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Complex ray analytical solutions for infrasound and
sonic boom propagation into shadow zones
Annie Zelias, Olaf Gainville, François Coulouvrat
To cite this version:
Annie Zelias, Olaf Gainville, François Coulouvrat. Complex ray analytical solutions for infrasound
and sonic boom propagation into shadow zones. e-Forum Acusticum 2020 - 9th edition, Dec 2020,
Lyon, France. pp.965-971, �10.48465/fa.2020.0240�. �hal-03229475�
COMPLEX RAY ANALYTICAL SOLUTIONS FOR INFRASOUND AND
SONIC BOOM PROPAGATION INTO SHADOW ZONES
Annie Zelias
12,
Olaf Gainville
1Franc¸ois Coulouvrat
21
CEA, DAM, DIF F-91297 Arpajon, France
2
Institut Jean le Rond d’Alembert, Sorbonne Universite, UMR CNRS 7190, France
annie.zelias@cea.fr
ABSTRACT
In application of the Comprehensive nuclear-Test-Ban Treaty (CTBT), micro-barometric stations of the Interna-tional Monitoring System (IMS) network detect various powerful natural and artificial infrasound sources. Sources such as explosions and meteorites produce similar multi-arrival pressure signatures. In order to distinguish these sources from one another, it is necessary to model the long range propagation of infrasound. Operational three dimen-sional method is ray tracing. However, in relatively fre-quent cases, micro-barometric stations are located into the geometrical shadow zone where only diffracted waves are recorded. To simulate infrasound propagation in this zone, the ray tracing method is generalized to complex ray the-ory. The sources, the media and the ground parameters are extended to complex values using analytic continua-tion. Two analytical cases are used to validate this method. The first one concerns the diffraction at a fold caustic in the vicinity of the ground. The second one concerns creep-ing waves at the ground. The analytical solutions of these cases show that complex ray theory is able to obtain in-frasound wave arrival times, azimuths and apparent veloc-ities in shadow zones. Our documented case study is the Carancas meteorite which has impacted the ground in Peru on September 15th, 2007. Comparisons between Caran-cas records and simulations of its sonic boom are thus per-formed.
1. INTRODUCTION
Meteorites can produce infrasound by three phenomena: sonic boom, fragmentation in the atmosphere and through the impact on the ground. It is not simple to distinguish infrasound generated by sonic boom and fragmentation. The Carancas meteorite is of special interest as it did not fragment significantly before impacting the ground [2, 24]. Hence, recorded signals emanate most likely from sonic boom. This meteorite fell on 2007 September 15th at 16h40 UTC near the frontier between Peru and Bolivia in the locality of Carancas and created a well documented crater [17]. The meteorite fell beside one infrasound sta-tion and several seismic stasta-tions (cf. Map of the Carancas region in [11–13]) from the IMS (International Monitoring System) network, which recorded wave signals associated
to the Carancas meteorite sonic boom and impact. In pticular Henneton et al. [11–13] have shown that the two ar-rivals at infrasound station are due to sonic boom emission at tropospheric and stratospheric altitudes. The propaga-tion of the first emission reached the stapropaga-tion by creeping waves on the ground, while the second emission resulted from diffraction at a caustic formed by the convergence of rays in altitude.
However, these situations cannot be simulated using geometrical propagation, according to which infrasound propagates along acoustical rays [14, 29]. This theory al-lows simple and fast computation taking into account 3D sources, topographies and atmospheric data [10]. How-ever, the ray theory is a simplifying asymptotic theory based on high frequency approximation. It leads to appari-tion of shadow zones, where no ray penetrates, but where signal can be recorded as for the Carancas meteorite at in-frasound station. Nevertheless, rays linking the source to the shadow zone point can exist if we extend the usual ray concept in the complex plane. The idea of complex rays (and diffraction rays, which is not considered here) was first introduced by Keller in 1958 [16] and often used in op-tics by Kravtsov [8, 18–20, 22] and others [4, 5], as well as in electromagnetics [3, 21], but rarely in acoustics [7, 32]. The main goal of this work is to demonstrate the applica-bility of complex ray theory for atmospheric propagation for realistic infrasound issues, including meteorite sonic boom and wind effects.
2. SUMMARY OF COMPLEX GEOMETRICAL ACOUSTICS THEORY
2.1 Geometrical ray equations
The mathematical development of the geometrical acous-tics in moving media have been performed by Milne [28]. The geometrical approximation assumes that the propa-gation medium and the wave amplitude are slowly vary-ing functions relatively to the wavelength. This approx-imation was succesfully applied to harmonic infrasound signatures [23, 32, 35] and to impulsive infrasound signa-tures [31]. The geometrical method is the standard way to compute sonic booms from superonic aircraft [36, 37] and has been compared succesfully with flight tests data (for a recent review, see [27]). The leading order provides the
eikonal equation which describes the wavefront propaga-tion implicitly defined by Φ(x, t) = 0:
(∇Φ)2= (−ω + v0.∇Φ)
2
c2 0
, (1)
with ω = −∂Φ/∂t, c0(x, t) the sound speed and v0(x, t)
the wind velocity. The characteristics of the eikonal equa-tion (1) are the ray equaequa-tions [6]:
dX
dt = c0n + v0, (2) dK
dt = −K∇c0− ∇v0.K, (3) where X is the position of the wavefront, K is the wave vector, K = √K.K (positive sign for forward propaga-tion [31]), and n = K/K is the unit normal to the wave-front.
Complex rays are then obtained by solving (2) and (3) in complex space from the initial source wavefront carac-terized by one parameter φ in 2D and two parameters in 3D. In this framework, each parameter is deemed complex. This implies that the source, the sound speed profile c0, the
wind profile v0and the topography have to be holomorphic
functions. For 2D problems, complex rays are surfaces de-scribed by X(φ, t), in the four dimensional complex space. For 3D geometries, rays would be surfaces in a 6D space. However, along complex rays only real points are physical. Then, to obtain complex rays reaching a physical receiver xrit is necessary to search for wavefront parameters φ and
propagation time t such that |X(φ, t) − xr| = 0. This is
numerically performed with a Levenberg-Marquardt algo-rithm [33].
2.2 Complex ray amplitude
The second order analysis leads to the transport equation [14, 29]: d dt A ρ0c30K +1 2 A ρ0c30K ∇.w = 0, (4) where A is the overpressure wave amplitude and w = c0n + v0is the group velocity. By introducing
the infinitesimal ray cross section ν = (Xφ× ey).n, this
equation is reduced in: A
ν ρ0c30K2
1/2
= Cst. (5)
The derivatives Xφ = ∂X/∂φ and Kφ = ∂K/∂φ with
respect to the wavefront parameter follow a system of or-dinary differential equations which may be solved together with (2) and (3). The constant in (5) is obtained analyti-cally at the source. In the frequency domain, geometrical acoustics overpressure is given by:
pGA(ω, xr) = Ae−jωt, (6)
for the ray such that X(φ, t) = xr. Note that the
imagi-nary part of complex propagation time t yields an exponen-tially decaying term adding to the geometrical amplitude A and describing for instance the wave decay in the shadow zone.
3. ANALYTICAL CASE FOR A SUPERSONIC SOURCE
The simple case of an incident plane wave is considered with a sound speed profile allowing an analytical solution. This configuration case, as illustrated in Fig. 1, is analo-gous to the Mach cut-off problem of the primary sonic boom of an aircraft flying at slightly supersonic Mach number M . We consider the bilinear sound speed pro-file [14, 18]:
c0(z) =
cref
√
1 + bz, (7)
with b = 2, 5.10−5m−1and cref = 340 km/s. The source
wavefront is the plane:
xs= φex+ zsez, ks= ks(cos αex+ sin αez),
ts=
φ(zs)
c0(zs)
cos α
in the orthonormal Cartesian base (ex, ey, ez) with the real
flight altitude zs = 17 km, α = −20◦ the real Mach
an-gle (sin α = 1/M ), ks = ω/c0(zs) and φ the complex
emission position.
Real rays are upward refracted by the negative sound speed gradient. This induces the existence of a horizontal fold caustic at the altitude of approximatively zc= 9694 m
and a shadow zone below (cf. Fig. 1).
Figure 1. Supersonic source flying at constant alti-tude zs = 17 km with negatif gradient of celerity profile
represented on the right. Triangles represent virtual re-ceivers above ground reach by real rays. Red lines are direct rays and blue lines are refracted ones which form a caustic (dashed orange) at zc = 9694 m. The shadow
zone is situated below this caustic.
All eigenrays reaching the real receiver xr = (xr, zr)
satisfy equations of Thom’s catastrophe theory for fold caustics [30]: b 4c2 ref τ2+ks ω sin ατ + zs− z = 0 (8) φ = x −ks ω cos ατ, (9) where τ = Rt tsc 2
0dt0. This problem is stratified so that
x = 80 km. In sonified zone above the caustic, Eq. (8) has two real roots for τ , one associated to the direct ray and the other one to the refracted ray, respectively red and blue lines on Fig. 1. Associated φ is given by Eq. (9). For re-fracted rays this parameter is smaller for direct rays. In the shadow zone below the caustic, Eq (8) has two complex conjugate roots, but only one is physical and represented on Fig. 2 (red lines in shadow zone). In addition, due to the ground defined by the holomorphic condition zg= 0,
com-plex rays are also reflected. The condition for the reflec-tion of complex rays follows the Snell-Descartes’ law ex-tended to complex plane [34]. The complex rays such that Im(τ ) ≤ 0 that make exponentially large contributions to the acoustics field must be discarded according to the rule of selection of nonphysical complex rays [20]. Addition-ally, forward propagation asumption implies Re(τ ) > 0. For each receiver, roots τ of Eq. (8) and associated φ are
Figure 2. Real and complex rays for the sound speed pro-file (7) propro-file. Real rays are the same as in Fig. 1. Red lines in the shadow zone are direct complex rays and green are ground reflected complex rays.
represented on Fig. 3. All rays in the shadow zone have the same arrival time Re(τ ) equal to the arrival time right on the caustic.
The knowledge of all eigenrays at receiver allows to compute the amplitudes and the arrival times. Here, the resulting field is the sum of all physical complex rays ar-riving at the observation point xr:
pGA(ω, xr) = Aie−jωti+ Are−j(ωtr+π/2)+ Age−jωtg,
(10) with subscripts i, r, g for respectively real or complex di-rect rays, real refracted rays and complex reflected rays. It is worth noting that for the real refracted ray, a π/2 phase shift occurs when the ray tangents the caustic [29]. The complex geometrical pressure for all receivers is repre-sented on Fig. 4 as black points. This solution is compared to the analytical solution of Helmholtz equation with the plane source ps(x, zs) = e−jks(cos αx+sin αzs):
p = Ai(¯z) + aBi(¯z) Aiinc(¯zs) + aBiinc(¯zs)
e−jks(cos αx+sin αzs) (11)
Figure 3. Emission parameters φ and τ for direct (red), re-fracted (blue) and reflected (green) rays, for each receiver.
where a takes into account the ground reflection [29]. Ai and Bi are the Airy functions of respectively first and sec-ond kind with the dimensionless altitude argument: ¯z = −(k2
0b)−2/3(k02− kr2+ k0bz). The asymptotic expansions
for large arguments, for the incident wave only, are respec-tively: ˜ Aiinc(¯z → −∞) ≈ ej(−2/3|¯z|)3/2+π/4) 2j√π|¯z|1/4 , (12) ˜ Biinc(¯z → −∞) ≈ ej(2/3|¯z|)3/2+π/4) 2j√π|¯z|1/4 . (13)
Solution (11) is represented at 0.1 Hz frequency on Fig. 4 for two different angles of emission α. The asymp-totic solution far from the caustic, noted ˜p, is obtained by replacing the Airy functions by their asymptotic expan-sion [29]. This approximation is also plotted in Fig. 4 and is different from the exact solution p arround the caustic. Finally the complex ray solution Eq. (10) is compared to these solutions. Complex rays match the asymptotic solu-tion in the shadow zone with low contribusolu-tion of reflected complex rays on the ground for α = −20◦ but is sig-nificantly different for α = −31◦. We noticed that the contribution of reflected complex rays is good only if the caustic is located far from the ground. The error between asymptotic Airy solution and complex rays solution is rep-resented in Fig. 5 according to the caustic altitude (Mach number) and frequency. This limit of the use of reflected complex rays is obtained for an error is the order of 10−3. The Airy’s function ground argument ¯zgassociated to this
value is of 2.54, limit shown as a black line in Fig. 5. Nev-ertheless, the singularity at the caustic remains. To circum-vent it, the uniform theory of diffraction is a method used for solving this type of discontinuities [9, 15, 16, 26].
Figure 4. Pressure associated to each receiver (black point) with airy solution for this case (blue) and related airy asymptotical solution (red) for α = −20◦on the left and α = −31◦on the right.
Figure 5. Airy’s function ground argument ¯zgfor different
caustic altitudes and corresponding Mach number accord-ing to frequency. Black line corresponds to the limit of validity of the use of reflected complex rays ¯zg= 2.54.
This analytical case shows the efficiency of complex ray theory to get rays in the shadow zone and get the ampli-tude there. The theory remains limited in the close vicinity of caustics and when the altitude caustic is to close to the ground when we consider complex reflected rays. This the-ory will now be applied to realistic long range infrasound propagation. This requires to develop numerical tools used in the next part.
4. APPLICATION TO INFRASOUND PROPAGATION
The selected realistic case is based on a sound speed profile described by a rational function, as proposed by Lingevitch et al. [25]. This profile is shown on Fig. 6.
For a ground source this profile induces a thermospheric waveguide. Note a source in altitude, around 20 km for instance, could also lead to both stratospheric and tropo-spheric waveguides. Such a situation is also recovered for a ground source by adding the Gaussian wind profile used by Blom [1], with a peak wind speed of 50 m/s realstic for stratospheric jets.
4.1 Thermospheric rays
To study the infrasound propagation in the thermosphere, we consider a point source on the ground located at xs=
(0, 0). The initial wavefront is defined by the complex emission angle φ such that ks = ks(cos φex+ sin φez).
Real rays are plotted on Fig. 6 as grey lines. For the chosen sound speed profile, the propagation forms a caustic. Its cusp point is at about zc= 210 km and one of its branches
reaches the ground at x = 325 km. At the ground level no real rays arrive between the source and the caustic i.e. in the shadow zone.
Figure 6. Geometrical theory for a point source and for Lingevitch et al. sound speed profile. Grey lines are real rays. Colored lines are the projection in the real plane of complex rays linking the source to a real receiver (identi-fied by triangles) in the shadow zone.
As the sound speed profile is defined by rational functions, it is an analytical function in the whole complex space. This allows to integrate complex rays. Some of them arriv-ing at physical receivers at the ground in the shadow zone are represented by colored lines in Fig. 6. Eigenray param-eters φ (complex emission angle) and t (complex arrival time) are numerically determined by Levenberg-Marquardt otpimization algorithm. This case shows once again the capability of complex rays to obtain information in the shadow zone. However, the complex sound speed profile has a pole which does not allow to compute complex rays between the source and x = 180 km at the ground.
4.2 Stratospheric and thermospheric rays in a windy atmosphere
The method ability to incorporate wind effects is now illus-trated by incorporating the Gaussian function representing
a jet wind, fitted from [1]: v0= vw (z − zw) σ2 w exp −(z − zw) 2 σ2 w ex (14) with vw = 50 m/s, zw = 50 km and σw = 17.5 km.
This case presents a shadow zone between the source and
Figure 7. Same as Fig. 6 with Gaussian wind profile Eq. (14)
x = 230 km. The existence of both thermospheric and stratospheric arrivals is obvious. A thermospheric caustic appears similarly to the previous case but now reaches the ground at z = 420 km. A second stratospheric caustic ap-pears between x = 120 km and x = 230 km. Note that on the figure rays after reflection are not represented for the sake of clarity. In this case, information in the whole first shadow zone can be computed (visualised as colored lines) and analysed. As expected the amplitude is infinite at the caustics. In the insonified zones there are two arrivals: a direct one and a refracted one. (cf. Fig. 8). In the shadow zones there is only one arrival (the unphysical complex conjugate ray is discarded). When taking into account the exponential decay (imaginary part of arrival time) one ob-serves the sharp decay inside the shadow zone. From an
Figure 8. Amplitude for the rays shown on fig Fig. 8 for each receiver on the ground.
operational point of view, a receiving station, such as the IMS ones, also determined the apparent speeds. With the
complex ray theory, it is also possible to obtain this quan-tity in the shadow zone as illustrated by Fig. 9 (Re(va)).
Once again, one observes two branches in the insonified zones and one in the shadow zone with much larger vari-ations corresponding to high elevation angles, up to 45◦, while at the caustic it is more horizontal with an angle of 19◦.
Figure 9. Apparent speed at receivers computed for rays shown on Fig. 7. Real and imaginary parts are separate.
5. CONCLUSION AND PROSPECTS Complex ray theory has been here applied in 2D atmo-spheric propagation cases. We show the ability of the method to predict wave arrivals in the shadow zone as re-sulting from diffraction around caustics, both without and with wind effects. The method can be applied to both stratospheric and thermospheric phases, for point or mov-ing sources. In further work, we will: i) extend the theory to creeping waves, ii) generalize the algorithms to 3D, iii) incorporate real atmospheric data, iv) include topography. The method will then be applied to the study of sonic boom from meteorites, such as the Carancas one.
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