Suspension Acoustics: An Introduction to the Physics of Suspensions
Written by a leading expert in the field, this book is an introduction to the physics of suspensions of bubbles, droplets, and solid particles in both gases and fluids.
TABLE OF CONTENTS Suspension Acoustics: An Introduction to the Physics of Suspensions
Appendix D: Some Properties of the Spherical Bessel Functions
Overview
For n equal to an integer, the solution to
may be written in terms of the spherical Bessel functions of the first kind:
the spherical Bessel functions of the second kind:
and the spherical Bessel functions of the third kind:
The pairs j n, y n and 
, 
are linearly independent.
Explicit Expressions
When the order n is small, the spherical Bessel functions may be expressed in terms of elementary functions:
n=0
n=1
Recurrence Relations
For n>2, the spherical Bessel functions, and their derivatives, may be obtained from the following relations, applicable to j n, y n, 
, and 
:
Wronskians
Expressions for Small Arguments
When z ?0, the leading terms in the series expansion of j n (z) and y z (z) are
For the cases n=0 and n=1, these give
Copyright Cambridge University Press 2005 under license agreement with Books24x7
