dp p v= = ON SHOCK WAVES AT LARGE DISTANCES FROM THE PLACE OF THEIR ORIGIN By Lev D. Landau J. Phys. U.S.S.R. 9, 496 (1945).

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1 ON SHOCK WAVES AT LARGE DISTANCES FROM THE PLACE OF THEIR ORIGIN By Lev D. Landau J. Phys. U.S.S.R. 9, 496 (1945). It is shown that at lage distanes fom the body, moving with a. veloity exeeding that of the sound, thee exists not one (as is usually assumed) but two subsequent shok waves. The shae of these waves and the law of the deease of thei intensity with distane is detemined. The oagation of a sheial shok (exlosive) wave at lage distanes fom the lae of exlosion is examined. At lage distanes fom the lae of thei oigin shok waves ae faint and have, theefoe, the haate of sound waves. Howeve, fo ou uose the odinay linea aoximation is insuffiient and it is neessay to onside the oeties of sound waves with a small amlitude in the seond aoximation. We shall have to do below with ylindial and sheial waves; sine, howeve, at lage distanes ylindial o sheial waves an be onsideed in evey small egion as a lane one, we shall dwell eliminaily on some oeties of lane waves. As is well known, a one-dimensional ogessive wave with an abitay amlitude is desibed by the so-alled Riemann solution of the equations of motion x = t[ v+ ( v)] + f ( v), whee f(v) is an abitay funtion of the veloity v of the gas, while the loal veloity of sound, onneted with v by the elation d V v= = d ρ (ρ - density, V - seifi volume of the gas). These two fomulae detemine, imliitly, the veloity v (and along with it the othe quantities haateizing the wave) as a funtion of :x and t, i.e., the shae of the wave at any given instant. Fo t = we have x = f(v), i.e., the funtion invese to f(v) detemines the shae of the wave at the initial moment. The quantity u = v + (v) (1) is the veloity at whih the oints of the wave ofile ae tavelling. This 1 1/

2 veloity is diffeent fo diffeent oints of the ofile, and that is why it does not emain onstant and vaies its shae with time. Exessing u as funtion, say, of the essue in the wave, we have fo the deivative du : d but = = V ρ du d 1 = +, d d ρ V The adiabati deivative so that alulation yields du 1 V = ρ 3 d. V S S (S - entoy) is ositive fo all gases, so dv that >. The veloity of the dislaement of a. given oint of the wave d ofile is thus the lage the lage is the essue at this oint; the omessions ae, theefoe, gadually advaning with eset to the aefations. Fo a wave with a small amlitude the dislaement veloity u of the oints of the ofile in the fist aoximation is obtained by utting in (1) v =, i.e., u = (the lettes with the index zeo denote the equilibium values of the oesonding quantities), whih oesonds to a. dislaement of the wave ofile without any hange of its shae. In the next aoximation we have o du u = + ', d ' u = 1 + α, () α = V V ( - denotes the vaiable at of the essue in the wave). Fo an ideal gas γ + 1 α = (fo ai α =. 86 ), whee γ at onstant essues and volume. S γ = is the atio of the seifi heats When the ofile of the wave is distoted to suh an extent that the uniqueness of the solution disaeas, a shok wave aises. The Riemann solution beomes, geneally seaking, inaliable afte the fomation of disontinuities. It is, howeve, natual that fo waves of small amlitude, in the seond aoximation unde onsideation, this solution emains valid in the v

3 esene of suh disontinuities. This an be seen in the following way. The jums in the veloity, essue, and seifi volume in the disontinuity ae onneted with eah othe by the elation v 1 ( 1 1 V v = )( V ). The hange of the veloity v ove a etain segment of the length of the x-axis in the Riemann solution is equal to the integal V v v1 = d. 1 A simle alulation with the hel of an exansion into a. seies shows that. the eeding exession diffe in the tems of the thid ode only (in aying out the alulation it must be ket in mind that the hange of entoy in the disontinuity is a quantity of the thid ode, while in the Riemann solution the entoy is onstant). Hene it follows that (in an auay u to tems of the seond ode) the motion in the tavelling wave in the esene of the disontinuity an be desibed on eah side of it by the Riemann solution with suitable bounday ondition on the disontinuity. In the following aoximations this will, howeve, no longe be the ase, whih is onneted with the aeaane of a efletion fom the sufae of the disontinuity. The osition of the disontinuity in the wave is detemined by a simle geometial ondition, whih an easily be deived with the hel of fomula () and the ondition of the ontinuity of the flow of the substane though the disontinuity. Namely, the disontinuity is haateized by the ondition that the aea of the uve, eesenting the ofile of the wave should emain the same as that of the multivalued uve, detemined by Riemann s solution. 1/ Let us now onside a body, efoming a steady motion with a veloity U exeeding that of sound. Let the x-axis be aallel to the dietion of the motion of the body and let be the distane fom this axis. At lage distanes fom the body the veloity otential φ (, z) of the gas is detemined, in the fist aoximation, by the wave equation 1 φ φ 1 φ = +. t x The steadiness of the motion of the body is exessed by the equation. φ φ + U =. t x Combining these two equations we get 3

4 U φ 1 φ 1 =. x Relaing x by the vaiable we obtain the equation τ = 1 U x φ 1 φ =, (3) τ i.e., the equation of a ylindial wave in whih the ole of the time is layed by the vaiable τ. At suffiiently lage distanes the ylindial wave within eah small egion an be teated as a lane one. The dislaement veloity of eah oint of the ofile of the wave will then be detemined by fomula (). If we wish, howeve, to follow with the hel of this fomula the dislaement of. a oint of the wave ofile duing lage time intevals, it is neessay to take into aount the fat that the amlitude of the ylindial wave deeases with the distane aleady in the fist aoximation as ' = and substituting in equation (), we get χ 1 /. Witing, (4) αχ u = 1 +. (5) The fist tem oesonds to a. dislaement of the wave without hange of ofile {aside fom the geneal deease of the amlitude as seond leads to a. distotion of the ofile. The magnitude 4 1 / ), while the δ of the additional dislaement of the oint of the ofile at a distane fom a. etain given (lage) to is obtained by multilying by d / and integating fom to fo a. onstant χ : δ = αχ ( ). If the ofile of the wave is defined by the hange of ' with τ, fo a given then the distotion δτ of the ofile will be δτ = δ /, i.e., δτ = αχ ( ). (6) A diveging ylindial wave an be witten, in the linea aoximation, in the following way:

5 The ositive sign in f ( τ + ξ / ) dξ φ =. (7) ξ τ + ξ / oesonds to the fat that in the esent ase the wave is oagated fom ositive values to the negative ones (hee and below the index zeo, oesonding to the equilibium values of the diffeent quantities, is doed fo the sake of bevity). In ou ase the time τ means in eality the o-odinate x. We shall take the oigin of the o-odinates inside the body (at a given instant), in this ase the egions in font of the body oesond to ositive x. Sine in the ase of: a motion with a veloity exeeding that of sound the etubations ae not oagated in the egion of sae, lying in font of the body, it an, in any ase, be asseted that φ fo τ. Futhe, at suffiiently lage distanes behind the body, whee the etubations aused by it ae small, even on the axis itself, the otential of the divegent wave detemined by fomula (7) must emain finite fo τ =. Fo the onvegene of the integal = dξ φ(, τ ) f ( τ + ξ / ) ξ at the lowe limit (fo lage negative τ ) it is neessay, that f ( τ ) fo lage negative τ. Hene it is easy to onlude that fo φ. τ we have also On the othe hand, the vaiable at of the essue in the linea aoximation is onneted with φ by the equation with eset to φ we get, onsequently, φ ' = ρ. Integating τ + ' dτ =. (8) This means that if thee is a omession in the gas (the egion with > ), then thee must also exist a egion of aefation, whee < (in this eset a ylindial wave - the same efes to a sheial one - diffes substantially fom a lane wave, whih an onsists of omessions o of aefations only). As is well known, if the veloity of the body is lage than the veloity of sound, thee aises in the gas a shok wave: in the sae lying in font of this wave the gas emains at est and dietly behind the wave thee is a egion of omession. It follows fom (8) that the omession must neessay be followed by a aefation and that, onsequently, thee must exist a oint in whih the aefation eahes its maximum; owing to the gadual distotion of the ofile this oint will lag with eset to those situated behind it; this 5

6 will, finally, lead to a loss of the uniqueness of the solution and one moe shok wave will aise. We aive at the esult that, at least at lage distanes fom the moving body, thee exists not a single shok wave, as is usually assumed, but two shok waves following eah othe. In the fist wave the essue suffes a ositive jum; theeafte the essue is gadually deeased, the omession being elaed by a. aefation, the essue, finally, juming again in the seond shok wave. Figue 1 eesents shematially (by a ontinuous line) the esulting itue of the deendene of essue on τ (i.e., on the o-odinate x) fo a given (lage) value of ; ab is the fist shok wave, de - the seond wave. In the latte the essue ineases u to a etain negative value only, tending to zeo asymtotially when τ. Passing to the quantitative alulation of the ofile eesented in Fig. 1, let us onside the egion between the two shok waves. Let the funtion τ = f (χ) (whee χ denotes ' / distane. ) detemine the ofile at a etain Taking into aount the effet of the distotion of the ofile, we obtain the ofile at a distane >> by adding to τ the dislaement δτ fom (6): α τ = f ( χ) + ( ) χ. (9) Fo lage χ is small, and one an with a suffiient auay wite in (9) the value of the funtion f (χ) fo χ =, while an be negleted when omaed with : α τ = χ+ onst. (1) Denoting by x the value of the o-odinate x at the oint (Fig. 1), whee χ = (it deends, of ouse, on :, aoding to the law 6

7 x = onst ) and assing to and x instead of χ and τ, we get hene ' 1 = U α x x 1. (11) The segment db of the ofile thus oves to be etilinea. The ofile, whih is obtained dietly by the aliation the Riemann solution thoughout the whole inteval, is eesented in Fig.1 by a dotted line. In eality thee exists a disontinuity at a etain oint a. The osition of this oint is detemined by the above mentioned geometial ondition of the equality of the aeas a b and ab. Noting that in the oints a and χ =, we find with the hel of (9) fo the ae a b : χ dτ = χf '( χ) dχ, ( ' b' a') ( ' b' a') i.e., a value, whih is indeendent of τ. The same must, onsequently, hold fo the aea ab. Taking into aount the deendene (1) χ on τ we find without diffiulty that the length l 1 of the segment fom the oint (whee = ) u to the font shok wave (' = 1 ) is ootional to l 1/ 4 1 ~. (1) Hene the deendene of the jum ' 1 of the essue in the font shok wave on the distane is exessed by onst ' 1 =. (13) 3/ 4 As egads the seond disontinuity (ed in Fig. 1), it an easily be shown that the atio of the essue behind it (essue at the oint e) to the jum ' of the essue in the disontinuity {length of the segment ed) tends slowly to unity fo. The essue behind this disontinuity an be egaded as equal to zeo at vey lage distanes only; the jum ' is equal hee to ' (the total aea of the ofile must be equal to zeo in vitue of (8). Let us onside now the sheially symmeti oagation of a shok wave aising in the ase of an exlosion (and investigated at lage distanes fom the lae of the exlosion). All the aguments hee ae exatly simila to those onsideed above. In the ase of a sheial oagation the amlitude of the wave deeases in the fist aoximation, as onst, whee is now the distane fom the 7

8 ente. We, theefoe, get fo the veloity u of the dislaement of the oints of the ofile instead of (5) whee χ denotes now the odut αχ u = 1 +, (14) ' χ =. (15) We find aodingly fo the dislaement the ath fom a etain to δ = αχ ln. δ of the oints of the ofile on If, howeve, the ofile is onsideed as the ouse of the hange of with the time τ, the distotion δ is αχ δτ = ln. (16) The distotion of the ofile of a sheial wave is thus ineased with the distane, aoding to the logaithmial law, i.e., muh moe slowly than in the ase of a lane o a ylindial wave (whee it is ootional esetively to the fist owe, o to the squae oot of the distane). Inasmuh as the oagation of an atual wave in gas is aomanied by usual sound absotion, whih is onneted with the visosity and the themal ondutivity, then, beause of the slowness of the inease of distotion, the sheial sound wave an be absobed, befoe the distotion of the ofile will lead to the fomation of disontinuities. In the atiula ase of the oagation of an exlosive shok wave it may haen that the seond shok wave by whih (just as in the ylindial ase) it should be followed, may not aea beause of the lak of time neessay fo its fomation. Instead of equation (9) we now have the equation α t = f ( χ) ln χ. (17) Exanding f (χ) into seies of owes of χ and limiting ouselves to the tems of the fist ode we get α t = χ ln onst, (18) a whee a is a. etain onstant. Hene we again obtain fo a linea deendene of t in the fom ' 1 ( t t) =. (19) α ln( / a) 8

9 Taking into aount the law of onsevation of the aea, we get in the sheial ase l1 ~ ln, () a 1 ' 1 ~. ln( / a) 9