Acoustic wave propagation and attenuation in lunar space environmen

Canup, R. E.

Wolf, R. L.

English

NASA Technical Reports Server

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9 February 1962 
A THEORETICAL STUDY OF THE 
PROPAGATION AND ATTENUATION O F  
ACOUSTIC WAVES IN THE LUNAR SURFACE 
Interim Report 
TM-1325 
R.  L. Wolf and R. E. Canup 
TEXACO EXPER WENT INCORPORATED 
Richmond 2, Virginia 
Submitted to 
Jet Propulsion Laboratory 
under Contract No. 950155 
https://ntrs.nasa.gov/search.jsp?R=19660006189 2020-03-16T21:41:39+00:00Z
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SUMMARY 
INTRODUCTION 
STATEMENT OF THE PROBLEM 
, APPROACH 
DISCUSSION OF RESULTS 
RECOMMENDATIONS 
ACKNOWLEDGMENTS 
-2- 
CONTENTS 
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11 
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SUMMARY 7 
Computations were made of the decrease in displacement of waves with 
frequencies from 25 to 3500 cps in passing through an attenuating medium. 
The usual attenuation equations were used with constants which have been 
determined by seismological experiments, Calculations were made at 2-ft , 
intervals for 2 to 20 f t  of wave travel, 
/ INTRODUCTION 
To help resolve some of the problems concerning the accuracy of the de- 
vices for measuring sonic velocity in the lunar surface Texaco Experi- 
ment Incorporated has made a short mathematical study as a part of con- 
tract No, 950155 of the attenuation of sonic waves in solid media, This 
study can serve as a starting point for a more complete study if such a 
study should be required. 
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STATEMENT OF THE PROBLEM 
The over-all problem is to determine theoretically the output of a trans- 
ducer as a function of (a) the amount and type of explosive used to generate 
a gas wave which will  strike the lunar ,surface and start  a surface wave 
which will activate the transducer and (b) the distance of the transducer 
from the explosive source. 
a complete study are  listed below. 
Some of the factors which must be included in 
A. The 
1. 
2. 
3. 
4. 
5. 
B. The 
I .  
2. 
3. 
4. 
5. 
6. 
C.  The 
1. 
2. 
3. 
4. 
D. The 
1. 
2. 
3. 
effect on the gas wave of  
Type of explosive 
Amount of explosive 
Shape of the explosive charge 
Geometry of the source holder 
The near-zero ambient pressure of the lunar surface 
effect on the coupling of the gas wave to the lunar surface of: 
Velocity of the gas wave 
Pressure across the gas wave 
Duration of pressure pulse from wave 
Surface a rea  contacted by the gas wave 
Type of surface (degree of roughness) 
Type of material in the surface (loose or compacted) 
effect on attenuation of the initial surface wave of: 
Type of surface material 
Horizontal uniformity of the surface 
Vertical uniformity near the surface 
Harmonic composition of the surface wave 
effect on the coupling of the transducer to the surface wave of: 
Type of surface 
Type of material in surface 
Harmonic composition of surface wave 
E. The effect on the response of the transducer of: 
1. Harmonic content of the coupled wave 
2. Type of transducer (geophone or accelerometer) 
3. Mechanical and electrical noise level 
-4 - 
If information on all of these factors were available, it sho& then be pos- 
sible to work from the required transducer output back to the required 
source or from a source to the output of the transducer. One of the difficult 
steps in the middle is the determination of the amplitude and phase relation- 
ship of the various frequencies in the wave that is formed in the surface. 
The factors included in A and B above must be coordinated in such a way as 
to  permit this determination. 
There is obviously no theoretical solution to most of these factors. 
means that empirical information from many sources will have to be used. 
To make a complete study, part of the necessary empirical information will  
probably have to be obtained by experimentation. 
This 
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APPROACH 
From the previous statement of the complete problem it is obvious that 
only a small start could be made in the three weeks alloted to the present 
study. 
yield the most useful information in the time alloted. 
To determine the velocity of the surface wave, one must measure the time 
it takes the wave to travel a given distance. To make an accurate determi- 
nation when, as in this case, the travel distance is small, a very accurate 
measurement of the elapsed time is required. 
A s  a basis for the present study it was assumed that the elapsed time would 
be determined by noting the time at which the wave arrived at two trans- 
ducers a known distance apart. 
ment is a function of the rate of rise of the output of the transducer as  the 
front of the wave passes under it. 
quencies present in the wave, their phase relation, and their amplitude. 
It is known that most formations rapidly attenuate the high-frequency com- 
ponents of the wave. Also it is known that high frequencies are  required for 
fast r ise  time; Le.,  a 25-cps wave takes 1 msec to rise to only 10% of its 
maximum amplitude, while the equivalent time for  a 2500-cps wave is 0.01 
msec. It was decided, therefore, that the first step in the study should be 
to  determine the attenuation of selected frequencies at  various distances from 
the source of the wave. 
In accordance with the above approach the attenuation was calculated by the 
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Therefore, it was necessary to select those factors which would 
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With this method the accuracy of the measure- 
This rate of rise is a function of the f re-  
c 
(1) C.  A. Heiland, Geophysical Exploration, p. 480, New York, Prentice- 
Hall, Inc, 1946. c 
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where 
I, = displacement at distance x 
= initial displacement 
k = attenuation constant (0.0049 at 25 cps and varied as 
the square and the 3/2 power of the frequency, f)  
x = distance from initial pulse 
Ir was calculated at 2-ft intervals from 2 to 20 f t  for frequencies of 25 
to 3600 cps with an attenuation constant, k, varying with both f2  and f3’2, 
Excerpts of the results of these calculations are  given in Tables I and 11. 
Calculation of an actual wave form at a known distance from the source re- 
quires knowledge of the frequencies present in the initial wave, the phase 
relationship of these frequencies, and the displacement for each frequency, 
Also it is necessary to know the velocity of the wave in the media and the 
exact distance. These latter items are  needed to calculate the phase re-  
lationship of the various frequencies at the desired distance from the source. 
Because the information required is not readily, if at all, available, it was 
not possible to continue the calculations to show the actual wave form that 
would arrive at the transducers. 
-7- 
TAB= I 
DLSPLACEMENT, 11, AT SELECTED DISTANCES, X, WHERE 
IS TAKEN AS 1.0 AND k VARIES WITH f2 
Frequency, 
cps 
25 
50 
75 
100 
1 50 
200 
2 50 
300 
400 
500 
600 
700 
800 
900 
1000 
Dis t a m e  
2 ft  
, 99024791 
,96155854 
.915578 10 
,85487560 
,70271813 
, 53408559 
.37531127 
, 24385059 
, 08136593 
,01984111 
-00353586 . 00046050 
,00004383 
, 00000305 . 000000 15 
- 4 f t  
.98059092 
,92459483 
.838 28325 
, 7308 1229 
4938 1278 
.28524742 
. I4085855 
.05946311 . 00662041 
.00039367 
,0000 1250 
, 00000021 
-
- 
- 
- 
8 ft 
.96155854 
, 85487560 
, 70271880 
.53408 66 1 
,24385106 
.08136609 
. 0 1984 1 13 
.00353586 
, 00004383 . 000000 15 
-
- 
- 
- 
20 ft -14 ft 
.93370037 90664929 
, 76002891 a 67570527 
, 53934637 ,41395577 
.33367253 , 2084631 1 
, 08461905 . 02936376 
,01239584 .00188846 
.00104892 .00005545 
,00005127 ,00000074 
00000002 - 
- - 
-8 - 
DISPLACEMENT, Ir, AT SELECTED DISTANCES, x, WHERE 
In IS TAKEN AS 1.0 AND k VARIES WITH f3’2 
Frequency, 
cps 
25 
50 
75 
100 
150 
200 
2 50 
300 
400 
500 
60 0 
7 00 
800 
900 
1000 
1 I00 
1200 
1300 
1400 
1500 
1600 
1700 
1800 
1900 
2000 
Distance 
2 ft 
9901488 
97238 69 
9498588 
9238553 
-8645906 
-7993066 
7312029 
6626333 
5306793 
.4125163 
3122370 
2306628 
.1666126 
1 1  78443 
0817153 
0556078 - 0371694 
0244223 
0157844 
0100409 
0062901 
.0038823 
0023619 
,0014170 
0008386 
- 4 ft -
9803948 
945 5364 
9022320 
853 508 9 
7475170 
. 6388912 
.5346578 
-28 16205 
-1701698 
097491 9 
0532053 
.0277597 
.O 1388 73 
-0066774 
0030922 
-0013816 
0005964 
Q002491. 
000 1008 
0000396 
0000151 
0000056 
0000020 
0000007 
-4390828 
8 ft 
.9611739 
-8940394 
8140228 
7284774 
-5587817 
408 1820 
-2858589 
1927937 
-0793101 
0289578 
.0095047 
0028308 
0007706 
.0001929 
.0000446 
0000096 
00000 19 
0000004 
0000001 
-
- 
- 
- 
- 
- 
20 ft -14 ft -
-9330468 - 9057429 
-8220045 -7557737 
.6976122 5978489 
574418 1 -4529395 
-361 1385 2334025 
2084463 -1064473 
1117546 ,0436897 
-0560935 -0163205 
0118529 (I017714 
0020328 0001427 
0002893 0000088 
.0000347 -0000004 
0000036 
0000003 - 
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TABLE II (cont'd) 
Frequency , 
cps 
2100 
2200 
2300 
2400 
2500 
2600 
2700 
2800 
2900 
3000 
3100 
3200 
3300 
3400 
3500 
Distance 
14 ft 20 ft -8 ft -4 ft -2 ft -
0004897 0 0000002 - - - 
-00028 23 0 0000001 - - - 
0001 607 - - - - 
0000903 - - - - 
0 0000502 - - - - 
0000275 
0000 149 
00000080 
0000042 
0 0000022 
0 0000012 
0000006 
0000003 
.0000002 
0 0000001 
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DISCUSSION OF RESULTS 
A n  inspection of the data in Tables I and II shows that especially in the 
high-attenuation media the high-frequency components of a wave are  
very rapidly attenuated to less than of their original displacement. 
This means that unless the displacement of these high frequencies in 
the initial wave is of the order of lo9 times that of the low frequencies, 
the high frequencies will have little or no effect on the shape of the wave 
at the measuring station. It seems very unlikely that this ratio of dis- 
placements will be generated in a high-attenuation medium. Therefore, 
the rise time of the wave at the transducer will be slow, since only low 
frequencies will arrive a t  the measuring station. Also, since accelerom- 
eters a re  insensitive at low frequencies, it means that they are  not suited 
for measurements in the high-attenuation media and that a geophone should 
be used. In the low-attenuation media some higher frequencies are  present 
especially at the shorter wave travels. 
to obtain useful data from an accelerometer. 
In these cases it may be possible 
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RECOMMENDATIONS 
Since a complete study of all the factors involved in developing methods 
for determining transducer output as a function of the explosive source, 
or vise versa, is a major undertaking, it is recommended that a meet- 
ing of representatives from JPL, Texaco, and TEI be held in the near 
future. At this meeting J P L  should explain the information they expect 
the study to provide and what they think should be included in the study. 
Texaco and TEI should present their opinion on whether a study could 
be made which would provide the information J P L  needs and, i f  so, an 
estimate of the time and costs of the study. 
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With this information it will 
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then be possible to make a logical decision on how to proceed. 
ACKNOWLEDGMENTS 
The authors wish to express their appreciation to Dr ,  A. L. Parrack, 
Supervisor, Seismic Research, a n d  K, D. Savage, Senior Electrical 
Engineer, Logging and Computational Research, of Texaco Inc.'s 
Bellaire Research Laboratories for their guidance and advice in the 
planning of this study and in the interpretation of the results. 
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A theoretical study of the propagation and attenuation of acoustic waves in the lunar surface Interim report

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