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Critical time for asymptotic acoustic waves in a gravitational atmosphere
WAVE MOTION 2 (1980) 267-275 @ NORTH-HOLLAND PUBLISHING
COMPANY
CRITICAL TIME FOR ASYMPTOTIC ACOUSTIC IN A GRAVITATIONAL ATMOSPHERE A.M. ANILE, Seminario
G. MULONE
Matematico,
WAVES
and S. PLUCHINO
Universitri di Catania,
Catania,
Italy
Received 18 December 1979
Non-linear wave propagation in a gravitational atmosphere is studied by the method of asymptotic waves. In particular the critical time for breaking is computed within this approximation scheme. Excellent agreement is found between the values obtained in this way and those coming from a direct numerical integration.
1. Introduction
In the framework of wave propagation one distinguishes, broadly speaking, two kinds of waves, the impulsive ones and periodic or superposition of periodic ones. The former waves are usually treated by means of the theory of surfaces of discontinuity while for the latter one has often to resort to asymptotic methods. In the case of high-frequency waves Choquet-Bruhat has developed the so-called method of asymptotic waves which extends to non-linear hyperbolic systems the standard W. K. B. approximation [ 11. Similar techniques have also been extensively investigated by Taniuti [2], Jeffrey and Kakutani [3] and several other authors in the case of dispersive and dissipative systems. In this paper we shall apply the method of asymptotic waves to the study of non-linear wave propagation in a gravitational atmosphere. Problems of this sort arise frequently in various areas of Astrophysics (i.e. in theories of the solar chromosphere [4]) and Atmospheric Physics [5,6]. The aim of this work is not only to obtain explicit expression for the waves’ evolution law but also to test the method of asymptotic waves by comparing the results so obtained with those reached by other methods. In particular our results are compared with those obtained by Ulmschneider and coworkers by numerical integration as well as with exact solutions to some problems (though perhaps not completely physical) examined by Kuznetsov, Nelepin and Staniukovich [7]. The plan of the paper is as follows. In Section 2 we review briefly the method of asymptotic waves. In Section 3 we treat the case of an acoustic wave propagating in an isothermal gravitational atmosphere; the results thus obtained are then compared with those obtained by Ulmschneider and collaborators in the same context. In Section 4 we study isentropic flows in two particular cases, where it is possible to achieve an exact solution; these are then compared with the approximate solutions obtained by the method of asymptotic waves. In both Sections 3 and 4 wave propagation is studied both in the framework of the initial value problem and in that of the mixed initial-boundary value problem.
2. Asymptotic
waves
Let us consider the quasi-linear hyperbolic system
(2.1)
&@(XY, u)ut3, + BA(XY, u) = 0, 267
268
A.M. Anile et al. / Critical time for acoustic waues
. ., N is the unknown field vector and & 2” the identity matrix. Greek indices run whereu=(UA),A=l,. from 1 to 4 and x4 = t, while Latin indices run from 1 to 3. We seek solutions to the system (2.1) in the form of asymptotic waves [l, 81, as u = f w-qu4(xy,& q=o
(2.2)
Here w >>1 is a real parameter, ,$= WC#I with C#J a real scalar function, and u. is a solution of (2.1) independent of e which is assumed to be known. Substituting the formal series (2.2) into (2.1) and proceeding as in [8] one finds at the zeroth-order in l/w, the eikonal equation for the phase 4, (2.3)
4,4+AIV4I=O,
where A is an eigenvalue of the matrix dAni, ni = 4,i/lV4 1,&d s &‘( x, uo). Here A is to be interpreted as the kinematic propagation speed of the wavefronts C$= constant [8]. Henceforth, for the sake of simplicity, we assume A to be a simple eigenvalue of the matrix -c4hni.We note that this is the case which occurs in the applications considered in this paper. Then if r and 1 denote the right and left eigenvectors corresponding to A, one obtains ui = r(xy, C$)r.
(2.4)
In (2.4) an arbitrary integration function has been set equal to zero. Furthermore at the first order one finds the following transport equation for r: arjag
+ NI’ ar/at
Here (T is a parameter N =
+Mr = 0.
(2.5)
along the rays defined by dx&/dr =
kw,
with
k’” =
ld$,
{Da(lAd~‘)~,i}orQrB,
and N, M are given by
(2.6) (2.7)
where Do = a/au O, We assume that for (T= 0 the values of r, 5, x’ have been assigned, i.e. 5(0=0)=&
xY(C7=O)=x;;,
r((+=~)=r~(~).
(2.8)
The implicit solution of (2.5), (2.8) is
r(a)=ro(p)P(u), 6 = p +ro(p)[ouN(d)ewacu”dc’,
(2.9) (2.10)
where Q(a) = lo”M(cr’) du’. In (2.9) it is understood that p = ~(5) from (2.10). The infimum value of the (T’Ssuch that (2.10) is not invertible with respect to p is called the critical parameter. Obviously the conditions (2.8) will be interpreted differently in the case of initial or mixed initialboundary value problems.
A.M. Anile et al. 1 Critical time for acoustic waves
269
3. The case of an isothermal atmosphere Let us consider now the case of an isothermal background atmosphere. For one-dimensional along the z-axis the field equations read
propagation
u;+&3U::+C%A=0,
(3.1)
where
for the arbitrary state equation p = p(p, s). The unperturbed solution is taken to be one-dimensional, p. = b. e-‘lh,
t)fj=o,
static and isothermal (To = const.):
so = so(r ),
(3.2)
where h = STo/g is the density scale height and the fluid has been assumed to be polytropic (p = exp(s/c”)p’) and to obey the perfect gas law p = WpT. Here, for the sake of simplicity, we will consider upward propagating acoustic waves, which correspond to the choice A = -(u +co), c$,=< 0, q5,t> 0, cg = (a~/@)~. Then we obtain rj = t-z/co 1 r=po/co, i
N=-z,
0
= const.
1
Y+l
(eikonal equation), ap
lco 1 l=---( 2’2po’2poco
M_lcoapo
( as >>o ”
lco
2p. a2 - -2X
*
The rays are the straight lines defined by 2 = cot + const.
C-7 = t,
Let us consider first the initial value problem: 6 = iu
and
I’,&) = ]I’o]sin cc
at t =O.
The implicit solution of (2.5) along the ray z = cof + .ZOis z = ~ +
Ir,(e’,‘2h~l_
e(z-r,)/2h}
sin
g
cL,
(3.3)
r = jr01 e(z-zo)‘2h sin CL,
(3.4)
where 5 = WC$= w(t - z/co). When (3.3) is invertible for p we obtain [9] E _ Ifo] e@h{e(z-zOVZh_ 1) g
where Jq is the Bessel function of order 4.
,
(3.5)
270
A.M. Anile et al. / Critical time for acoustic waves
Also, from (3.3) one obtains the critical height .zc which occurs on the z. = 0 ray zc = 2h In{1 + g/lPoj}.
(3.6)
Now we turn our attention to the classical mixed initial-boundary r=O
at t=O,
6(O)=p
=wr
R=2r/P and
value problem for which one has
atz=O, (3.7)
PO(~) = If01 sin &.
This corresponds to an atmosphere excited by a velocity field sinusoidally varying with period P at the lower boundary. In this case the rays z = cot + const. can be distinguished into two classes, according to whether they originate from t = 0 (I) or from the lower boundary z = 0 (II). Along the rays belonging to Class I one gets P = 0, which shows that no perturbation propagates along these rays. On the Class II rays (separated from Class I by the ray C*: z = cot) one finds 5 =
(’ + ‘jh If 0I{
2
1 _
eC,(‘-r)l2h}
E ~ +p
sin
CO
(3.8)
0
and (3.9) with 8 = w(t - z/co). For the sake of physical interpretation @o=ll
it is convenient
to introduce the initial average flux [4]
p POCOU:dr,
(3.10)
w2P I 0
where all the quantities are evaluated at the lower boundary z = 0. The relationship between Ii01 and Go is easily seen to be
Then one obtains for the critical height
zc=2h
c: In ( l+(y+l)h
1 (aco)i’2). n 240
(3.11)
In Tables 1 and 2 we display the critical heights for various values of P and Go in the case of an isothermal atmosphere. Table 1 corresponds to To = 4000 “K, Table 2 to To = 5000 “K. In both cases b. = 1.0 E-5 g/cm3, B = 6.3956923 E7, g = 2.736 E4 cm/s2, y = 5/3, appropriate for the conditions in the solar atmosphere, where L%is the gas constant and the notation En means 10”. The results for the critical height shown in Tables 1 and 2 are in good agreement with those of [4]. Also, (3.11) is quite close to that derived by Stein and Schwartz [lo] on the basis of the simple intuitive picture of the crest of the wave catching its trough.
4. Some exact solutions In this section we consider the system (3.1) for a gravitational p = p(p), i.e. an isentropic flow.
atmosphere
with an equation of state
271
A.M. Anile et al. / Critical time for acoustic waves Table 1 p
00
20 25 30 40 80 150 180
1.6 E8 406.17 443.59 474.75 524.82 648.71 763.53 797.10
8.0 E7 ES E5 ES E5 ES E5 E5
464.63 503.22 535.19 586.30 711.81 827.41 861.14
4.0 E7 E5 E5 E5 E5 E5 E5 E5
524.82 564.28 596.84 648.71 775.40 891.57 925.40
2.0 E7 E! E5 E5 E5 E5 E5 ES
586.30 626.39 659.39 711.81 839.34 955.92 989.82
E5 E5 E5 E5 E5 E5 E5
Table 2 P
d
20 25 30 40 80 150 180
8.0 E7
1.6 E8 519.31 566.34 605.48 668.31 823.54 967.24 1009.24
E5 E5 E5 E.5 E5 E5 E5
592.78 641.21 681.30 745.36 902.52 1047.16 1089.34
E5 E5 E5 E5 E5 E5 E5
4.0 E7
2.0 E7
668.31 ES 717.77 E5 758.58’E5 823.54 E5 982.09 E5 1127.40 E5 1169.70 E5
745.36 795.59 836.91 902.52 1062.08 1207.86 1250.26
E5 E5 65 E5 E5 E5 ES
For some special equations of state it is then possible [7] to find exact solutions which represent non-linear plane waves. In particular we shall consider the cases of the following state equations: (i) p = -A/p +B, A = const., B = const., (ii) p = Ap3 + B, A = const., B = const. Case (i) corresponds to transonic aerodynamics [ 111. Case (ii) is not very physical. However the solutions so obtained permit an easy comparison with the corresponding asymptotic waves. As we have already stated in the Introduction this is a useful check upon the validity of the method of asymptotic waves. Case (i): By introducing the Lagrangian mass coordinate
where 5 is the Lagrangian coordinate,
au/at -A
the system (3.1) can be reduced [7] to the following:
a v/a9 = -g, (4.1)
av/aq -a v/at = 0, where V = l/p. The general solution of the linear system (4.1) is
(4.2) V= l/p = co+gq/A
+{&(q +vh)-&(q-fit)}/fi,
where Fi, F2 are two smooth arbitrary functions.
(4.3)
272
A.M. Anile et al. / Critical time for acoustic waves
The initial value problem for an upward progressive wave, v(q, 0) = (&Jw) sin wq
(4.4)
has the solution u = (t&/w) sin 0 (q -&it),
(4.5)
V= ~~+gq/A-(~~/JAw)sinw(q-bat).
(4.6)
It is easy to see that the asymptotic wave for the system (4.1) corresponding to the same initial value problem (4.4) coincides with the exact solution at all orders. The same conclusion can be drawn in the case of the mixed initial-boundary value problem. We note that this result is a straightforward consequence of the system (4.1) being linear with constant coefficients. Case (ii): By the change of variables u = $((Y+p),
c=J&&=&-p)
the system (3.1) reduces to [7]
as/at + ff a.ff/at = -g,
(4.7)
ap/at + p ap/az = -g.
(4.8)
Henceforth we shall restrict ourselves to (4.7). It is easy to see that the solution to the initial value problem of (4.7) for cu(z, 0) =f(z) is a (2, t) = -gt +f(V),
(4.9) (4.10)
2 =-$gt2+f(77)t+7/, where 77= z(O). Now we specialize f(7) as follows f(q) = aO(q) +;
PO1
sin f so(n),
]rO]= const.,
w = const.,
(4.11)
where so(n) = JZi - 2gq, ai0 = const. We remark that (4.12)
(Y&r) = JG; - 2gz
corresponds to the static solution of (4.7) and therefore should be interpreted as the unperturbed state. Of course this static solution is defined only in the half;plane --oOdz a&$2g. However, for the sake of physical interpretation, we restrict it to the strip 0 G z s &:/2g. It is easy to see that the domain of existence of the solution (4.9)-(4.11) can exceed the above strip for some particular values of w. The critical time and critical height for the solution (4.9)-(4.11) occur on the characteristic issuing from n=O,t=Oandare tc = Go/(g +
v-01)
(4.13)
and
lro12
2
zc=E
(
l-(g+(z#
a?olrol
I +o(g+,T,()Sin~~O~
(4.14)
A.M. Anile et al. / Critical time foracoustic waves
Now we seek the asymptotic wave for (4.7) corresponding
273
to ‘the initial value (4.11). One writes (4.15)
~(~,~)=~0(z)+~~(~,f,s)/w+0(1/~*) with the following initial values at t = 0: (Y1 = lroj sin CL.
5=cC,
(4.16)
The phase q5 is then easily seen to be 4 = t + m/g
+ const.,
const. = -Go/g
and the rays are given by z = -$gt*+mt+E
(4.17)
with z (0) = E. We note that the domain of existence of the asymptotic wave (4.15) is restricted to the strip 0 s z d f = &2g, which is at variance with the exact solution. Along each ray issuing from (E, 0) the solution is given implicitly by ai(z, &5)=
(4.18)
Ir~l(~o(E)l~uo(z))sin(u,
6 = CL+ Irol((~o(z) - &E))/ga&))
(4.19)
sin CL
where in (4.18) p must be expressed as a function of 5 through (4.19), and 6 = w&. The critical time and the critical height occur on the ray issuing from E = 0 and are seen to be t: = Go/@ + lrol),
(4.20.)
-2 z"=Z
1
PO12 1-(g+Irol)2
(4.21)
* I
We note that the critical time for the asymptotic wave coincides with that for the exact solution. However the critical heights differ by the quantity
a;olrol . W I w (g + IcJl) s1ng a0 which vanishes as o + co. Let us now consider the following mixed initial-boundary a(z, 0) = so(z),
cy(0,1) = ai, + (Irol/o) sin of.
value problem for (4.7): (4.22)
Analogously to the previous case the characteristics can be distinguished as belonging to two classes. On the Class I characteristics the implicit solution of (4.7) and (4.22) is a(z, t) = -gt + @o(V), (4.23) z = -~gr2+(Y0(n)f + n, while on Class II characteristics (Y(z, t) = -g(t-7)+a(O, z = -$g(t-7)2+(Y(0,
7), 7)(f-7),
LY (0,~) = ai0+ (IFol/w) sin 0f.
(4.24)
214
A.M. Anile et al. / Critical time for acoustic waves
Formally one could evaluate the critical time on the Class I characteristics, which turns out to be t, = 0, corresponding to the critical height f = &?$2g. Of course this non-physical result was to be expected because it is due to the non-existence of the initial data beyond Z = cZ?j/2g. The critical time and critical height for the Class II characteristics occur on the characteristic issuing from T = 0 and are given by G = &/(g + lrcllnlw),
(4.25)
-2
zc=
2(g + ,;&2,co)2
( + 2Irolwo). g
(4.26)
Now we seek the asymptotic wave for (4.7) corresponding problem (4.22). We write Q =ao(z)+al(z,
to the same mixed initial-boundary
t, 6)/w +O(1/02)
values
(4.27)
with CX~(Z,O,CL)=O, CL=~ (f=O)
and
czi(O,f,~)=]r~lsin-v,
n w
v=[
(z=O).
The rays can be divided into two classes: -Class I: t = --(Y&)/g + m(q)/g, 77= z(O), -Class II: g(t-T)=c&-aO(z). On the Class I rays one has r=o,
1~= const.
On the Class II rays one has (4.28)
(4.29) where F(z) = (&-%l(z))lg&l(z). The critical time and critical height occur on the ray issuing from the r = 0 ray and are identical to those for the exact solution.
5. Conclusion
The main result of this work is that the technique of asymptotic waves yields a reliable method for calculating the critical time. Whenever a comparison is feasible we find excellent agreement with results obtained by other methods. In particular in the case of Section 3, the method of asymptotic waves yields good results also for periods P which are not small compared with the natural period PI = 47rh/c0, as can be seen by comparing Tables 1 and 2 with Fig. 5 of [4]. Therefore this method is rather promising even in those cases (P = PI), where a priori one would not expect it to yield good results.
275
A.M. Anile et al. / Critical time for acoustic waves
Acknowledgment This research was partly supported by C.N.R., G.N.F.H.
References [l] Y. Choquet-Bruhat, “Ondes asymptotiques et approchees pour des systemes d’tquations aux derivees partielles non lintaires”, J. Math. Pures Appl. 48, 117 (1969). [2] T. Taniuti, “Reductive perturbation method and far fields of the wave equations”, Suppl. to Progr. Theoret. Phys. 55, 1 (1974). [3] A. Jeffrey and T. Kakutani, “Weak nonlinear dispersive waves; a discussion centered around the Korteweg-De Vries equation”, SIAMRev. 14,582 (1972). [4] P. Ulmschneider, W. Kalkofen, T. Nowak and U. Bohn, “Acoustic waves in the solar atmosphere I, The hydrodynamic code”, Astron. Astrophys. T4, 61 (1977). [S] F. Einaudi, “Shock formation in acoustic gravity waves”, J. Geophs. Res. 75, 193 (1970). [6] P.A. Bois, “Propagation lineaire et non lintaire d’ondes atmosphtriques”, J. M&anique 15,781 (1976). [7] AI. Kuznetsov, V.F. Nelepin and K.P. Staniukovich, “Propagation of acoustic waves in a medium located in a gravitational field”, Prikl. Mat. Mech. 38, 99 (1974). [8] G. Boillat, “Ondes asymptotiques non lintaires”, Ann. Mat. Pura Appl. (IV), CXI, 31 (1976). [9] A.M. Anile, “Non-linear high-frequency waves in relativistic cosmology”, Rend. Accad. Naz. Lincei (VIII), LXIZZ (S), 375 (1977). [lo] R.F. Stein and R.A. Schwartz, “Waves in the solar atmosphere II, Large-amplitude acoustic pulse propagation”, Astrophys. J. 177, 807 (1972).
[l l] A. Greco, “On the strict exceptionality for a subsonic flow”, Proc. II Congress0 AZMETA,
Napoli (1974).
COMPANY
CRITICAL TIME FOR ASYMPTOTIC ACOUSTIC IN A GRAVITATIONAL ATMOSPHERE A.M. ANILE, Seminario
G. MULONE
Matematico,
WAVES
and S. PLUCHINO
Universitri di Catania,
Catania,
Italy
Received 18 December 1979
Non-linear wave propagation in a gravitational atmosphere is studied by the method of asymptotic waves. In particular the critical time for breaking is computed within this approximation scheme. Excellent agreement is found between the values obtained in this way and those coming from a direct numerical integration.
1. Introduction
In the framework of wave propagation one distinguishes, broadly speaking, two kinds of waves, the impulsive ones and periodic or superposition of periodic ones. The former waves are usually treated by means of the theory of surfaces of discontinuity while for the latter one has often to resort to asymptotic methods. In the case of high-frequency waves Choquet-Bruhat has developed the so-called method of asymptotic waves which extends to non-linear hyperbolic systems the standard W. K. B. approximation [ 11. Similar techniques have also been extensively investigated by Taniuti [2], Jeffrey and Kakutani [3] and several other authors in the case of dispersive and dissipative systems. In this paper we shall apply the method of asymptotic waves to the study of non-linear wave propagation in a gravitational atmosphere. Problems of this sort arise frequently in various areas of Astrophysics (i.e. in theories of the solar chromosphere [4]) and Atmospheric Physics [5,6]. The aim of this work is not only to obtain explicit expression for the waves’ evolution law but also to test the method of asymptotic waves by comparing the results so obtained with those reached by other methods. In particular our results are compared with those obtained by Ulmschneider and coworkers by numerical integration as well as with exact solutions to some problems (though perhaps not completely physical) examined by Kuznetsov, Nelepin and Staniukovich [7]. The plan of the paper is as follows. In Section 2 we review briefly the method of asymptotic waves. In Section 3 we treat the case of an acoustic wave propagating in an isothermal gravitational atmosphere; the results thus obtained are then compared with those obtained by Ulmschneider and collaborators in the same context. In Section 4 we study isentropic flows in two particular cases, where it is possible to achieve an exact solution; these are then compared with the approximate solutions obtained by the method of asymptotic waves. In both Sections 3 and 4 wave propagation is studied both in the framework of the initial value problem and in that of the mixed initial-boundary value problem.
2. Asymptotic
waves
Let us consider the quasi-linear hyperbolic system
(2.1)
&@(XY, u)ut3, + BA(XY, u) = 0, 267
268
A.M. Anile et al. / Critical time for acoustic waues
. ., N is the unknown field vector and & 2” the identity matrix. Greek indices run whereu=(UA),A=l,. from 1 to 4 and x4 = t, while Latin indices run from 1 to 3. We seek solutions to the system (2.1) in the form of asymptotic waves [l, 81, as u = f w-qu4(xy,& q=o
(2.2)
Here w >>1 is a real parameter, ,$= WC#I with C#J a real scalar function, and u. is a solution of (2.1) independent of e which is assumed to be known. Substituting the formal series (2.2) into (2.1) and proceeding as in [8] one finds at the zeroth-order in l/w, the eikonal equation for the phase 4, (2.3)
4,4+AIV4I=O,
where A is an eigenvalue of the matrix dAni, ni = 4,i/lV4 1,&d s &‘( x, uo). Here A is to be interpreted as the kinematic propagation speed of the wavefronts C$= constant [8]. Henceforth, for the sake of simplicity, we assume A to be a simple eigenvalue of the matrix -c4hni.We note that this is the case which occurs in the applications considered in this paper. Then if r and 1 denote the right and left eigenvectors corresponding to A, one obtains ui = r(xy, C$)r.
(2.4)
In (2.4) an arbitrary integration function has been set equal to zero. Furthermore at the first order one finds the following transport equation for r: arjag
+ NI’ ar/at
Here (T is a parameter N =
+Mr = 0.
(2.5)
along the rays defined by dx&/dr =
kw,
with
k’” =
ld$,
{Da(lAd~‘)~,i}orQrB,
and N, M are given by
(2.6) (2.7)
where Do = a/au O, We assume that for (T= 0 the values of r, 5, x’ have been assigned, i.e. 5(0=0)=&
xY(C7=O)=x;;,
r((+=~)=r~(~).
(2.8)
The implicit solution of (2.5), (2.8) is
r(a)=ro(p)P(u), 6 = p +ro(p)[ouN(d)ewacu”dc’,
(2.9) (2.10)
where Q(a) = lo”M(cr’) du’. In (2.9) it is understood that p = ~(5) from (2.10). The infimum value of the (T’Ssuch that (2.10) is not invertible with respect to p is called the critical parameter. Obviously the conditions (2.8) will be interpreted differently in the case of initial or mixed initialboundary value problems.
A.M. Anile et al. 1 Critical time for acoustic waves
269
3. The case of an isothermal atmosphere Let us consider now the case of an isothermal background atmosphere. For one-dimensional along the z-axis the field equations read
propagation
u;+&3U::+C%A=0,
(3.1)
where
for the arbitrary state equation p = p(p, s). The unperturbed solution is taken to be one-dimensional, p. = b. e-‘lh,
t)fj=o,
static and isothermal (To = const.):
so = so(r ),
(3.2)
where h = STo/g is the density scale height and the fluid has been assumed to be polytropic (p = exp(s/c”)p’) and to obey the perfect gas law p = WpT. Here, for the sake of simplicity, we will consider upward propagating acoustic waves, which correspond to the choice A = -(u +co), c$,=< 0, q5,t> 0, cg = (a~/@)~. Then we obtain rj = t-z/co 1 r=po/co, i
N=-z,
0
= const.
1
Y+l
(eikonal equation), ap
lco 1 l=---( 2’2po’2poco
M_lcoapo
( as >>o ”
lco
2p. a2 - -2X
*
The rays are the straight lines defined by 2 = cot + const.
C-7 = t,
Let us consider first the initial value problem: 6 = iu
and
I’,&) = ]I’o]sin cc
at t =O.
The implicit solution of (2.5) along the ray z = cof + .ZOis z = ~ +
Ir,(e’,‘2h~l_
e(z-r,)/2h}
sin
g
cL,
(3.3)
r = jr01 e(z-zo)‘2h sin CL,
(3.4)
where 5 = WC$= w(t - z/co). When (3.3) is invertible for p we obtain [9] E _ Ifo] e@h{e(z-zOVZh_ 1) g
where Jq is the Bessel function of order 4.
,
(3.5)
270
A.M. Anile et al. / Critical time for acoustic waves
Also, from (3.3) one obtains the critical height .zc which occurs on the z. = 0 ray zc = 2h In{1 + g/lPoj}.
(3.6)
Now we turn our attention to the classical mixed initial-boundary r=O
at t=O,
6(O)=p
=wr
R=2r/P and
value problem for which one has
atz=O, (3.7)
PO(~) = If01 sin &.
This corresponds to an atmosphere excited by a velocity field sinusoidally varying with period P at the lower boundary. In this case the rays z = cot + const. can be distinguished into two classes, according to whether they originate from t = 0 (I) or from the lower boundary z = 0 (II). Along the rays belonging to Class I one gets P = 0, which shows that no perturbation propagates along these rays. On the Class II rays (separated from Class I by the ray C*: z = cot) one finds 5 =
(’ + ‘jh If 0I{
2
1 _
eC,(‘-r)l2h}
E ~ +p
sin
CO
(3.8)
0
and (3.9) with 8 = w(t - z/co). For the sake of physical interpretation @o=ll
it is convenient
to introduce the initial average flux [4]
p POCOU:dr,
(3.10)
w2P I 0
where all the quantities are evaluated at the lower boundary z = 0. The relationship between Ii01 and Go is easily seen to be
Then one obtains for the critical height
zc=2h
c: In ( l+(y+l)h
1 (aco)i’2). n 240
(3.11)
In Tables 1 and 2 we display the critical heights for various values of P and Go in the case of an isothermal atmosphere. Table 1 corresponds to To = 4000 “K, Table 2 to To = 5000 “K. In both cases b. = 1.0 E-5 g/cm3, B = 6.3956923 E7, g = 2.736 E4 cm/s2, y = 5/3, appropriate for the conditions in the solar atmosphere, where L%is the gas constant and the notation En means 10”. The results for the critical height shown in Tables 1 and 2 are in good agreement with those of [4]. Also, (3.11) is quite close to that derived by Stein and Schwartz [lo] on the basis of the simple intuitive picture of the crest of the wave catching its trough.
4. Some exact solutions In this section we consider the system (3.1) for a gravitational p = p(p), i.e. an isentropic flow.
atmosphere
with an equation of state
271
A.M. Anile et al. / Critical time for acoustic waves Table 1 p
00
20 25 30 40 80 150 180
1.6 E8 406.17 443.59 474.75 524.82 648.71 763.53 797.10
8.0 E7 ES E5 ES E5 ES E5 E5
464.63 503.22 535.19 586.30 711.81 827.41 861.14
4.0 E7 E5 E5 E5 E5 E5 E5 E5
524.82 564.28 596.84 648.71 775.40 891.57 925.40
2.0 E7 E! E5 E5 E5 E5 E5 ES
586.30 626.39 659.39 711.81 839.34 955.92 989.82
E5 E5 E5 E5 E5 E5 E5
Table 2 P
d
20 25 30 40 80 150 180
8.0 E7
1.6 E8 519.31 566.34 605.48 668.31 823.54 967.24 1009.24
E5 E5 E5 E.5 E5 E5 E5
592.78 641.21 681.30 745.36 902.52 1047.16 1089.34
E5 E5 E5 E5 E5 E5 E5
4.0 E7
2.0 E7
668.31 ES 717.77 E5 758.58’E5 823.54 E5 982.09 E5 1127.40 E5 1169.70 E5
745.36 795.59 836.91 902.52 1062.08 1207.86 1250.26
E5 E5 65 E5 E5 E5 ES
For some special equations of state it is then possible [7] to find exact solutions which represent non-linear plane waves. In particular we shall consider the cases of the following state equations: (i) p = -A/p +B, A = const., B = const., (ii) p = Ap3 + B, A = const., B = const. Case (i) corresponds to transonic aerodynamics [ 111. Case (ii) is not very physical. However the solutions so obtained permit an easy comparison with the corresponding asymptotic waves. As we have already stated in the Introduction this is a useful check upon the validity of the method of asymptotic waves. Case (i): By introducing the Lagrangian mass coordinate
where 5 is the Lagrangian coordinate,
au/at -A
the system (3.1) can be reduced [7] to the following:
a v/a9 = -g, (4.1)
av/aq -a v/at = 0, where V = l/p. The general solution of the linear system (4.1) is
(4.2) V= l/p = co+gq/A
+{&(q +vh)-&(q-fit)}/fi,
where Fi, F2 are two smooth arbitrary functions.
(4.3)
272
A.M. Anile et al. / Critical time for acoustic waves
The initial value problem for an upward progressive wave, v(q, 0) = (&Jw) sin wq
(4.4)
has the solution u = (t&/w) sin 0 (q -&it),
(4.5)
V= ~~+gq/A-(~~/JAw)sinw(q-bat).
(4.6)
It is easy to see that the asymptotic wave for the system (4.1) corresponding to the same initial value problem (4.4) coincides with the exact solution at all orders. The same conclusion can be drawn in the case of the mixed initial-boundary value problem. We note that this result is a straightforward consequence of the system (4.1) being linear with constant coefficients. Case (ii): By the change of variables u = $((Y+p),
c=J&&=&-p)
the system (3.1) reduces to [7]
as/at + ff a.ff/at = -g,
(4.7)
ap/at + p ap/az = -g.
(4.8)
Henceforth we shall restrict ourselves to (4.7). It is easy to see that the solution to the initial value problem of (4.7) for cu(z, 0) =f(z) is a (2, t) = -gt +f(V),
(4.9) (4.10)
2 =-$gt2+f(77)t+7/, where 77= z(O). Now we specialize f(7) as follows f(q) = aO(q) +;
PO1
sin f so(n),
]rO]= const.,
w = const.,
(4.11)
where so(n) = JZi - 2gq, ai0 = const. We remark that (4.12)
(Y&r) = JG; - 2gz
corresponds to the static solution of (4.7) and therefore should be interpreted as the unperturbed state. Of course this static solution is defined only in the half;plane --oOdz a&$2g. However, for the sake of physical interpretation, we restrict it to the strip 0 G z s &:/2g. It is easy to see that the domain of existence of the solution (4.9)-(4.11) can exceed the above strip for some particular values of w. The critical time and critical height for the solution (4.9)-(4.11) occur on the characteristic issuing from n=O,t=Oandare tc = Go/(g +
v-01)
(4.13)
and
lro12
2
zc=E
(
l-(g+(z#
a?olrol
I +o(g+,T,()Sin~~O~
(4.14)
A.M. Anile et al. / Critical time foracoustic waves
Now we seek the asymptotic wave for (4.7) corresponding
273
to ‘the initial value (4.11). One writes (4.15)
~(~,~)=~0(z)+~~(~,f,s)/w+0(1/~*) with the following initial values at t = 0: (Y1 = lroj sin CL.
5=cC,
(4.16)
The phase q5 is then easily seen to be 4 = t + m/g
+ const.,
const. = -Go/g
and the rays are given by z = -$gt*+mt+E
(4.17)
with z (0) = E. We note that the domain of existence of the asymptotic wave (4.15) is restricted to the strip 0 s z d f = &2g, which is at variance with the exact solution. Along each ray issuing from (E, 0) the solution is given implicitly by ai(z, &5)=
(4.18)
Ir~l(~o(E)l~uo(z))sin(u,
6 = CL+ Irol((~o(z) - &E))/ga&))
(4.19)
sin CL
where in (4.18) p must be expressed as a function of 5 through (4.19), and 6 = w&. The critical time and the critical height occur on the ray issuing from E = 0 and are seen to be t: = Go/@ + lrol),
(4.20.)
-2 z"=Z
1
PO12 1-(g+Irol)2
(4.21)
* I
We note that the critical time for the asymptotic wave coincides with that for the exact solution. However the critical heights differ by the quantity
a;olrol . W I w (g + IcJl) s1ng a0 which vanishes as o + co. Let us now consider the following mixed initial-boundary a(z, 0) = so(z),
cy(0,1) = ai, + (Irol/o) sin of.
value problem for (4.7): (4.22)
Analogously to the previous case the characteristics can be distinguished as belonging to two classes. On the Class I characteristics the implicit solution of (4.7) and (4.22) is a(z, t) = -gt + @o(V), (4.23) z = -~gr2+(Y0(n)f + n, while on Class II characteristics (Y(z, t) = -g(t-7)+a(O, z = -$g(t-7)2+(Y(0,
7), 7)(f-7),
LY (0,~) = ai0+ (IFol/w) sin 0f.
(4.24)
214
A.M. Anile et al. / Critical time for acoustic waves
Formally one could evaluate the critical time on the Class I characteristics, which turns out to be t, = 0, corresponding to the critical height f = &?$2g. Of course this non-physical result was to be expected because it is due to the non-existence of the initial data beyond Z = cZ?j/2g. The critical time and critical height for the Class II characteristics occur on the characteristic issuing from T = 0 and are given by G = &/(g + lrcllnlw),
(4.25)
-2
zc=
2(g + ,;&2,co)2
( + 2Irolwo). g
(4.26)
Now we seek the asymptotic wave for (4.7) corresponding problem (4.22). We write Q =ao(z)+al(z,
to the same mixed initial-boundary
t, 6)/w +O(1/02)
values
(4.27)
with CX~(Z,O,CL)=O, CL=~ (f=O)
and
czi(O,f,~)=]r~lsin-v,
n w
v=[
(z=O).
The rays can be divided into two classes: -Class I: t = --(Y&)/g + m(q)/g, 77= z(O), -Class II: g(t-T)=c&-aO(z). On the Class I rays one has r=o,
1~= const.
On the Class II rays one has (4.28)
(4.29) where F(z) = (&-%l(z))lg&l(z). The critical time and critical height occur on the ray issuing from the r = 0 ray and are identical to those for the exact solution.
5. Conclusion
The main result of this work is that the technique of asymptotic waves yields a reliable method for calculating the critical time. Whenever a comparison is feasible we find excellent agreement with results obtained by other methods. In particular in the case of Section 3, the method of asymptotic waves yields good results also for periods P which are not small compared with the natural period PI = 47rh/c0, as can be seen by comparing Tables 1 and 2 with Fig. 5 of [4]. Therefore this method is rather promising even in those cases (P = PI), where a priori one would not expect it to yield good results.
275
A.M. Anile et al. / Critical time for acoustic waves
Acknowledgment This research was partly supported by C.N.R., G.N.F.H.
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Napoli (1974).