the for of: d v v (1.9) dt T where the value equal to [1] d v (1.1) dt t the dienion of eed quare (1.11) T Thu, (1.9)-(1.11) ake aarent that in non-hoogeneou edia, when they are incoreible v denity i all the ae changed in the circutance of entroy erturbance. In light of the aforeentioned the yte of equation of ga-dynaic (hydrodynaic) for non-hoogeneou edia hould have the for: 1 v vv t g v v v t 3. Generalied Equation of Gravitational Wave (1.1) At wave otion v v are conidered all quantitie reure denity are rereented in the for of the u of their tationary Pr, r erturbed r, t, r, t value, after which the linearied yte of Equation (1.1) take the for v P g t vp v v t (.1) By uing the equation of equilibriu of ediu in gravitational field P g equation of the tate of ediu, where (ee [1]), the equation yte (.1) can eaily be reduced to the equation of ga-dynaic (hydrodynaic) wave in gravitational field of the Earth. 1 1 g g 1 g (.) t The Equation (.) ay be called the generalied equation of gravitational wave in ideal (nondiiative) adiabatic edia. In articular cae, under correonding value of coefficient, the equation of ound wave in the Earth atohere can be obtained fro thi equation. onequently, the ound wave i a hort eriod (high-frequency) gravitational wave ( 1 H) the oyright 1 SciRe.

3 V. G. KIRTSKHALIA 117 current oinion that the gravitational field ha no influence on generation roagation of ound wave i incorrect. The firt attet of correct derivation of the equation of gravitational field were taken in the work [3] [4], however the equation of continuity of a wa ued in the for of (1.) therefore their reult differ fro the Equation (.). 4. Equation of Sound Wave in the Earth Atohere It wa entioned above that the ound eed in the whole atohere i defined by forula kt 1 which doe not take into conideration the effect of gravitational field. In actual ractice the ound eed in the Earth atohere hould be identified fro Equation (.) for thi uroe it i neceary to deterine the here of it alication. Let firt call a quare of a certain eed which i reduced fro quare of certain eed, where (ee [1]) kt RT (3.1) M Then, for we will have ckt (3.) g 3 kt g 1 ckt (3.3) Here R = J/(ol K) ga contant, = kg a of one olecular of air = kg/ole a of one ole of air, while the adiabatic index c cv = 1.4 i the relation of heat caacity of air under contant reure volue. Exreion (3.1) (3.) ily that the Earth atohere rereent the ideal ga nkt deendence of denity on altitude i decribed by Lalace baroetric forula ex g However, a it aear fro exeriental data [5] at the altitude = 7.5 k, where air denity i calculated according to the forula (3.4) dro e tie, ratio error between calculated exeriental value of denity equal to 3% at the altitude = 11 k it contitute 4%, oreover the exeriental data exceed the calculated one. At the altitude = 35 k they equalie then error again increae at the altitude = 85 k reache 95%. It follow that the forula (3.4) work kt (3.4) oorly in the Earth atohere, eecially in the uer art of it. Beide, a it i aarent fro the chart of deendence of teerature on altitude ([1,5,6]), contructed on the bai of exeriental eaureent, in the interval of altitude fro = to = 1 k, teerature dro onotonouly according to the linear law 3 T aroxiately according 3 3 to the law T in interval of altitud e fro = 51 k to = 85 k. In the interediate interval (1-51 k) the teerature i either contant or increae, which allow to aue that either heat exchange or diiative rocee occur there. In the firt cae the ediu i nonadiabatic while in the econd cae it i not ideal ga. Thu, alication of forula (.) to the whole atohere would be incorrect. We aue that it decribe in good aroxiation wave rocee in the Erath atohere only in the interval of altitude fro = to = 11-1 k, i.e. in the troohere, where teerature dro trictly obey the linear law in which ratio error between exeriental calculated value of denity i inignificant. A hown in [1] the uer boundary of thi interval coincide with the altitude at which i defined by the exreion ckt =11.6 k (3.5) g where = 1 3 J/kg K T = 5.4 K i average value of teerature. We are not ready to dicu here the reaon of thi coincidence however we aue that it ha a dee hyical eaning. oefficient etiation at ' in Equation (.) how that in the ecified interval it change within the range thu thi u ay not be conidered. onequently, wave rocee in the troohere ( - 1 k) are decribed by equation t g (3.6) Let conider the Equation (3.6) for the wave reading in horiontal k kx ky vertical k k direction. In the firt cae ' i rereented in the for i k xk y t (3.7) ex x y after which g the Equation (3.6) tranfer into equation of lane wave in which 1 t, T (3.8) (3.9) k oyright 1 SciRe.

4 118 V. G. KIRTSKHALIA i deterined according to the forula (3.3) ha the eaning of the quare of hae eed of the wave, which i called ound or acoutic wave. Hence, eed deterined according to the forula (3.1) (3.) can alo be called adiabatic iobaric eed of ound. For wave reading in vertical direction (along the axi ) the Equation (3.6) ha to be olved by ethod of geoetrical otic, ince in thi direction the atohere i non-hoogeneou a a reult of influence of gravitational field. Following thi rocedure [7] reure erturbation will be reented in a for ex i t where i a certain dienionle function which atify the equation d k (3.11) d Then fro Equation (3.6) we will get (3.1) 1 1 i ig (3.1) The rie ark i herein taken to ean derivative with reect to. By denoting the uer boundary of the troohere through, which i deterined by forula (3.5) it will be eay to deontrate that (3.13) (3.14) (3.15) onequently, the Equation (3.1) ay be written a g 1 i i (3.16) Let ex along all araeter, where λ i the wave length i the ie of non-hoogeneity of ediu 1 (3.17) olve the Equation (3.16) in null aroxiation for which uroe we et ignore the econd derivative thereafter we will obtain g i 1 fro which for ' we ll find 1/ 1 ig g 1 4 (3.18) (3.19) Fro (3.19) it i een that for roagation of ound along axi the exreion under the radical ut be nonnegative, i.e. it i neceary to fulfill the condition 1/ g 4 1 / The function 1/ 1 / (3.) i cloe to unit in the whole interval of change of therefore fro (3.) it follow that g H (3.1) A we ee ound roagation condition i fulfilled with big reerve for ound frequencie ( 1 H), after neglect of the ter above the firt order of allne (which i equivalent to neglect of the econd u in Equation (.18)) the Equation (3.19) will reult in ' 1 (3.) Thu, ound wave reading in vertical direction i alo flat. Let now find ' 1, with thi uroe let inert 1 in (3.16). Having ignored the all quantity of the econd order taken into conideration th e Equation (3.18) we will obtain i d 1' ln' (3.3) d Fro (3.3) it i obviou that in (3.) the ign + hould be taken. Suoe the ound ource i in oint 1 ( 1 ) aue that in the tated interval teerature looely deend on altitude, et cont. Then fro (3.) (3.3) for 1 reectively we will obtain d i ln dln ' i 1 ln 1 (3.4) (3.5) Fro (3.1), (3.4) (3.5) for reure erturbation in firt aroxiation we will have oyright 1 SciRe.

5 V. G. KIRTSKHALIA 119 where t 1 ', t Pexi 1 P 1 1 P 1 14 (3.6) (3.7) Fro (3.7) we ee that when the ound i roagated fro the botto to the to the wave alitude dro while when it i roagated fro the to to the botto it increae. Thu, for intance if the ound ource i at ea level 1, then by reaching altitude =, wave alitude P.84 P. Let now find k, which by convention i k ' 1' uing exreion (3.) (3.3) we will find k 1 i 1 1 (3.8) Module of iaginary coonent k achieve larget extreu in the oint = equal to 4 I k 1 4 ax 1, which allow to ignore it a coared to the real art finally we will get k 1 (3.9) A we ee, the ound wave equation in any direction ha the for of the equation of lane wave the wave nuber in vertical direction i analytically exreed by the forula iilar to the horiontal direction. However, there i difference between the. In the econd cae the wave nuber at the tated altitude ha a contant value while in the firt cae it increae when ound i roagated fro the botto to the to decreae in cae of to-to-botto roagation. Such behavior of k erve a a reaon for the henoenon know a ound refraction which i that in the coure of roagation of the ound wave the direction of wave vector change to the ide of decreae of the ound eed. In the current theory thi henoenon i exlained by gradient along the axi of adiabatic eed of ound which i condi- tioned by it deendence on teerature T, which in it turn deend on according to the aforeentioned law. Adhering to our theory, gravitational field of the Earth alo contribute to ound refraction which i exreed by eergence of facient G1 1 in (3.9). For aeent of thi contribution it i neceary to calculate dk d, which equal to 1 3 dk T 3.61 G (3.3) d TG Effect of gravitational filed take into conideration the firt ter in nuerator (3.3), influence of which becoe deterining when G 3.31 (3.31) T Here we aue that G 43и T = 5.4 K are the average value of thee volue. Aarently thi i the altitude u to which the ound can be conidered adiabatic above which influence of gravitational filed cannot be ignored. 5. oncluion The ot iortant reult of the work i generaliation of the equation of continuity of a for non-hoogeneou edia after which it acquire clearer finihed hyical eaning. According to the exiting notion change of denity in incoreible edia i ioible ince it i ioible to change the a of liquid article of tated volue. Develoed equation of continuity of a (1.9) doe not contradict to the univerally acceted definition of incoreibility of ediu v, however it deontrate that for uch edia change of denity i all the ae oible at the cot of change of the volue of elected liquid article a a reult of entroy ocillation which certainly occur in the roce of echanical ocillation in non-hoogeneou edia. Thu there exit two echani of erturbation of denity: adiabatic V cont iobaric cont. When the firt echani revail, i.e. the ediu i coreible. In cae of revalence of the econd echani, i.e. the ediu i incoreible. Thi reult ha great ignificance, ince in incoreible aroxiation which i often alied in olving alied rob- le, the equation of tate of ediu ' ' i often reent in the yte of equation in it abence the yte i unable to decribe the real hyical r oce adequately. Beide, in the work it i deontrated that the ound wave i lane the equation of which i obtained fro the generalied equation of gravitational wave on the aution of adiabaticity ideality of the ediu. onequently, etablihed inference that acoutic wave i not gravitational i erroneou. Gravitational effect aear at altitude > which in the firt lace i exreed in growth of ound refraction. Thi i the very reaon due to which ound o far ha been conidered adiabatic in the whole atohere. Our calculation alo deontrate that they are true u to the uer boundary of the troohere above oyright 1 SciRe.

6 1 V. G. KIRTSKHALIA which anoalou rocee occur in the atohere. Preceding fro the aforeentioned it i afe to ay that the new theory of ound wave uggeted by u will trigger qualitatively new reearche in ga dynaic a well a in hydrodynaic. The author exree rofound gratitude to Profeor A. Rukhade for ueful advice recoendation. REFERENES [3] V. G. Kirtkhalia A. A. Rukhade, The Influence of Effective Gravity Field on the Develoent of Intability Tangential Dicontinuity, Kratkie Soobhchenya o Fiike, No. 4, Mocow, 6. [4] V. G. Kirtkhalia A. A. Rukhade, On the Quetion of Hydrodynaic Tangential Ga, Georgian International Journal of Science Technology, Vol. 1, No. 3, 8. [5] US Stard Atohere, National Aeronautic Sace Adinitration, [6] E. E. Goard W. H. Hooke, Wave in the Ato- [1] V. G. Kirtkhalia, Seed of Sound in Atohere of the here, Elevier, New York, Earth, Oen Journal of Acoutic, Vol., No.. 1, [7] A. F. Alekrov, L. S. Bogdankevich A. A. Ruk- hade, Onovi Electrodinaiki Plai, Mocow, [] L. D. Lau E. N. Lifhit, Theoretikal Phyic, Hydrodynaic, Vol. 6, oyright 1 SciRe.