Table Of ContentSPHERICAL WAVE MOTION
AND
DYNAMIC STRAIN MEASUREMENTS
by
S* Norman Domenico
ProQuest Number: 10795916
All rights reserved
INFORMATION TO ALL USERS
The quality of this reproduction is dependent upon the quality of the copy submitted.
In the unlikely event that the author did not send a complete manuscript
and there are missing pages, these will be noted. Also, if material had to be removed,
a note will indicate the deletion.
uest
ProQuest 10795916
Published by ProQuest LLC (2018). Copyright of the Dissertation is held by the Author.
All rights reserved.
This work is protected against unauthorized copying under Title 17, United States Code
Microform Edition © ProQuest LLC.
ProQuest LLC.
789 East Eisenhower Parkway
P.O. Box 1346
Ann Arbor, Ml 48106 - 1346
A thesis submitted to the Faculty and Board of Trustees
of the Colorado School of Mines in partial fulfillment of the
requirements for. the degree of Doctor of Science#
Signed: 5~
S» Norman Domenico
Golden, Colorado
February 1, 1951
Approved
J. C. Hollister
ACKNOWLEDGMENTS
The writer wishes to acknowledge his indebtedness to
Professor J. C. Hollister, Head of the Department of Geo
physics, Colorado School of Mines, whose direct and sincere
supervision made this investigation possible and also to
Dr. G. T. Merideth for his assistance with many aspects of
the theory.
The writer further wishes to express his deep appre
ciation to the Socony-Vacuum Oil Company, whose financial
assistance aided materially in the completion of this work.
CONTENTS
Page
INTRODUCTION ........................................ 1
THEORETICAL CONSIDERATIONS .......................... 5
General Wave Equation • • • • • • • • • • • • • • 5
Spherical Wave Equation ...... ............... 8
Minimum Oscillatory Requirement .............. 11
Change of Pulse Shape • • • • ................ .. 1®
Energy Density. ........ ........... .. 2?
Resumd. 33
EXPERIMENTAL RESULTS ................................ 3k
Work Site ................ « 3^
Field Equipment and Procedure........ .. 35
Field Data and Computations............ .. 39
Résumé. ^9
CONCLUSION.......................................... 51
BIBLIOGRAPHY ................................ $k
APPENDIX: _
INSTRUMENTATION....................... .............. la
General . . . . . . . .
Strain Gage . . . . . . . 5&
Pre-Amplifier • • • • • • • • • • • • * • • • • • 1 ^
Central Control Unit. . . . . . . . . . . . 18a
Tuning Fork ........ . . . . . . . . . . . . . . 23a
Camera. . . . . . . . . . . ^5a
Oscillographs . . . . . . . . . . . . . . 29a
INTRODUCTION
For quite some time geophysicists have employed the
concepts of seismic wave propagation in many successful
attempts to delineate subsurface structural features of the
earth's outermost crust. This has been accomplished without
specific knowledge of the form of the seismic pulse and of
the manner in which this form changes with distance. The
important factors in exploration geophysics are the deter
mination of velocities of the seismic waves through various
types of sediments, and the detection at the surface and
accurate timing of the reflected or refracted waves. To
this end the geophysicist has enjoyed considerable success
and at present the seismic method of prospecting is perhaps
the most effective known, although the most expensive. The
physicist has attempted to keep pace with the very rapid de
velopments in the field by explaining as best he can the
various empirical observations of the field worker and, in
many respects, he also has been successful. There are, how
ever, numerous aspects of seismic wave propagation that are
only partially understood and, undoubtedly, many others that
remain to be discovered.
2
The problem presented here Is to examine the spherical
seismic pulse resulting from an explosion in a small spheri
cal cavity in an infinite, homogeneous, isotropic, and solid
medium. The disturbance from the explosion does not cease
when the pressure at the scene of the explosion becomes zero
or negligible but continually progresses until all of its
energy has been dissipated. We may, .at least theoretically,
assign a definite length to the wave and properly term the
wave a ntransient.*1 In all probability this is the simplest
case of wave propagation, for we have not assigned any sur
faces of discontinuity or free surfaces td the problem.
Furthermore, the source of energy is spherical and the dis
placement in the medium must be symmetric about this source.
To attempt a theoretical solution of the problem is difficult
enough, but to verify this solution by sound empirical obser
vations is a matter of field or laboratory technique which,
to this writer’s knowledge, has not been developed to the
point that the results are free from serious extraneous fac
tors. The measuring device, whether it be a geophone or some
type of gage, must be placed at a considerable depth below
the surface of the medium without appreciably changing the
physical properties of the medium. This in itself is not a
task easily accomplished. However, before we can fully under
stand the effect of a free surface or of a discontinuous sur
face on a seismic pulse, or before we can definitely verify
the theoretical solutions concerning the wave motion at such
surfaces, it does seem that we must first establish the form
3
of the seismic pulse in the absence of these surfaces.
For the above reasons the writer has treated in theory
only the case of a spherical seismic pulse in an infinite and
homogeneous medium. Since suitable equipment for placing the
gages at depth was not available, the experimental work con
sisted of the measurement of strain at the surface of a rock
outcrop and in the neighborhood of a rather shallow dynamite
blast. Because the measurements were at the surface, the em
pirical results are not a verification of the theoretical con
clusions, but it is believed that the design of the equipment
is such that dynamic strain in a solid medium may be measured
accurately. The remaining problem is to measure this strain
at depth and, when this is accomplished satisfactorily, many
avenues of approach to the problem become apparent. For in
stance, the exact form of the spherical pulse and the amount
and type of energy loss may be determined. Also, the total
energy and the maximum stress and strain at various distances
from the source may be computed.
The following dissertation has been divided into three
general parts. The first of these is a theoretical discussion
of spherical wave motion in which several physical features of
a transient seismic pulse in a homogeneous and solid medium
are treated. The spherical wave equation is developed from
the general wave equation, and the dilatation and displacement
functions in a spherical wave are shown to have a minimum os
cillatory character. Also, it is demonstrated that the form
of the displacement wave must alter in a continuous manner as
the wave progresses and that the energy density at any point
in the wave also must change in a continuous but quite com
plicated way.
The second portion is devoted to the experimental data
and results. After the design and construction of suitable
instruments, strain records were taken on the surface of a
sandstone outcropping. Small charges of dynamite were placed
in shallow bore holes, and the undistorted and unfiltered
strain waves resulting from a dynamite blast were recorded.
Most of the direct spherical wave was obscured by surface
waves, but it was possible to observe how the maximum dis
placement and maximum strain decrease with distance and also
how the first compressional pulse in the direct wave changes
in form as the wave progresses.
The third and last part deals exclusively with the de
sign and arrangement of the Instruments used in the measure
ment of dynamic strain. Briefly, the components of the field
equipment are (1) resistance-wire strain gages or the "pick
ups," (2) a two-channel pre-amplifier, (3) two cathode-ray
oscillographs, (4) a recording camera, (5) a central control
unit, and (6) a 1000-cps tuning fork and drive amplifier.
All of this equipment was mounted in a small panel truck and
the power was supplied by a motor-generator and various dry
cell batteries. The back of the truck was made light-tight
and the records were developed immediately after they were
taken. It was possible to record the strains occurring at two
locations or in two directions at the same location.
5
THEORETICAL CONSIDERATIONS
The following is a theoretical discussion of spherical
wave motion in an infinite and homogeneous solid medium, , The
differential equation of motion is developed from the familiar
classical equation of wave motion and it is shown that the
spherical wave equation is not as general as might be supposed
in that the wave motion must fulfill certain basic require
ments. It is shown further that the shape of the displace
ment wave resulting from an explosion in a solid medium must
change as the wave progresses. By assuming a particular form
for the transient displacement wave, the writer has computed
the corresponding dilatation, strain, and energy-density wave.
The energy-density equation is developed for the case of
spherical wave motion and it is shown how this energy density
must vary with distance from the source.
General Wave Equation
For a medium under stress the relationship between the
stress and strain may be expressed in vectorial form as
f = >v?-y0 l + 2 /J$ (1)
in which $ and $ are the stress and strain dyadics respec-
