Computer Algebra Recipes

6.1. WAVE EQUATION MODELS 259
Then S3 is converted to trig form and the result expanded.
> S4:=expand(convert(S3,trig));
S4 := C1 BesselJ(m; ® r) C5 C3 cos(qz)cos(mμ)
¡ C1 BesselJ(m; ® r) C5 C3 sin(qz)sin(mμ)
+ ¢¢¢¢¢¢¢¢¢¢¢¢¢¢¢¢¢¢¢¢¢¢¢¢
Approximating the metallic walls of the cylinder as being rigid, the normal
component of the °uid velocity must vanish at the walls. Since the velocity
is equal to minus the gradient of the potential, this implies that the normal
derivative of the potential must vanish at the walls. In the z direction, one
must have dZ=dz =0atz = 0, so only terms involving cos(qz)areselected.
> S5:=select(has,S4,cos(q*z));
S5 := C1 BesselJ(m; ® r) C5 C3 cos(qz)cos(mμ)+¢¢¢
To satisfy the derivative boundary condition on the top face of the cylinder at
z = h, one must have sin(qh)=0,orq = n¼=h,withn =0; 1; 2;::::
For the angular part to remain single-valued as μ increases by 2 ¼, onemust
have cos(m (μ +2¼)) = cos(μ), which is satis¯ed if m =0; 1; 2; :::: Negative
integer values of m need not be considered because the minus sign can be
absorbed in the arbitrary constants. Similar remarks apply to sin(mμ), which
also appears in S5 . So the allowed Bessel functions are J0; J1; J2; etc.
To satisfy the derivative condition on the cylindrical wall, one must have
(d=dr)Jm(®m;s r)jr=a =0,whereslabels the zeros of the derivative of the mthorder
Bessel function. For later convenience, Vectoria sets ®m;s = ¼pm;s=a,
the values of pm;s still to be determined. The allowed values of q and ® are
subsituted into S5 and the result factored.
> S6:=factor(subs(fq=n*Pi/h,alpha=Pi*p[m,s]/ag,S5));
S6 := (¡cos(mμ) C3 I ¡ cos(mμ) C4 I +sin(mμ) C3 ¡ sin(mμ) C4)
³
n¼z
´ ³
cos BesselJ m;
h
¼pm; s r
´
C1 ( C5 + C6 ) I
a
Replacing the awkward coe±cient combinations with An;m;s and Bn;m;s gives
normal modes of the following form. For each m value, except m =0,thereare
actually two modes corresponding to cos(mμ) and sin(mμ).
> NM:=(A[n,m,s]*cos(m*theta)+B[n,m,s]*sin(m*theta))
*select(has,S6,fcos(n*Pi*z/h),BesselJg);
³
n¼z
´ ³
NM := (An; m; s cos(mμ)+Bn; m; s sin(mμ)) cos BesselJ m;
h
¼pm; s r
´
a
Noting that the boundary condition (d=dr)(Jm(¼pm;s r=a)jr=a = 0 may be
rewritten as (d=dp)(Jm(¼p)jp = pm;s = 0, Vectoria creates a functional operator
f to apply the derivative boundary condition at the cylinder wall.
> f:=m->diff(BesselJ(m,Pi*p),p)=0:
Then using f, she employs a do loop to numerically determine pm;s for m =0
and 1 and s =0tos =4.