328 Chapter 5 Higher Dimensions and Other Coordinates
Figure 9 Nodal curves: The curves in these graphs represent solutions of
φmn(r,θ)=0. Adjacent regions bulge up or down, according to the sign. Only
thoseφ’s containing the factor cos(mθ)have been used. (See the cover photo-
graph.)
3.List the five lowest frequencies of vibration of a circular membrane.
4.Sketch the functionJ 0 (λnr)forn= 1 , 2 ,3.
5.What boundary conditions must the functionφof Eq. (18) satisfy?
6.Justify the derivation of Eqs. (19) and (20) from Eqs. (12)–(14) and (18).
7.Show that
∫a0Jm(λmnr)Jm(λmqr)rdr= 0 , n=q,if
Jm(λmsa)= 0 , s= 1 , 2 ,....8.Sketch the nodal curves of the eigenfunctions Eq. (21) corresponding to
λ 31 ,λ 32 ,andλ 33.
9.In the simple case of symmetric vibrations, we found the eigenfunctions
φ 0 n(r,θ)=J 0 (λ 0 nr),whereJ 0 (λ 0 na)=0 forn= 1 , 2 , 3 ....Thenodal
curves ofφ 03 are concentric circles. What are their radii (as multiples