www.pudn.com > NumericalComputingwithMatlabCode.zip > waves.m, change:2003-10-25,size:7593b

```function waves
% WAVES  Wave equation in one and two space dimensions.
%   Solutions of the one- or two-dimensional wave equation are expressed
%   as a time-varying weighted sums of the first four eigenfunctions.
%   The one-dimensional domain is an interval of length pi, so the k-th
%   eigenvalue and eigenfunction are lambda(k) = k^2 and u(k) = sin(k*x).
%   The two-dimensional domains include a pi-by-pi square, a unit disc,
%   a three-quarter circular sector and the L-shaped union of three squares.
%   The eigenfunctions of the square are sin(m*x)*sin(n*y).  With polar
%   coordinates, the eigenfunctions of the disc and the sector involve Bessel
%   functions.  The eigenfunctions of the L-shaped domain also involve
%   Bessel functions and are computed by the MATLAB function membranetx.m.
%   The first eigenfunction of the L-shaped domain is the MathWorks logo.

% 2-D eigenvalues and eigenfunctions

m = 11;   % Determines number of grid points
speed = 1;
bvals = [1; 0; 0; 0; 0];
t = 0;

while bvals(5) == 0

% Initialize figure

shg
clf reset
'numbertitle','off','name','Waves','colormap',hot(64))
for k = 1:5
b(k) = uicontrol('style','toggle','value',bvals(k), ...
'units','normal','position',[.15*k .01 .14 .05]);
end
set(b(1),'style','pop','string', ...
{'1-d','square','disc','sector','L','bizcard'})
set(b(2),'string','modes/wave')
set(b(3),'string','slower')
set(b(4),'string','faster')
set(b(5),'string','close')
if bvals(3)==1
speed = speed/sqrt(2);
set(b(3),'value',0);
end
if bvals(4)==1
speed = speed*sqrt(2);
set(b(4),'value',0);
end
bvals = cell2mat(get(b,'value'));
region = bvals(1);
modes = bvals(2)==0;

if region == 1

% 1-D

x = (0:4*m)/(4*m)*pi;
orange = [1 1/3 0];
gray = get(gcf,'color');
if modes

% 1-D modes

for k = 1:4
subplot(2,2,k)
h(k) = plot(x,zeros(size(x)));
axis([0 pi -3/2 3/2])
set(h(k),'color',orange,'linewidth',3)
set(gca,'color',gray','xtick',[],'ytick',[])
end
delta = 0.005*speed;
bvs = bvals;
while all(bvs == bvals)
t = t + delta;
for k = 1:4
u = sin(k*t)*sin(k*x);
set(h(k),'ydata',u)
end
drawnow
bvs = cell2mat(get(b,'value'));
end

else

% 1-D wave

h = plot(x,zeros(size(x)));
axis([0 pi -9/4 9/4])
set(h,'color',orange,'linewidth',3)
set(gca,'color',gray','xtick',[],'ytick',[])
delta = 0.005*speed;
a = 1./(1:4);
bvs = bvals;
while all(bvs == bvals)
t = t + delta;
u = zeros(size(x));
for k = 1:4
u = u + a(k)*sin(k*t)*sin(k*x);
end
set(h,'ydata',u)
drawnow
bvs = cell2mat(get(b,'value'));
end
end

elseif region <= 5

switch region

case 2
% Square

x = (0:2*m)/(2*m)*pi;
y = x';
lambda = zeros(4,1);
V = cell(4,1);
k = 0;
for i = 1:2
for j = 1:2
k = k+1;
lambda(k) = i^2 + j^2;
V{k} = sin(i*y)*sin(j*x);
end
end
ax = [0 pi 0 pi -1.75 1.75];

case 3
% Disc, mu = zeros of J_0(r) and J_1(r)

mu = [bjzeros(0,2) bjzeros(1,2)];
[r,theta] = meshgrid((0:m)/m,(-m:m)/m*pi);
x = r.*cos(theta);
y = r.*sin(theta);
V = cell(4,1);
k = 0;
for j = 0:1
for i = 1:2
k = k+1;
if j == 0
V{k} = besselj(0,mu(k)*r);
else
V{k} = besselj(j,mu(k)*r).*sin(j*theta);
end
V{k} = V{k}/max(max(abs(V{k})));
end
end
lambda = mu.^2;
ax = [-1 1 -1 1 -1.75 1.75];

case 4
% Circular sector , mu = zeros of J_(2/3)(r) and J_(4/3)(r)

mu = [bjzeros(2/3,2) bjzeros(4/3,2)];
[r,theta] = meshgrid((0:m)/m,(3/4)*(0:2*m)/m*pi);
x = r.*cos(theta+pi);
y = r.*sin(theta+pi);
V = cell(4,1);
k = 0;
for j = 1:2
for i = 1:2
k = k+1;
alpha = 2*j/3;
V{k} = besselj(alpha,mu(k)*r).*sin(alpha*theta);
V{k} = V{k}/max(max(abs(V{k})));
end
end
lambda = mu.^2;
ax = [-1 1 -1 1 -1.75 1.75];

case 5
% L-membrane

x = (-m:m)/m;
y = x';
lambda = zeros(4,1);
V = cell(4,1);
for k = 1:4
[L lambda(k)] = membranetx(k,m,9,9);
L(m+2:2*m+1,m+2:2*m+1) = NaN;
V{k} = rot90(L,-1);
end
ax = [-1 1 -1 1 -1.75 1.75];
end

if modes

% 2-D modes

p = [.02 .52 .02 .52];
q = [.52 .52 .02 .02];
for k = 1:4
axes('position',[p(k) q(k) .46 .46]);
h(k) = surf(x,y,zeros(size(V{k})));
axis(ax)
axis off
view(225,30);
caxis([-1.5 1]);
end
delta = .08*speed;
mu = sqrt(lambda(:));
bvs = bvals;
while all(bvs == bvals)
t = t + delta;
for k = 1:4
U = 1.5*sin(mu(k)*t)*V{k};
set(h(k),'zdata',U)
set(h(k),'cdata',U)
end
drawnow
bvs = cell2mat(get(b,'value'));
end

else

% 2-D wave

h = surf(x,y,zeros(size(V{1})));
axis(ax);
axis off
view(225,30);
caxis([-1.5 1]);
delta = .02*speed;
mu = sqrt(lambda(:));
a = 1.25./(1:4);
bvs = bvals;
while all(bvs == bvals)
t = t + delta;
U = zeros(size(V{1}));
for k = 1:4
U = U + a(k)*sin(mu(k)*t)*V{k};
end
set(h,'zdata',U)
set(h,'cdata',U)
drawnow
bvs = cell2mat(get(b,'value'));
end

end

elseif region == 6

figure
bizcard
set(b(1),'value',1)

end

% Retain uicontrol values

bvals = cell2mat(get(b,'value'));

end
close

% -------------------------------

function z = bjzeros(n,k)
% BJZEROS  Zeros of the Bessel function.
% z = bjzeros(n,k) is the first k zeros of besselj(n,x)

% delta must be chosen so that the linear search can take
% steps as large as possible without skipping any zeros.
% delta is approx bjzero(0,2)-bjzero(0,1)
delta = .99*pi;

Jsubn = inline('besselj(n,x)','x','n');
a = n+1;
fa = besselj(n,a);
z = zeros(1,k);
j = 0;
while j < k
b = a + delta;
fb = besselj(n,b);
if sign(fb) ~= sign(fa)
j = j+1;
z(j) = fzerotx(Jsubn,[a b],n);
end
a = b;
fa = fb;
end
```