Bongo1.html

Worksheet: Bongo.mw

Solution to the Wave Equation in a Disc

In this worksheet we solve the radial wave equation for u(r,θ,t):

DE: = + + r<1

BC: u(1,θ,t)=0

IC: u(r,,0)=f(r,θ) (r,,0)=0

> restart:with(plots):

Cylindrical coordinates

> addcoords(z_cylindrical,[z,r,theta],[r*cos(theta),r*sin(theta),z]);

I don't like typing BesselJZeros! Maple has a nice alias feature:

> alias(Z=BesselJZeros):

Eigenfunctions in the Disc

> p:=3;q:=2; p is the angular mode number; q is the radial mode number

> R:=(p,q)->BesselJ(p,Z(p,q)*r); R(p,q) is the (p,q)-eigenfunction

> T:=(p,q)->cos(Z(p,q)*t); T(p,q) is the (p,q)- time function

> u:=(p,q)->R(p,q)*T(p,q)*cos(p*theta);

> Tmax:=evalf(2*Pi/Z(p,q));Let's graph a solution R[j]*T[j]

> animate3d(u(p,q),r=0..1,theta=0..2*Pi,t=0..Tmax,shading=zhue,style=patchnogrid,coords=z_cylindrical);

Plot_2d

Orthogonal Expansion of an Initial Condition

Now, let's expand the solution for a given initial condition f(r)*cos(theta)

> f:=r->r^3*(1-r)^2;

> plot(f(r),r=0..1,thickness=2);

Plot_2d

Compute the coefficients:

> A:=n->int(r*f(r)*R(1,n),r=0..1)/int(r*R(1,n)^2,r=0..1);

proc (n) options operator, arrow; `/`(`*`(int(`*`(r, `*`(f(r), `*`(R(1, n)))), r = 0 .. 1)), `*`(int(`*`(r, `*`(`^`(R(1, n), 2))), r = 0 .. 1))) end proc

> M1:=3; M is the number of terms in the expansion

> a:=seq(evalf(A(n)),n=1..M1);

Approximate the Initial Condition

> F:=sum(a[n]*R(1,n),n=1..M1);

To see if we're on track lets plot the real initial data with its solution approximation (at t=0)

> plot({f(r),F},r=0..1,thickness=2);

Plot_2d

Solution to the Wave Equation

> u:=cos(theta)*sum(a[n]*T(1,n)*R(1,n),n=1..M1);

> animate3d(u,r=0..1,theta=0..2*Pi,t=0..5*2*Pi/Z(1,1),shading=zhue,style=patchnogrid,coords=z_cylindrical,frames=120);

Plot_2d

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