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The generation of detonation by coalescing spherical deflagration waves
Acta Astronautica. VoL 1, pp. 1187-1199. PergamonPress 1974. Printedin the U.S.A.
The generation of detonation by coalescing spherical deflagration waves E. S C H M O L I N S K E Ernst-Mach-Institut, Freiburg, W. Germany
(Received 12 September 1973) AImtraet--The generation of a spherical detonation was verified experimentally by two coalescing deflegration waves. Numerical results of treating the three different types of deflagration of detonation transition with Hugoniot curves are in full agreement with our experimental observations.
Introduction THE TRANSITIONfrom combustion to detonation is a phenomenon of considerable importance in many practical applications of explosives. Of major interest, therefore, is the experimental and theoretical elucidation of this gasdynamic event. Up to now the experiments were carried out in narrow channels of varying cross sections. Afterwards it was the task to explain the various types of waves and their origin and to find an analytical solution [1]. As was reported repeatedly, e.g. Ref. [2], the transition process is considerably influenced by the geometry of the duct, the roughness of the walls and site and kind of ignition. In consequence, the flame acceleration model [3] can give only an approximate description of the development of a detonation from a laminar combustion. This model assumes that due to turbulences at the walls of the duct the flame area is increased and accelerated until a shock wave is formed in front of the flame by coalescence of blast waves. Hence, a detonation is initiated. An essential purpose of this paper was to avoid strong boundary effects which was possible by using spherical waves. This allows us to give an exact description of the gasdynamic events by means of Hugoniot curves for we have a flow without friction and heat conduction. As is well known the introduction of a solid wall in the meridian plane of the spherical geometry does not derange the latter and gives rise to only negli~'bly small disturbances. On the other hand, it permits the simultaneous observation of wave propagations by means of soot patterns and schlieren pictures. Only the use of two or more independent registration methods seems to exclude the possibility of a misinterpretation as on several occasions wrong conclusions were made from the analysis of single photographic records.
Experimental results All our experiments were carried out in a cubic chamber of 15 cm lateral length. A perspex plate which had two pairs of electrodes placed 10 mm apart was mounted in the centre of the chamber. Above this plate lay another one, sooted 1187
1188
E. SCHMOLINSKE
and with orifices for the electric spark ignition. Each pair of electrodes was charged by a 0.5/~F capacitor with 14 kV and ignition of the explosive gas was initiated by a triggered spark gap. The framing frequency of the rotating mirror camera was 0.94 MHz and the exposure time 0.38 fts. The detonating mixture used in this study was equimolar acetylene-oxygen and the initial pressure range of the experiments was 50 to 200 torr. From all experiments it became clear that the transition from deflagration to detonation was originated in a Mach stem which in our case was formed by the coalescence of two spherical waves. Three different types of deflagration to
4~IS
7~S
9gS
llgS
13Us
16Us a f t e r
lqnltlc~
Fig. 1. Spherical transition from deflagration to detonation by a Mach stem in the subcritical energy regime. 50% C=H~--O, 65 torr, ignition energy 49 J.
The generationof detonationby coalescingsphericaldeflagrationwaves
1189
detonation transition were observed, depending on the appearance of a subcritical, critical or supercritical energy regime [4] in the Mach stem. The subcritical type is characterized by the fact that the Mach stem starts as a deflagration wave. Then, after a limited time, the detonation is initiated. Figure 1 shows a typical sequence of spark schlieren pictures illustrating this gasdynami~ event. At the moment of its generation and the following 9/zs the Mach stem consists of two discontinuities, i.e., a flame front with a preceding shock wave. At a later instant the Mach stem induced deflagration wave becomes irregular whereas the primary deflagration fronts do not show any alteration. Due to increasing instabilities in the form of reaction centers the Mach stem gets accelerated until a detonation is initiated. Figure 2 shows the soot pattern of this experiment. Neither the combustion process nor the shock wave itself affects the soot and no structure is visible at first. Only when single irregularities are occuring will the soot be wiped away. At a later instant the Mach stem initiates a detonation which is propagating into the quiescent gas as well as into the shock compressed zone of the deflagration. Clearly one can recognize these two areas by different structures of the soot imprint, the scale of which becomes, as is well known, smaller the higher the initial pressure of the explosive gas is. The critical type of a deflagration to detonation transition is shown in Fig. 3. Here, the Mach stem has an irregular structure from the beginning, whereas the primary deflagration waves have well defined contours. With increasing time the Mach stem front becomes more and more irregular until finally a detonation is initiated. The soot pattern of this experiment is shown in Fig. 4. Shock and combustion fronts again do not affect the soot. The Math stem, however, immediately after formation initiates a fast chemical reaction which is characterized by an irregular, nonreproducible soot imprint. Whilst the fast chemical reaction slowly transforms into a detonation, marked by the regular soot pattern, the compressed gas between the shock wave and the combustion front is detonating from the beginning. The supercritical type of a Mach stem induced detonation, in contrast to the two other types mentioned before, shows no intermediate reactions during the transition process (Fig. 5). The Mach stem causes an explosive reaction from the outset, (producing on the photographic record) a relatively broad line due to the fast motion of the front and the long exposure time (as well as a) soot pattern. Figure 6 shows that the Mach stem at the moment of its generation initiates a very fast chemical reaction which at once transforms into a detonation, showing up as a regular structure all over the picture with the exception of the primary deflagration process. As mentioned before, a meridional wall does not disturb the propagation of a spherical wave. In order to prove this and to demonstrate that the deflagration to detonation transition is caused by coalescing waves and independent of wall effects, the deflagration process was observed in a direction perpendicular to that of the schlieren pictures shown before. In Fig. 7 it is clearly seen that the detonation is first induced behind the deflagration front on a side well separated from the meridional wall and only later appears at the latter. Figure 8 shows the very regular soot pattern of this experiment.
Fig. 2. Soot pattern of the same experiment as shown in Fig. i.
The generation of detonation by coalescing ~ c a l
6~s
8gs
12t~S
deflagration waves
7Us
9Na
l?pS a f t e r ignition
Fig. 3. Spherical transition from deflagration to detonation by a Mach stem in the critical energy regime. 50% C~-I=-O2, 70 tort, ignition energy 49 J.
1191
Fig. 4. Soot pattern of the same experiment as shown in Fig. 3.
The generation of detonation by coalescing spherical dellagration waves
5.S
6~m
7pa
9.a
iii .... lllJS
1193
1 151~S a f t e r i g n i t i o n
Fig. 5. Spherical transition from deflagration to detonation by a Mach stem in the supercritical energy regime. 50% C2H2-O2, 150 tort, i~tion energy 49 J.
Fig. 6. Soot pattern of the same experiment as shown in Fig. 5.
The generation of detonation by coalesciq spherical deflagration waves
Ius
4~S
9US
IluS
19US a f t e r J . ~ t l o n
Fig. ?. Spherical Mach stem induced detonation at a meridiona] wall, subcrifical energy regime. 5 0 ~ C ~ 2 , 68 torr, ignition e n e r ~ 49 J.
1195
Fig. 8. Soot pattern of the same experiment as shown in Fig. 7.
The generation of detonation by coalescing spherical deflagration waves
1197
Conclusions The mechanism of the Mach stem induced detonation may be explained when analyzing the soot and schlieren pictures by means of the description of the detonation wave structure, as given by Schultz-Grunow [5]. During the formation of a Mach stem, locally important density gradients are found which are greater
Pl
Huqonlot
O - Is
:
Is-
:
Ic
shock shock induced cumbustlon
O - Im
:mach
stem
Im-ldm
:mach
stem induced detonation
Curve
~A
\ \
Flanm and d e f l a g r a t i o n propagation ~the closed end of a tube \
Shock Adiabat~
from
\ \
I
Gene]~allzed Hugonlot Curve
C
I Vk Fig. 9.
Schematic diagram for a deflagration to detonation transition by a Math stem.
Points with the number ] represent the subcriticalenergy regime. Points with the number 2 represent the critical encTgy regime, Points with the number 3 represent the supcrcritical energy regime.
1198
E. S C H M O L I N S K E
than those behind a deflagration wave. Moreover, the very irregular slip stream of the triple shock configuration [6] obviously increases the chemical reactivity of the flow behind the M a t h stem. Consequently, the probability of generation of arbitrarily distributed reaction centers is increased. These centers start blast waves which when intersecting under favorable conditions give rise to a further Mach stem induced local increase in density. As a consequence of this, new and stronger reactions are produced. Because the strength of a Mach stem depends on the strength of the coalescing blast waves and the intersection angle, not every Mach stem generates a reaction. This seems to be the reason for the irregular structure of the soot imprints and of the turbulent wave fronts in the schlieren pictures at the beginning. The local reaction centers and the blast waves increase reciprocally until a detonation is initiated. In Fig. 4 it is clearly seen how a detonation is generated. At the beginning there are only a few reaction centers which cause the irregular structure. The more centers which exist the smaller will be the cells in the soot imprint until a constant size is reached, i.e., a detonation has developed. We arrived at an analytical elucidation of the Mach stem induced detonation by employing the normal Hugoniot curve ( D A B C ) and the generalized Hugoniot curve (D 3 cm 2 cm C)[7]. The latter curve is the geometrical locus of such flow processes the reaction products of which have the Mach number one with respect to the discontinuity (Fig. 9). For a first approximation it is assumed that the propagation of a sector of a spherical wave is the same as the propagation of a Table 1. Gasdynamic parameters corresponding to states represented by particular points on Fig. 9. Subcritical Energy Regime pl
Torr
D
m Id
ut
m/s
w2 m / s uM m / s wM m / s pJpt p,/p~ p31P,
point
1s point 1c point 1m point
p~/P3
p2M/pl pt/p2M
P~M/P~ p,lp3•
1 dm
65, 2775 673 482 1072 935 4.43 0.3431 3.81 4.115 11.42 0.2123 7.21 2.1825
Subscripts: 1 2 3 M
Initial gas Shock compressed gas Reaction products Refers to Mach stem
Critical Energy Regime
point 2s point 2c point 2m point 2 dm
70 2785 765 591 1240 1116 5.74 0.295 4.62 3.413 15.30 0.1908 8.36 1.8935
Supercritical Energy Regime
point 3s point 3c point 3m point 3 dm
150 2820 1010 865 1642 1539 10.1 0.2233 6.75 2.396 26.93 0.1634 10.68 1.520
Nomenclature: p D u w
Pressure Detonation velocity Shock front velocity Flame front velocity
The generation of detonation by coalescingspherical deflagration waves
1199
plane wave in a tube closed at one end. Consequently, such a flow process can be described b y a curve (B 2 c m ) on the premise that the reaction products have a subsonic velocity [8]. With the aid of the shock adiabatic one can define a deflagration in the subcritical energy regime by a shock (0-1s) and a combustion ( l s - l c ) . An increase of the chemical reactivity of the gas due to a Mach stem process, (0-1 m) and ( l m - l d m ) , leads to an acceleration of the deflagration, i.e., the points I m and 1 d m are shifted into the direction of the points d and D, representing a C h a p m a n - J o u g u e t detonation. Deflagrations in the critical and supercritical energy regime, represented by the points 2 and 3, respectively, are more energetic. Consequently, Mach stems which are f o r m e d by the coalescence of spherical deflagration waves in these regimes cause other chemical reactions, points 2 c m and 3 cm, than Mach stems in the subcritical energy regime. Numerical results of treating the deflagration to detonation transition with Hugoniot curves are in full agreement with our experimental observations (see Table 1).
References 1. Oppenheim, A. K. and Stern, R. A., On the development of gaseous detonation-analysis of wave phenomena, 7th Syrup. on Comb., pp. 837-850, Baltimore (1958). 2. Manson, N., Brochet, Ch., Brossard, J. and Pujol, Y., Vibratory phenomena and instability of self sustained detonation in gases, 9th Symp. on Comb., pp. 461-469, Academic Press (1963). 3. Laderman, A. J. and Oppenheim, A. K., Experimental study of the development of detonation, Techn. Note DR 9, p. 14, Univ. of Calif. (1960). 4. Bach, G. G., Knystautas, R. and Lee, J. H., Direct initiation of spherical detonations in gaseous explosives, 12th Syrup. on Comb., pp. 853-864,The CombustionInstitute, Pittsburg, Penn. (1969). 5. Schultz-Cn'unow,F., Mikrostruktur der Detonationswellen in Gasen, Bericht Nr. 6/71, Ernst-MachInstitut, Freiburg. 6. Semenov, A. N., Syshchikova, M. P. and Berezkina, M. K., Experimental investigation of Math reflection in a shock tube, Soy. Phys. Techn. Phys. 15 (5), 797 (1970). 7. Troshin, Ya. K., The generalized Hugoniot adiabatic curve, 7th Syrup. on Comb., pp. 789-798, Baltimore (1958). 8. Shchelkin-Troshin, Gasdynamics of combustion, Mono Book Corp., p. 130, Baltimore (1965).
The generation of detonation by coalescing spherical deflagration waves E. S C H M O L I N S K E Ernst-Mach-Institut, Freiburg, W. Germany
(Received 12 September 1973) AImtraet--The generation of a spherical detonation was verified experimentally by two coalescing deflegration waves. Numerical results of treating the three different types of deflagration of detonation transition with Hugoniot curves are in full agreement with our experimental observations.
Introduction THE TRANSITIONfrom combustion to detonation is a phenomenon of considerable importance in many practical applications of explosives. Of major interest, therefore, is the experimental and theoretical elucidation of this gasdynamic event. Up to now the experiments were carried out in narrow channels of varying cross sections. Afterwards it was the task to explain the various types of waves and their origin and to find an analytical solution [1]. As was reported repeatedly, e.g. Ref. [2], the transition process is considerably influenced by the geometry of the duct, the roughness of the walls and site and kind of ignition. In consequence, the flame acceleration model [3] can give only an approximate description of the development of a detonation from a laminar combustion. This model assumes that due to turbulences at the walls of the duct the flame area is increased and accelerated until a shock wave is formed in front of the flame by coalescence of blast waves. Hence, a detonation is initiated. An essential purpose of this paper was to avoid strong boundary effects which was possible by using spherical waves. This allows us to give an exact description of the gasdynamic events by means of Hugoniot curves for we have a flow without friction and heat conduction. As is well known the introduction of a solid wall in the meridian plane of the spherical geometry does not derange the latter and gives rise to only negli~'bly small disturbances. On the other hand, it permits the simultaneous observation of wave propagations by means of soot patterns and schlieren pictures. Only the use of two or more independent registration methods seems to exclude the possibility of a misinterpretation as on several occasions wrong conclusions were made from the analysis of single photographic records.
Experimental results All our experiments were carried out in a cubic chamber of 15 cm lateral length. A perspex plate which had two pairs of electrodes placed 10 mm apart was mounted in the centre of the chamber. Above this plate lay another one, sooted 1187
1188
E. SCHMOLINSKE
and with orifices for the electric spark ignition. Each pair of electrodes was charged by a 0.5/~F capacitor with 14 kV and ignition of the explosive gas was initiated by a triggered spark gap. The framing frequency of the rotating mirror camera was 0.94 MHz and the exposure time 0.38 fts. The detonating mixture used in this study was equimolar acetylene-oxygen and the initial pressure range of the experiments was 50 to 200 torr. From all experiments it became clear that the transition from deflagration to detonation was originated in a Mach stem which in our case was formed by the coalescence of two spherical waves. Three different types of deflagration to
4~IS
7~S
9gS
llgS
13Us
16Us a f t e r
lqnltlc~
Fig. 1. Spherical transition from deflagration to detonation by a Mach stem in the subcritical energy regime. 50% C=H~--O, 65 torr, ignition energy 49 J.
The generationof detonationby coalescingsphericaldeflagrationwaves
1189
detonation transition were observed, depending on the appearance of a subcritical, critical or supercritical energy regime [4] in the Mach stem. The subcritical type is characterized by the fact that the Mach stem starts as a deflagration wave. Then, after a limited time, the detonation is initiated. Figure 1 shows a typical sequence of spark schlieren pictures illustrating this gasdynami~ event. At the moment of its generation and the following 9/zs the Mach stem consists of two discontinuities, i.e., a flame front with a preceding shock wave. At a later instant the Mach stem induced deflagration wave becomes irregular whereas the primary deflagration fronts do not show any alteration. Due to increasing instabilities in the form of reaction centers the Mach stem gets accelerated until a detonation is initiated. Figure 2 shows the soot pattern of this experiment. Neither the combustion process nor the shock wave itself affects the soot and no structure is visible at first. Only when single irregularities are occuring will the soot be wiped away. At a later instant the Mach stem initiates a detonation which is propagating into the quiescent gas as well as into the shock compressed zone of the deflagration. Clearly one can recognize these two areas by different structures of the soot imprint, the scale of which becomes, as is well known, smaller the higher the initial pressure of the explosive gas is. The critical type of a deflagration to detonation transition is shown in Fig. 3. Here, the Mach stem has an irregular structure from the beginning, whereas the primary deflagration waves have well defined contours. With increasing time the Mach stem front becomes more and more irregular until finally a detonation is initiated. The soot pattern of this experiment is shown in Fig. 4. Shock and combustion fronts again do not affect the soot. The Math stem, however, immediately after formation initiates a fast chemical reaction which is characterized by an irregular, nonreproducible soot imprint. Whilst the fast chemical reaction slowly transforms into a detonation, marked by the regular soot pattern, the compressed gas between the shock wave and the combustion front is detonating from the beginning. The supercritical type of a Mach stem induced detonation, in contrast to the two other types mentioned before, shows no intermediate reactions during the transition process (Fig. 5). The Mach stem causes an explosive reaction from the outset, (producing on the photographic record) a relatively broad line due to the fast motion of the front and the long exposure time (as well as a) soot pattern. Figure 6 shows that the Mach stem at the moment of its generation initiates a very fast chemical reaction which at once transforms into a detonation, showing up as a regular structure all over the picture with the exception of the primary deflagration process. As mentioned before, a meridional wall does not disturb the propagation of a spherical wave. In order to prove this and to demonstrate that the deflagration to detonation transition is caused by coalescing waves and independent of wall effects, the deflagration process was observed in a direction perpendicular to that of the schlieren pictures shown before. In Fig. 7 it is clearly seen that the detonation is first induced behind the deflagration front on a side well separated from the meridional wall and only later appears at the latter. Figure 8 shows the very regular soot pattern of this experiment.
Fig. 2. Soot pattern of the same experiment as shown in Fig. i.
The generation of detonation by coalescing ~ c a l
6~s
8gs
12t~S
deflagration waves
7Us
9Na
l?pS a f t e r ignition
Fig. 3. Spherical transition from deflagration to detonation by a Mach stem in the critical energy regime. 50% C~-I=-O2, 70 tort, ignition energy 49 J.
1191
Fig. 4. Soot pattern of the same experiment as shown in Fig. 3.
The generation of detonation by coalescing spherical dellagration waves
5.S
6~m
7pa
9.a
iii .... lllJS
1193
1 151~S a f t e r i g n i t i o n
Fig. 5. Spherical transition from deflagration to detonation by a Mach stem in the supercritical energy regime. 50% C2H2-O2, 150 tort, i~tion energy 49 J.
Fig. 6. Soot pattern of the same experiment as shown in Fig. 5.
The generation of detonation by coalesciq spherical deflagration waves
Ius
4~S
9US
IluS
19US a f t e r J . ~ t l o n
Fig. ?. Spherical Mach stem induced detonation at a meridiona] wall, subcrifical energy regime. 5 0 ~ C ~ 2 , 68 torr, ignition e n e r ~ 49 J.
1195
Fig. 8. Soot pattern of the same experiment as shown in Fig. 7.
The generation of detonation by coalescing spherical deflagration waves
1197
Conclusions The mechanism of the Mach stem induced detonation may be explained when analyzing the soot and schlieren pictures by means of the description of the detonation wave structure, as given by Schultz-Grunow [5]. During the formation of a Mach stem, locally important density gradients are found which are greater
Pl
Huqonlot
O - Is
:
Is-
:
Ic
shock shock induced cumbustlon
O - Im
:mach
stem
Im-ldm
:mach
stem induced detonation
Curve
~A
\ \
Flanm and d e f l a g r a t i o n propagation ~the closed end of a tube \
Shock Adiabat~
from
\ \
I
Gene]~allzed Hugonlot Curve
C
I Vk Fig. 9.
Schematic diagram for a deflagration to detonation transition by a Math stem.
Points with the number ] represent the subcriticalenergy regime. Points with the number 2 represent the critical encTgy regime, Points with the number 3 represent the supcrcritical energy regime.
1198
E. S C H M O L I N S K E
than those behind a deflagration wave. Moreover, the very irregular slip stream of the triple shock configuration [6] obviously increases the chemical reactivity of the flow behind the M a t h stem. Consequently, the probability of generation of arbitrarily distributed reaction centers is increased. These centers start blast waves which when intersecting under favorable conditions give rise to a further Mach stem induced local increase in density. As a consequence of this, new and stronger reactions are produced. Because the strength of a Mach stem depends on the strength of the coalescing blast waves and the intersection angle, not every Mach stem generates a reaction. This seems to be the reason for the irregular structure of the soot imprints and of the turbulent wave fronts in the schlieren pictures at the beginning. The local reaction centers and the blast waves increase reciprocally until a detonation is initiated. In Fig. 4 it is clearly seen how a detonation is generated. At the beginning there are only a few reaction centers which cause the irregular structure. The more centers which exist the smaller will be the cells in the soot imprint until a constant size is reached, i.e., a detonation has developed. We arrived at an analytical elucidation of the Mach stem induced detonation by employing the normal Hugoniot curve ( D A B C ) and the generalized Hugoniot curve (D 3 cm 2 cm C)[7]. The latter curve is the geometrical locus of such flow processes the reaction products of which have the Mach number one with respect to the discontinuity (Fig. 9). For a first approximation it is assumed that the propagation of a sector of a spherical wave is the same as the propagation of a Table 1. Gasdynamic parameters corresponding to states represented by particular points on Fig. 9. Subcritical Energy Regime pl
Torr
D
m Id
ut
m/s
w2 m / s uM m / s wM m / s pJpt p,/p~ p31P,
point
1s point 1c point 1m point
p~/P3
p2M/pl pt/p2M
P~M/P~ p,lp3•
1 dm
65, 2775 673 482 1072 935 4.43 0.3431 3.81 4.115 11.42 0.2123 7.21 2.1825
Subscripts: 1 2 3 M
Initial gas Shock compressed gas Reaction products Refers to Mach stem
Critical Energy Regime
point 2s point 2c point 2m point 2 dm
70 2785 765 591 1240 1116 5.74 0.295 4.62 3.413 15.30 0.1908 8.36 1.8935
Supercritical Energy Regime
point 3s point 3c point 3m point 3 dm
150 2820 1010 865 1642 1539 10.1 0.2233 6.75 2.396 26.93 0.1634 10.68 1.520
Nomenclature: p D u w
Pressure Detonation velocity Shock front velocity Flame front velocity
The generation of detonation by coalescingspherical deflagration waves
1199
plane wave in a tube closed at one end. Consequently, such a flow process can be described b y a curve (B 2 c m ) on the premise that the reaction products have a subsonic velocity [8]. With the aid of the shock adiabatic one can define a deflagration in the subcritical energy regime by a shock (0-1s) and a combustion ( l s - l c ) . An increase of the chemical reactivity of the gas due to a Mach stem process, (0-1 m) and ( l m - l d m ) , leads to an acceleration of the deflagration, i.e., the points I m and 1 d m are shifted into the direction of the points d and D, representing a C h a p m a n - J o u g u e t detonation. Deflagrations in the critical and supercritical energy regime, represented by the points 2 and 3, respectively, are more energetic. Consequently, Mach stems which are f o r m e d by the coalescence of spherical deflagration waves in these regimes cause other chemical reactions, points 2 c m and 3 cm, than Mach stems in the subcritical energy regime. Numerical results of treating the deflagration to detonation transition with Hugoniot curves are in full agreement with our experimental observations (see Table 1).
References 1. Oppenheim, A. K. and Stern, R. A., On the development of gaseous detonation-analysis of wave phenomena, 7th Syrup. on Comb., pp. 837-850, Baltimore (1958). 2. Manson, N., Brochet, Ch., Brossard, J. and Pujol, Y., Vibratory phenomena and instability of self sustained detonation in gases, 9th Symp. on Comb., pp. 461-469, Academic Press (1963). 3. Laderman, A. J. and Oppenheim, A. K., Experimental study of the development of detonation, Techn. Note DR 9, p. 14, Univ. of Calif. (1960). 4. Bach, G. G., Knystautas, R. and Lee, J. H., Direct initiation of spherical detonations in gaseous explosives, 12th Syrup. on Comb., pp. 853-864,The CombustionInstitute, Pittsburg, Penn. (1969). 5. Schultz-Cn'unow,F., Mikrostruktur der Detonationswellen in Gasen, Bericht Nr. 6/71, Ernst-MachInstitut, Freiburg. 6. Semenov, A. N., Syshchikova, M. P. and Berezkina, M. K., Experimental investigation of Math reflection in a shock tube, Soy. Phys. Techn. Phys. 15 (5), 797 (1970). 7. Troshin, Ya. K., The generalized Hugoniot adiabatic curve, 7th Syrup. on Comb., pp. 789-798, Baltimore (1958). 8. Shchelkin-Troshin, Gasdynamics of combustion, Mono Book Corp., p. 130, Baltimore (1965).