WAVE GENERATIONS FROM

CONFINED EXPLOSIONS IN ROCKS

C� L� Liu and Thomas J� Ahrens

Seismological Laboratory� California Institute of Technology� Pasadena� CA �����

In order to record P� and S�waves generated from con�ned explosions in rocks in the labo�

ratory� a method is developed based on the interactions between incident P� and SV�waves

and free�surfaces of rocks� The relations between particle displacements of incident P� and

SV�waves� and the strains measured using strain gauges attached on free�surfaces of rocks are

analytically derived� P� and SV�waves generated from con�ned explosions in Bedford limestone

are recorded�

INTRODUCTION

Virtually all the methods that have been pro�posed for discrimination of underground explo�sions � mb � �� from earthquakes and mining ex�plosions�� and �� are based on various P�to�S am�plitude ratios� Although there are many previousstudies of seismic radiation patterns from decou�pled explosions�� � �� it is still unclear what con�trols the radiation pattern of S�waves in tampedand decoupled explosions ��� Therefore studyof P� and S�wave generation from con�ned explo�sions is important for discrimination purposes� Inorder to investigate waves generated from small�scale laboratory explosions in rocks a measure�ment method is required to monitor both P� andS�waves� Conventional seismic recording systemsand methods for laboratory scale high strain rateexperiments��� can not be utilized� Based onthe analysis of the interactions between P� andSV�waves and free surfaces we have developed amethod to monitor P� and SV�wave pro�les usingtwo perpendicular strain gauges attached to thefree�surfaces of samples� The method and someinitial experimental data are presented below�

MEASUREMENT METHOD

When elastic P� and S� waves generated fromexplosions in rocks re ect at free surfaces thesegenerate di�erent displacement�time history� We

determine P� and SV�wave displacements usingthe strains measured along two perpendicular di�rections at a series of stations on free�surfaces ofthe rock samples�

Data reduction method

The strain gauges are attached at positionsalong the intersection of the plane containing theaxis of the spherical wave front and the sam�ple free�surface as shown in Fig��� The strainsrecorded by the gauges include the contributionsfrom incident P� or S�waves and re ected P� andS�waves� The relation between the measuredstrains and incident P� and S�wave particle dis�placements are derived as follows�

P�wave re�ections at free surfaces

The displacement re ection coe�cients forincident planar P�waves at free surfaces ���are PP � �B � A���B � A� PS ���� sin���� cos��j���A � B�� where PP and PSare re ection coe�cients of P� and SV�wave dis�placements due to incident P�waves and � and� are P� and S�wave velocities respectively� A �cos���j� and B � ��� �

� sin��j� sin����� where �and j are P�wave incident angle and S�wave re� ection angle respectively �Fig����The displacements of particles on free surfaces

after P�wave re ections are

uppar � u

Ip��� � PP � sin � � PS cos j� � Hparu

Ip� ���

: in Shock Compression of Condensed Matter-1997, edited by S.C. Schmidt, D.P. Dandekar, and J.W. Forbes, pp. 859-862, AIP Press, Woodbury, NY, 1998 859-862.

upper � u

Ip���� PP � cos � � PS sin j� � Hperu

Ip� ���

where uIp is the particle displacement of in�cident P�waves� uppar and upper are the parti�cle displacements along the directions shown inFig�� respectively� Having substituted PP andPS into Eqs���� and ��� the two coe�cientsare Hpar � � cos � sin��j���A�B� and Hper �� cos � cos��j���A�B��

SV�wave re�ections at free surfaces

For incident SV�waves the re ection coe��cients for P�waves �SP � and SV�waves�SS����are SP � �

� sin������As � Bs�� SS � �As �

Bs���As � Bs�� where As � cos������ Bs ���� �

� sin���� sin��j�� � and j are SV�wave incidentangle and P�wave re ected angle respectively�The displacements of particles on free surfaces

after SV�wave re ections are

usvpar � u

Isv ����SS� cos ��SP sin j� � Gparu

Isv� ���

usvper � u

Isv��SS� �� sin ��SP cos j� � Gperu

Isv� ��

where usvpar and usvper are particle displacements af�

ter re ection along the directions shown in Fig��and uIsv is the particle displacement of incidentSV�waves� The two coe�cients are determinedto be Gpar � � cos���� cos ���As � Bs�� Gper �

���� cos j sin������As �Bs��

Particle displacements of incident P�waves

Because upper is perpendicular to free surfacesand incident waves are assumed to be sphericalthe strain due to upper along direction � �per� issimply expressed as

�per� � HperuIp�r�� ��

where r� is the distance between the center ofexplosive source and the free surface of samplesat � � ��uppar does not result in any strains along direc�

tion � so the total strain induced by the incidentP�waves is determined to be �p� � H�u

Ip�r�� where

H� � Hper� Therefore the strain along direction� yields the particle displacement of incident P�waves as

uIp � r��p��H�� ���

Since both upper and uppar have contributions to

strains along direction � we need to consider the

resultant displacements� The length of the gauge

after re ection �s is equal to �r�n � ��rn�� ����

�

���

where �� � ls cos �rn

ls is the initial length of thestrain gauge rn is the distance between the ex�plosive source center to the position of a gaugeupon P�wave re ection� Here rn can be expressedas rn � r�u cos������where r is the distance be�tween the explosive source center and the gaugebefore P�wave re ection u is the resultant dis�placement of the point at � on the free surfaceand � is the angle between u and upper� u and �

are given by u � uIp� cos�����A�B� and � � �j�From the expressions above �s is given as

�s � �rn ��

�rn�rn�

��

���� ���

BecauseuIpr � � �s is rewritten as

�s � �r���tan����

���uIp�W ������

tan����

���tan���

dW

d������

���

where W ��� � � cos��� cos�� � ����A�B��From the de�nition of strains

�p� ��s� ls

ls�

H�uIp

r�� ���

where H� � cos����W �� � tan������� �tan � dWd� ���� � tan��������The particle displacement of an incident P�

wave determined from a gauge along direction �is

uIp � r��p��H�� ����

Particle displacements of incident SV�waves

Using the same above formulation the rela�tions between incident SV�wave particle displace�ments and strains along the two directions areobtained�The particle displacements of incident SV�

waves from the strains along direction � and �are

uIsv � r��sv� �G�� ����

anduIsv � r��

sv� �G�� ����

where G� � Gper

: in Shock Compression of Condensed Matter-1997, edited by S.C. Schmidt, D.P. Dandekar, and J.W. Forbes, pp. 859-862, AIP Press, Woodbury, NY, 1998 859-862.

Rock sample

θ

θj

Incident P- or SV-wave

Reflected SV- or P-wave

Reflected P- or SV-wave

12

3

4

56

7

8

Figure 1. Schematic arrangement of experiments

Rock joint

Explosives

Spherical cavity

Gauge

u per

par

Direct

ion 2

(u

)

Direction 1

Gauges (7, 8)

(5,6)

(3, 4)

(1,2)

G� � cos����Ws���tan����

�� � tan �

dWs

d����� �

tan����

���

Ws �� cos�� � �� cos���

As �Bs

�cos����� � ��

���

cos� j sin�����

�

�

�

and tan � � �� cos ���� cos j�� tan�����

Characteristics of strains in di�erent directions

Figure � shows the dependence of H�� H�� G�

and G� on incident angle that were calculatedusing the equations derived above� For incidentP�waves the constant H� is relatively insensi�tive to � and H� changes from ��� at � � �o

to ��� at � � ��o for Bedford limestone� Theconstant H� is very sensitive to � and it variesfrom � to ���� when � varies from � to ��o� Itcan be seen that the strains induced by compres�sional P�waves along direction � are always posi�tive but the strains along direction � are positivewhen � is less than ��o and negative when � islarger than ��o� This change in polarity is con�trolled by the ratio of the projection of P�wavedisplacements along direction � to that along thedirection that is perpendicular to the free sur�face� If the strain induced by the displacementalong direction � is less than that due to the dis�placement along the perpendicular direction thestrain is positive otherwise the strain is nega�tive�

From the above calculation the gauges alongdirection � are not sensitive to an incident SV�wave however the gauges along direction � arevery sensitive to an incident SV�wave� The po�larities of the strains along direction � are alwaysnegative and the polarities of the strains alongdirection � are determined by the direction of SV�wave particle motion �Eqs���� ���� ���� and ���� give the relations

between strains along the two directions and in�cident P� and SV�wave particle displacements� Ifstrains along the two directions can be recordedthe P� and SV�wave amplitudes can be deter�mined experimentally�

EXPERIMENTS AND RESULTS

The method described above was used to mon�itor P� and SV�waves generated from con�ned ex�plosions in rocks� The rock sample �Bedford lime�stone� was assembled with two blocks as shownin Fig��� The rock sample with strain gauges wasplaced inside a tank that was pressurized to ��bars�The recorded strains for one of the experiments

are shown in Figs�� and �� The characteristics ofthe strains recorded by the gauges are the sameas predicted from our derived equations� Thestrains along direction � induced by incident P�waves are always positive while the strains alongdirection � change polarities as P�wave incidentangle increases �Figs�� and ��� The strains in�

: in Shock Compression of Condensed Matter-1997, edited by S.C. Schmidt, D.P. Dandekar, and J.W. Forbes, pp. 859-862, AIP Press, Woodbury, NY, 1998 859-862.

duced by incident SV�waves along direction � arenegative and are much larger than that along di�rection ��From the records it is straightforward to de�

termine P and SV�wave amplitudes for the ex�periment using the expression given above� Fromthe P� and SV�wave velocities of Bedford lime�stone the expected S�wave arrivals are labeledon the records� The time di�erence between theexpected and the recorded is less than �s for theexperiment�

CONCLUSIONS

In this work a method has been developed formeasuring P� and SV�wave amplitudes generatedfrom explosions in rocks� The relations betweenthe strains given by gauges placed on the free sur�faces of rocks and incident particle displacementsare derived analytically� The experimental re�sults showed that the characteristics of recordedstrains along the two directions are in good agree�ment with the predictions�

ACKNOWLEDGMENTS

Research was sponsored by Air Force Techni�cal Applications Center� Contribution ���� Di�vision of Geological and Planetary Science Cali�fornia Institute of Technology�

REFERENCE

�� Blandford� R� R�� AFTAC�TR�������� �����

�� Helmberger� D� V� and Woods� B�� Monitoring a

Comprehensive Test Ban Treaty� edited by Husebye�E�S� and Dainty� A�M�� Kluwer Academic Publishers�the Netherlands� ����� ����

� Glenn L� and Goldstein P�� J� Geophys� Res�� �������� � ��������� ��

� Glenn L�� Ladd A�� Moran B�� and Wilson K�� Geo�phys� J� R� astr� Soc�� ��� ��� � �������

�� Murphy J� � op� cit in Ref����� pp � �����

�� Sykes L�� op� cit in Ref����� pp ����� ��

�� Kim S�� Clifton R� and Kumar P�� J� Appl� Phys��

��� ��� ���������

� Aki� K� and Richards� P�� Quantitative Seismology

Theory and Methods� W� H� Freeman and Company����� ch��� ���

-4.0

-2.0

0.0

2.0

4.0

0 15 30 45 60

H1

H2

G1

G2

Figure 2. Incident angle (degree)

H o

r G

Gauge-2

4

68

Stra

in(1

00µε

)

Time ( µ s)

Figure 3. Strains induced by P-wave along direction 1

20 40 60 100800

2

4

3Gauge-1

5

7

Time( µs)

Str

ain

(100

µε)

" " ------ Indicates SV-wave arrival

Figure 4. Strains induced by incident SV-waves along direction 2

20 40 60 80 100

0

-2

: in Shock Compression of Condensed Matter-1997, edited by S.C. Schmidt, D.P. Dandekar, and J.W. Forbes, pp. 859-862, AIP Press, Woodbury, NY, 1998 859-862.