Interpolation between values given in table of zeros is permitted provided curve is traced between at least three values from table. Zeros have been obtained on digital computer and results were rounded off to the fourth decimal point

Rostafinski, W. A.

English

NASA Technical Reports Server

April 1974	 i	 B74-10012 
•	 NASA TECH BRIEF 
Lewis Research Center 
NASA Tech Briefs announce new technology derived from the U.S. space program. They are issued to encourage 
commercial application. Tech Briefs are available on a subscription basis from the National Technical Information 
Service, Springfield, Virginia 22151. Requests for individual copies or questions relating to the Tech Brief program may 
be directed to the Technology Utilization Office, NASA, Code KT, Washington, D.C. 20546. 
Zeros of Certain Cross Products 
of Bessel Functions of Fractional Order 
' 0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5 
1.5 0 0. 1,027 1.2070 2.0000 2.80140 3.5930 4.3730 5.145 5.9040 6.6537 7.39314 8.1239 
I 6.3355 6.14 1450 6.6580 6.9660 7.3600 7.8300 8.365 8.9535 9.5834 10. 2612 10.9649 
2.0
0 0.3396 1.0115 1.6633 2.2870 2.8817 3.'e523 14.0065 4.5500 5.0669 5.6193 6.1489 
I 3.182 3.3877 3.7099 4.1575 46980 5.2992 5.9320 6.5771 7.2169 7.81439 8.14553 
2 
F3
6.3226 6.40o 6.56148 6.8011 7.1000 7.14810 7.9153 8.14058 8.9466 9.5293 10. 1443 
9.14512 14.50148 9.61114 9.7698 9.9780 10.23140 10.5350 10.8800 11.2665 11.6926 12.1572 
0 0.29145 0.8100 1.3950 1.8810 2.3309 2.7810 3.2t50 3.6 1 e50 4.0719 4. 1;966 4.9156 
2.5 1 2.18140 2.3950 2.7760 3.2717 3.8150 4.3610 4.8850 5.3870 5.8706 6.31430 6.8632 
2 14.2360 4.3350 4.5320 4.8225 5.1991 5.6530 6. 1650 6.7078 7. 2537 7.7857 8. 2590 
3 . 3150 6.3800 6.5090 6.7015 6.9532 7.2650 7.6320 8.0594 8.5379 9.0575 9.5995
The s in the table is the index of the zero. S
Zeros of cross products of the derivatives of Bessel 
functions of fractional order of the form 
J 1 (x) Y1 (Xx) - J, (Xx) Y (x) 
are needed in problems related to mathematical physics 
and engineering acoustics. Higher zeros of such cross 
products, that is, higher values of x for which the cross 
product vanishes, may be calculated by McMahon's 
expansion, but lower zeros, which are of interest in 
engineering applications are not calculable by any known 
method. 
Using Bessel functions of order v = n + ½, n = 0,1 ,2 
which are characterized by closed form solutions, a set 
of zeros was obtained for a range of parameter X. 
Interpolation between the values given in the table is 
permitted provided that a curve is traced between at least 
three values from the table. The zeros have been obtained 
on a digital computer and the results were rounded off to 
the fourth decimal point. The table can also be used to 
give by interpolation zeros v for any given x and X, that is, 
values of p for which the cross product vanishes. 
Notes: 
. 1. Zeros of the cross product of the derivatives of Bessel 
functions of fractional order have application in the 
area of acoustics.
Further information is available in the following 
report: 
NASA TM-X-2698 (N73-15705), Propagation of 
Waves of Acoustic Frequencies in Curved Ducts 
Copies may be obtained at cost from: 
Aerospace Research Applications Center 
Indiana University 
400 East Seventh Street 
Bloomington, Indiana 47401 
Telephone: 812-337.7833 
Reference: B74.10012 
3. Specific technical questions may be directed to: 
Technology Utilization Officer 
Lewis Research Center 
21000 Brookpark Road 
Cleveland, Ohio 44135 
Reference: B74.10012
Source: W. A. Rostafinski
Lewis Research Center
(LEW- 1222 1) 
Category 03 
This document was prepared under the sponsorship of the National 	 Government assumes any liability resulting from the use of the 
Aeronautics and Space Administration. Neither the United States 	 information contained in this document, or warrants that such use 
Government nor any person acting on behalf of the United States	 will be tree from privately owned rights. 
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Zeros of certain cross products of Bessel functions of fractional order

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