Detonation Diffraction

Issw22

Detonation DiractionJ.E. Shepherd, E. Schultz, R. Akbar

Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, CA 91125, USA.

Abstract: We contrast the problems of detonationand shock diraction over convex and concave cor-ners. Detonations are distinguished from shock wavesby the presence of an intrinsic length scale associ-ated with a reaction zone. The extent of this reac-tion zone depends on both the nature of the chemicalsystem and the detonation wave speed. The depen-dence of the reaction front thickness on wave speedcauses the diraction of detonations to be essentiallydierent from shock waves. Shock waves diract in aself-similar manner around both concave and convexcorners, but detonations do not. The departure of det-onations from self-similarity is manifested most dra-matically in diracting around convex corners. Undercertain conditions, a detonation can fail by degenerat-ing into a weak shock followed by a low-speed ame.Diraction around concave corners also departs fromself-similarity but the behavior is bounded by the lim-its of frozen and equilibrium ow.

Key words: detonation, diraction, Mach reection

1. Introduction

The study of shock wave diraction (Hornung 1985,Skews 1967) is greatly simplied by the use of similar-ity solutions. This is possible when there are no lengthscales introduced by either the geometry or physicalprocesses within the shocked gas. As long as the shocklayer structure is unimportant on the scale of the ex-periment (Hornung and Smith 1979), often the case inmany shock tube studies of diraction, self-similar be-havior is evident as shown in Fig. 1.

In the case of detonations, an intrinsic length scale isalways present due to the intense chemical reactionsthat occur in a layer just behind the leading shockwave (Fig. 2). The thickness of the reaction zoneis a function of the chemical system and the propor-tions of fuel and oxidizer as illustrated in Fig. 3 for thehydrogen-oxygen and hydrogen-air system. A secondlength scale is created as a result of the nonlinearcoupling between the chemicial reaction rates and thegas dynamics that creates a turbulent structure onand behind the detonation front. The nature of theturbulent structure is shown in Fig. 4 as manifestedin a shadowgraph (a) and on sooted foils (b) placed

Present address: Combustion Dynamics Ltd, Medicine Hat,Alberta, CANADA

shock

reflected wave

diffracted

reflected wave

incident wave

Mach stem

(a) (b)Figure 1. Shock waves exhibit self-similar or conical owbehavior during diraction over sharp convex (a) and con-cave (b) corners. The ows are characterized by anglesthat are a function of the incident shock Mach number.

against the inside tube wall. An idealized sketch ofthe front structure is shown in Fig. 4c, illustrating howthe triple point tracks create the cellular pattern withcharacteristic spacing on the sooted foil. The length is between 10 and 100 times the computed ideal-ized reaction zone length depending on the chemicalsystem.

0.01

0.10

1.00

10.00

100.00

1000.00

0.0 1.0 2.0 3.0

Equivalence Ratio

Reaction Z

one L

ength

(m

m)

H2-O2

H2-Air

Figure 3. Computed idealized reaction zone thicknessfor detonations in hydrogen-oxygen and hydrogen-air mix-tures propagating at the Chapman-Jouguet (CJ) velocity(295 K and 1 bar).

In addition to the intrinsic chemical length scale,detonations have a length scale associated with theexpansion wave (Taylor-Zeldovich wave) ow immedi-

Paper 5410 22nd International Symposium on Shock Waves, Imperial College, London, UK, July 18-23, 1999

Detonation Diraction 2

Pressure

TemperatureShock ReactantsProducts

distance

Intermediatescon

cen

tratio

n

en

erg

y re

leas

e ra

te

InductionRecombination

Figure 2. Idealized reaction zone structure in detonations.

Triple-Point Tracks

Transverse Waves

Reaction Zone

Shear Layers

(a) (b) (c)

Figure 4. Cellular structure of detonation fronts. (a) shadowgraph visualization. (b) soot foil tracks. (c) schematicof front structure.

ately behind the detonation front. Unlike shock waves,in which it is relatively easy to generate a uniform owbehind the shock front, detonations almost invariablyare generated in a fashion that results in nonuniformow. The characteristic length L associated with theTaylor-Zeldovich ow for planar waves is about one-half the propagation distance from the origin of thedetonation wave. In the cases of interest for our study,the detonations have propagated several meters fromthe initiator, which is usually a factor of 10 to 100times larger than the observation length of a few cen-timeters associated with the gradients produced bydiraction. For this reason, we focus on the role ofow gradients associated with the diraction processrather than the initiation process.

1.1. Failure of Self-similarity

The eect of all of these length scales, < < L, isto spoil the possibility of a unique self-similar solutionfor diraction of detonation waves. The results of anyexperiment have to be analyzed considering the thick-ness of the front relative to the characteristic phys-ical length scale over which the measurements aretaken. From an experimental viewpoint, the appro-priate thickness measure of the front is the cell width while most theoretical considerations use the ideal-

ized reaction zone thickness . For diraction oversharp corners, the observation length scale is that as-sociated with the wave transit time, Ut, from thecorner itself.

Studies on shock waves in dissociating gases(Hornung and Smith 1979) have identied two limit-ing regimes in which self-similar behavior will be re-covered. The frozen limit, / , will occur atearly times and the diraction process is controlled bynonreactive or frozen dynamics of the leading shockfront. The equilibrium limit, / 0, will occur atlate times and the diraction process is controlled byfully-reacted or equilibrium dynamics of an ideal-ized, thin reaction front. Although this classicationseems reasonable to extend to detonations, this sim-ple picture is complicated by the dependence of thereaction zone thickness on the strength of the leadingshock front. The outcome of a diraction experimentwith a detonation can be dramatically dierent fromthe diraction of a shock wave in a dissociating gassince the speed of the detonation wave depends on thecoupling of the reaction zone to the shock front. De-coupling of the reaction zone from the shock front cancause the propagation mode to change from a detona-tion traveling at 3000 m/s to a sound wave followedby a subsonic ame.



We have systematically explored these tworegimes through experiments; varying the char-acteristic length by changing the amount ofdiluent and the initial pressure in the reactants.This is not a new subject (Bazhenova et al. 1965,Edwards et al. 1979, Gavrilenko and Prokhorov 1983,Zeldovich et al. 1956) but previous work is oftencontradictory since the role of the intrinsic lengthscale has not always been explicitly recognized.We have recently carried out high-resolution owvisualization studies in our laboratory (Akbar 1997,Schultz 1998) to examine some quantitative aspectsof the diraction process. Some preliminary resultsfor the cases of Mach reection and corner turningare given below.

2. Mach Reection

The eect of the chemical reaction length scale on det-onation Mach reection is shown in Fig 5. The keydierence between shocks and detonation waves is thehorizontal striations appearing behind the detonationwave. These striations, either shock waves or contactsurfaces, are evidence of the weak transverse shockwaves propagating along the main front. The interac-tion of the transverse waves with the main reectedwave is one of the essential dierences between shockand detonation diraction. This results in a blurringof the main triple point location and makes it di-cult to distinguish the associated reected wave andcontact surface. Note that the cellular wave spacingis smaller behind the Mach stem than the incidentwave. This is a consequence of decreasing reactionzone length with increasing wave speed.

The reection of shock waves from two-dimensionalwedges was systematically explored (Akbar 1997) forwedge angles between 15 and 50. The eect of cellwidth and the regularity of the cellular structurewas examined for three mixtures, given in Table 1.Some representative results are given for two wedgeangles in Fig. 6. The rst row of shadowgraphs is fora wedge angle of 20, much less than the critical an-gle of about 40 for transition from Mach to regularreection, shown in the second row of images. Thetriple point and reected wave features in the 40 caseare much more distinct at 40 than at 20 since theMach stem is much stronger. Images (a), (b), (d) and(e) have incident wave cell widths of 7-15 mm, compa-rable to or larger than the Mach stem height shown.The third image (c) has incident cell widths of 1-2mm, smaller than the Mach stem height. A secondaryeect is that the cellular structure of the highly Ar-diluted mixtures is very regular in comparison to theundiluted mixture.

Detailed analysis of the wavefront shapes and tra-

jectories is not completely consistent with either thefrozen or equilibrium limits. This is most evidentwhen the main triple point track trajectory angle (see Fig. 1) is plotted vs. wedge angle andcompared to the results of model computations forthe limiting cases of frozen and equilibrium inter-actions (Akbar 1997). The rst method used thethree-shock construction (assuming the Mach stemis straight and normal to the surface of the ramp)and matched the ow deection angle across the con-tact surface (Hornung 1985) using numerical solutionswith realistic thermochemistry for the gases. Thesecond method used the approximate shock dynam-ics method of Whitham, appropriately modied fordetonations (Akbar 1991). The present results areshown in Fig. 7 for all three mixtures. The mixture(2H2+O2+10.33Ar, 20 kPa) with the largest cell width appears to be close to the frozen ow limit. Themixture (C2H2+2.5O2+14Ar, 50 kPa) with the small-est cell width appears to be near equilibrium for largewedge angles but intermediate at smaller values. Themixture (2H2+O2, 20 kPa) appears to be closest toequilibrium although it has a cell width of 7-8 mm.The irregular nature of the cellular structure of thismixture may be a factor. Note the lack of data forwedge angles less than 20, a region where the dier-ence between predicted equilibrium and frozen behav-ior is quite pronounced. It is dicult to obtain data atsmaller wedge angles since the wave fronts were curvednear the triple point making the location ambiguous;also, the reected wave is weak and obscured by thetransverse waves.

(a) (b)Figure 5. Shadowgraph visualization of the diractionof (a) shock wave (M=1.7, air) on a 20 wedge and (b)detonation in Ar-diluted acetylene-oxygen mixture on a25 wedge (Akbar 1997).

In Fig. 7c, we compare the 2H2+O2 mixture withthe previous results (Meltzer et al. 1993) obtained us-ing soot foils. At 20 the values of inferred fromthe soot foils are signicantly higher than from theshadowgraphs. At 25 and 30, the data from thesoot foils are slightly higher than from the images. Inaddition, we can see from the images that the trans-



Mixture P0 UCJ Structure(kPa) (m/s) (mm)

2H2 + O2 20 2757 7 - 8 irregular2H2 + O2 + 10.33 Ar 20 1540 15 - 18 regular

C2H2 + 2.5 O2 + 14.0 Ar 50 1688 1.75 regular

Table 1. Mixture properties for Mach reection study (Akbar 1997)

(a) (b) (c)

(d) (e) (f)

Figure 6. Detonation Mach reection on a 20 wedge: (a) 2H2+O2, 20 kPa. (b) 2H2+O2+10.33Ar, 20 kPa.(c) C2H2+2.5O2+14Ar, 50 kPa. On a 40 wedge: (d) 2H2+O2, 20 kPa. (e) 2H2+O2+10.33Ar, 20 kPa. (f)C2H2+2.5O2+14Ar, 50 kPa. (Akbar 1997)

0

5

10

15

20

25

0 10 20 30 40 50

(degrees)

(

degre

es)

Soot Foils (Meltzer)

Shadowgraph

Equilibrium CCW

Equilibrium 3-shock

Frozen CCW

Frozen 3-shock

0

5

10

15

20

25

0 10 20 30 40 50

(degrees)

(

degre

es)

Shadowgraph

Equilibrium CCW

Frozen CCW

Frozen 3-shock

Equilibrium 3-shock

0

5

10

15

20

25

0 10 20 30 40 50

(degrees)

(

degre

es)

Shadowgraph

Equilibrium CCW

Frozen CCW

Frozen 3-shock

Equilibrium 3-shock

(a) (b) (c)

Figure 7. Triple-point trajectories inferred from laser schlieren photographs and compared to simple models of thereection process. (a) 2H2+O2, 20 kPa. (b) 2H2+O2+10.33Ar, 20 kPa. (c) C2H2+2.5O2+14Ar, 50 kPa.(Akbar 1997).

verse waves may propagate through the Mach reec-tion triple point so that changes in cell width may

not be truly indicative of the main triple point lo-cation. This is a particular problem with irregular



mixtures and weak reected waves for which it is dif-cult to distinguish on the sooted foil the Mach re-ection triple points from the instability wave triplepoints. All this suggests caution is in order when in-terpreting sooted foils in this type of experiment. Thecases studied are not close enough to the limiting situ-ations to provide denite tests of the frozen and equi-librium limits although it is clear we have approachedthose limits in some instances. The available data isbounded by the equilibrium and frozen limits. Fu-ture experiments should include consideration of thereaction length scale in analyzing detonation Mach re-ection.

3. Corner Turning

The breakdown of self-similarity and the inuence ofthe reaction zone length scale is much more strik-ing in diraction around convex corners. Exam-ples can be seen in Fig. 8, a detonation diract-ing about a corner created by a tube emerging intoa larger volume. The dependence of reaction layerthickness on shock strength can result in the catas-trophic increase of reaction zone thickness when theleading shock weakens as it diracts out of the tubeinto the surrounding region. In Fig. 8a, a region ofshocked but unreacted gas can be seen behind theshock front in the vicinity of the tube opening. Ahighly disturbed region near the front results fromthe reinitiation of the detonation. High-speed moviesshow that this region spreads from the front to therear of the wave, reinitiating the detonation processalong the entire wave front. Experimental obser-vations (Bazhenova et al. 1965, Edwards et al. 1979,Zeldovich et al. 1956) show that the outcome of thediraction process depends on the ratio of the reac-tion zone thickness or cell width to the tube diame-ter and also, the specics of the chemical system. Ageneral rule of thumb is that when D < A, reini-tiation of the detonation wave does not occur and ashock wave followed by a contact surface is the nal re-sult. The constant A is between 10 and 30 dependingon the shape of the opening and the type of mixture(Shepherd et al. 1986). The extremes of the behaviorobserved in initiation are shown in photographs (b)and (c) of Fig. 8. These two shadowgraphs illustratethe subcritical (b) and supercritical (c) situations thatbracket the critical case (a).

The origin of the dramatic behavior shown in Fig. 8is the strong dependence of the reaction time on theshock strength. Estimates of reaction time for thestoichiometric hydrogen-oxygen system are shown inFig. 9. Note the extremely rapid increase of reactiontime with decreasing shock strength below a shockspeed of 0.8UCJ . This is due to the Arrhenius depen-dence of reaction rates on postshock temperature and

1.E-08

1.E-07

1.E-06

1.E-05

1.E-04

0.5 1 1.5

U/UCJ

reaction t

ime (

s)

Figure 9. Reaction time behind shock waves in a stoi-chiometric hydrogen-oxygen mixture (Po = 20 kPa) as afunction of shock strength, UCJ =2760 m/s.

is responsible for the failure of the detonation waveobserved in Fig. 8b.

The decrease in shock strength in the diractedregion of the detonation can be understood by ex-amining the analogous process in shock diraction(Skews 1967). As shown in Fig. 10, diraction arounda sharp corner creates a self-similar, unsteady expan-sion fan that interacts with the shock. There is a dis-tinct point at which the boundary of the disturbedregion intersects the undisturbed shock front. Dueto the diracted front area increase and interactionwith the expansion wave, the shock strength decreasescontinuously along the shock front from the boundaryof the disturbed region up to the intersection of thediracted shock with the side wall. The location ofthe boundary of the disturbed region on the shockcan be determined by a simple acoustic construction(Fig. 11) that determines where the rst disturbancefrom the corner intersects the incident shock wave.

From the geometry of this gure, the transversespeed V of the boundary of the disturbed region alongthe shock front is

V =c2 (U u)2 (1)

where c is the sound speed behind the shock front, U isthe shock speed and u is the postshock velocity in thelab frame. The trajectory of the disturbance boundaryis a line with angle measured relative to the owdirection as shown in Fig. 1 . From the geometry ofFig. 11, this angle is

= tan1(V

U

)(2)



Figure 8. Shadowgraph visualization of the (a) critical (b) subcritical and (c) supercritical diraction of a detonationwave emerging from a tube of diameter D = 25 mm into a large volume (Schultz 1998).

self-similar unsteady expansion

wave head

Figure 10. Region of similarity solution for shock dirac-tion around a sharp corner.

ut

ct

Ut

Vt

Figure 11. Skews (1967) construction for determiningthe location of intersection of the disturbed region withthe incident shock front in corner signalling problem.

The shock jump conditions can be used to evaluatethe speed V and angle as a function of the incidentshock speed. The results of calculations for a perfectgas are shown in Fig. 12. These computations indicate

that the disturbance spreads at a nite rate that de-pends on the shock Mach number and ratio of specicheats. For large shock Mach numbers, Ms > 5 to 7,typical of detonation fronts, the angle asymptotes toa value between 15 and 30 degrees for 1.1 < < 1.67.Observations of diracting shock waves (Skews 1967)agree with these predictions.

0

5

10

15

20

25

30

35

0 1 2 3 4 5 6 7

Shock Mach No

Wave H

ead A

ngle 1.67

1.4

1.1

1.2

1.3

Figure 12. Disturbance angle computed using Skewsconstruction for a shock wave in a perfect gas, no reac-tion.

The propagation of disturbances and the magnitudeof the disturbance speed play a key role in determin-ing the critical condition in the detonation diractionproblem. Similar computations can be made for a det-onation on the basis of assuming an innitely thin re-action zone. This implies evaluating the sound speedand particle velocity at the product thermodynamicstate found at the end of the reaction zone. The resultsare shown in Fig. 13 for the stoichiometric hydrogen-oxygen case and the shock wave results are shown forcomparison. Inspection of this gure reveals that fora detonation traveling at the CJ speed, the transversesignalling wave speed based on the equilibrium prod-



uct state is zero, i.e., = 0. This leads to a paradoxrst articulated by Gvozdeva (Bazhenova et al. 1965)that CJ detonations should diract without any inu-ence of the diracting aperture size.

However, our observations and those ofprevious researchers (Bazhenova et al. 1965,Edwards et al. 1979, Zeldovich et al. 1956) indi-cate that not only does the aperture size have astrong inuence on the diraction but also thesignalling speed V is clearly nite. The resolution tothis paradox is to consider the role of the reactionzone structure (Fig. 3) in the signalling problem. Theresults shown in Fig. 13 indicate that there must bea continuous variation of transverse signalling speedbetween the shock and the end of the reaction zone.This arises because of the variation in the thermody-namic state and velocity through the reaction zone.The idealized one-dimensional approximation to adetonation (ZND model) has been used to evaluatethe variation of transverse signalling speed and arepresentative result is given in Fig. 14.

0

5

10

15

20

25

30

35

0 1 2 3 4 5 6 7

Shock Mach No

Wa

ve

He

ad

An

gle

shock, =1.4detonation

MCJ

Figure 13. Disturbance angle computed using Skewsconstruction for leading shock wave and the detonationproducts for a stoichiometric hydrogen-oxygen mixture(Po = 20 kPa).

Note that the signaling speed in Fig. 14 has a max-imum value close to the shock front and that themaximum signalling speed can be approximated bythe value at the front. This suggests that the re-sults of Fig. 12 can be used to estimate that rate atwhich the disturbance propagates into the front. Sig-nalling within a reaction zone is a complex problem(Barthels and Strehlow 1966) since the sound speedand uid velocity depend on position. However, in thepresent analysis, we are only concerned with how theleading disturbance propagates. For the case of thestoichiometric hydrogen-oxygen mixture, this is about1200 m/s, resulting in a signalling boundary angle of

0

200

400

600

800

1000

1200

1400

1600

0 0.5 1

distance (cm)

sig

nal speed (

m/s

)

CJ

Figure 14. Signaling speed computed using generaliza-tion of Skews construction for detonation reaction zoneregion, CJ detonation in stoichiometric hydrogen-oxygenmixture (Po = 20 kPa).

about 23. Experiments are in progress at our labora-tory to make measurements of the disturbance propa-gation in a variety of mixtures.

The signalling process and the propagation of dis-turbances along the detonation are the easiest under-stood aspects of the diraction process. Much morechallenging are the phenomena shown in Fig. 8a, inwhich the disturbance initiates the failure of the det-onation and then at some point, a spontaneous re-emergence of the detonation occurs. The process offailure can be readily explained through the combi-nation of the dependence of the reaction zone lengthon shock strength (Fig. 9) and the rapid decrease inshock strength from the disturbance head to the wall.However, additional considerations are needed to de-scribe the re-emergence of a detonation since onlya monotone increase in reaction zone length is pre-dicted with decreasing shock strength. This problemis very closely related to that of detonation initiationby spherical shock waves in which a critical energylevel is observed for initiating detonations. That criti-cal energy level is apparently a consequence of a secondtime scale associated with a critical level of unsteadi-ness (Eckett et al. 1998) that determines if reactionswill be quenched or continue to propagate. Similarconsiderations should apply to the critical tube prob-lem and work on these ideas is in progress.

4. Summary

Chemical reactions lead to the breakdown of self-similar behavior in diracting detonation and shockwaves. Experimental observations of detonationdiraction around convex and concave corners show



that this occurs in a distinct fashion for these twocases.

In the case of diraction around a concave corneror Mach reection, the phemonena are superciallysimilar to those of nonreactive shocks, exhibiting sim-ple Mach reection for small ramp angles and regularreection for large angles. However, for small rampangles, the triple point is diuse, apparently becausethe reected wave strength is comparable to that ofthe transverse waves. An alternative explanation isthat the reected wave is a compression wave of -nite thickness due to the reaction zone thickness. Thediraction process appears to be self-similar in thelimit of very large or very small Mach stems as com-pared to the detonation cell width (or reaction zonelength). The trajectory of the triple point is boundedby the limiting cases of frozen and equilibrium behav-ior. As a consequence, experiments with diering re-action zone lengths will result in dierent outcomes forboth the triple point trajectory and transition pointbetween regular and Mach reection. This can helpexplain the discrepancies between experiments carriedout with dierent fuel-oxidizer systems.

In the case of diraction around a convex corner,a detonation can fail during the diraction process.Detonation failure is apparently a consequence of un-steadiness quenching the reactions in the expansionwave that is produced by the ow around the cor-ner. Failure of the detonation means that the frontseparates into an essentially non-reactive shock wavefollowed by a contact surface that may evolve into aame. When the diraction occurs through a channelor tube, complete failure is only observed when thetube or channel width is smaller than some multiple ofthe detonation cell width. Near the critical condition adetonation wave spontaneously re-emerges within thepartially quenched reaction zone. The mechanism andprediction of this re-emergence of a detonation is oneof the key unsolved problems in detonation physicstoday. We are focusing on explanations that involvesignal propagation within the reaction zone and criti-cal levels of unsteadiness competing with reaction pro-cesses.

Acknowledgement. This work was partially supportedby the Powell Fund of the California Insitute of Technol-ogy and a NASA STTR contract in collaboration with Ad-vanced Projects Research Inc., La Verne, CA.

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