5.8 Some Applications of Bessel Functions 329
Figure 10 Exercise 10.ofa)? What are the radii of the circles that are the nodal curves ofφ 0 n(r,θ)
for generaln?
10.The nodal curves ofφmn(r,θ)are shown in Fig. 10.
a.By examining the figure, determine what valuesmandnhave.
b.What is the numerical value of the eigenvalueλmn(as a multiple ofa)
for this eigenfunction?
c. What is the formula for the functionφmn(r,θ)whose nodal curves are
shown?
d.What is the frequency of vibration for the drumhead when it is vibrat-
ing in this mode? (“In this mode” means “so that the displacementu
equals a product solution in which this eigenfunction is a factor.”)5.8 Some Applications of Bessel Functions
After the elementary functions, the Bessel functions are among the most useful
in engineering and physics. One reason for their usefulness is they solve a fairly
general differential equation. The general solution of
φ′′+1 − 2 α
x φ′+
[(
λγxγ−^1) 2
−
p^2 γ^2 −α^2
x^2]
φ=0(1)is given by
φ(x)=xα[
AJp(λxγ)+BYp(λxγ)