Matlab code examples. Appendix A. A.1 The simple harmonic oscillator

Transcription

1 Appix A Matlab code examples In this appix, various simple code fragments are provided. All can be viewed as prototypes for physical modeling sound synthesis. The coding style reflects something of a compromise between efficiency, on the one hand, and brevity and intelligibility, on the other. The choice of Matlab as a programming environment definitely reflects the latter sensibility, though the use of Matlab as an actual synthesis engine is not recommed. Some of these examples make use of constructs and features which need not appear in a code fragment inted for synthesis, including various calls to plotting functions, as well as the demonstration of energy conservation in some cases. It should be clear, in all cases, which elements of these examples may be neglected in an actual implementation. For the sake of brevity, these examples are too crude for actual synthesis purposes, but many features, discussed at various points in the texts and exercises, may be added. A.1 The simple harmonic oscillator % matlab script sho.m % finite difference scheme for simple harmonic oscillator f0 = 1000; TF = 1.0; u0 = 0.3; v0 = 0.0; % fundamental frequency (Hz) % duration of simulation (s) % initial displacement % initial velocity %%%%%% global parameters % check that stability condition is satisfied if(sr<=pi*f0) error( Stability condition violated ); N ume ric al Sound Sy nthe sis: Finite Diffe re nc e Sc he m e s and Simulation in M usic al Ac oustic s 2009 John Wiley & Sons, Ltd. ISBN: Stefan Bilbao

2 392 APPENDIX A % derived parameters coef = 2-k^2*(2*pi*f0)^2; NF = floor(tf*sr); % scheme update coefficient % initialize state of scheme u1 = u0+k*v0; u2 = u0; % last value of time series % one before last value of time series % initialize readout out = zeros(nf,1); out(1) = u2; out(2) = u1; u=coef*u1-u2; out(n) = u; u2 = u1; u1 = u; % difference scheme calculation % read value to output vector % update state of difference scheme %%%%%% main loop soundsc(out,sr); plot([0:nf-1]*k, out, k ); xlabel( t ); ylabel( u ); title( SHO: Scheme Output ); axis tight A.2 Hammer collision with mass spring system % matlab script hammermass.m % hammer collision with a mass-spring system xh0 = ; vh0 = 2; TF = 0.05; w0 = 2000; MR = 10; wh = 1000; alpha = 2; % initial conditions of hammer % duration of simulation (s) % angular frequency of mass-spring system % hammer/target mass ratio % stiffness parameter for hammer % hammer stiffness nonlinearity exponent %%%%%% global parameters % derived parameters NF = floor(tf*sr);

3 APPENDIX A 393 % initialization uh2 = xh0; uh1 = xh0+k*vh0; u2 = 0; u1 = 0; out = zeros(nf,1); f = zeros(nf,1); out(1) = u2; out(2) = u1; % hammer % mass-spring system if(uh1>u1) f(n-1) = wh^(1+alpha)*(uh1-u1)^alpha; else f(n-1) = 0; uh = 2*uH1-uH2-k^2*f(n-1); u = 2*u1-u2-w0^2*k^2*u1+MR*k^2*f(n-1); out(n) = u; u2 = u1; u1 = u; uh2 = uh1; uh1 = uh; %%%%%% main loop % plots of displacement of target mass and force subplot(2,1,1) plot([0:nf-1]*k, out, k ); title( Position of Target Mass ); xlabel( t ); axis tight subplot(2,1,2) plot([0:nf-1]*k, f, k ); title( Hammer Force/Mass ); xlabel( t ); axis tight A.3 Bowed mass spring system % matlab script bowmass.m % finite difference scheme for a bowed mass-spring system % soft friction characteristic w/iterative Newton-Raphson method f0 = 200; FB = 500; TF = 0.1; vb = 0.2; sig = 100; tol = 1e-4; % oscillator frequency (Hz) % bow force/mass (m/s^2) % simulation duration (s) % bow velocity (m/s) % friction law free parameter (1/m^2) % tolerance for Newton-Raphson method %%%%%% global parameters % derived parameters NF = floor(tf*sr); A = exp(1/2)*sqrt(2*sig);

4 394 APPENDIX A % initialize time series/iterative method u = zeros(nf,1); f = zeros(nf,1); vr = zeros(nf,1); qlast = 0; restrictions if(k>min(1/(pi*f0),exp(1)/(fb*sqrt(2*sig)))) error( Time step too large ); % Newton-Raphson method to determine relative velocity b = (2*pi*f0)^2*u(n-1)-(2/k^2)*(u(n-1)-u(n-2))+(2/k)*vB; eps = 1; while eps>tol q=qlast-(fb*a*qlast*exp(-sig*qlast^2)+2*qlast/k+b)/... (FB*A*(1-2*sig*qlast^2)*exp(-sig*qlast^2)+2/k); eps = abs(q-qlast); qlast = q; % update position of mass and relative bow velocity u(n) = 2*k*(q+vB)+u(n-2); vr(n-1) = q; %%%%%% main loop % plot mass displacement and relative bow velocity tax = [0:NF-1]*k; subplot(2,1,1); plot(tax, u, k ); title( Displacement of Mass ); xlabel( time (s) ); subplot(2,1,2); plot(tax, vr, k ); title( Relative Bow Velocity ); xlabel( time (s) ); A.4 The 1D wave equation: finite difference scheme % matlab script waveeq1dfd.m % finite difference scheme for the 1D wave equation % fixed boundary conditions % raised cosine initial conditions f0 = 330; % fundamental frequency (Hz) TF = 1; % duration of simulation (s) ctr = 0.7; wid = 0.1; % center location/width of excitation u0 = 1; v0 = 0; % maximum initial displacement/velocity rp = 0.3; % position of readout (0-1) lambda = 1; % Courant number %%%%%% global parameters

5 APPENDIX A 395 % begin derived parameters gamma = 2*f0; NF = floor(sr*tf); % wave equation free parameter % stability condition/scheme parameters h = gamma*k/lambda; N = floor(1/h); h = 1/N; lambda = gamma*k/h; s0 = 2*(1-lambda^2); s1 = lambda^2; % readout interpolation parameters rp_int = 1+floor(N*rp); rp_frac = 1+rp/h-rp_int; % rounded grid index for readout % fractional part of readout location % create raised cosine xax = [0:N] *h; ind = sign(max(-(xax-ctr-wid/2).*(xax-ctr+wid/2),0)); rc = 0.5*ind.*(1+cos(2*pi*(xax-ctr)/wid)); % initialize grid functions and output u2 = u0*rc; u1 = (u0+k*v0)*rc; u = zeros(n+1,1); out = zeros(nf,1); u(2:n) = -u2(2:n)+s0*u1(2:n)+s1*(u1(1:n-1)+u1(3:n+1)); % scheme calculation out(n) = (1-rp_frac)*u(rp_int)+rp_frac*u(rp_int+1); % readout u2 = u1; u1 = u; % update of grid variables %%%%%% main loop plot([0:nf-1]*k, out, k ); xlabel( t ); ylabel( u ); title( 1D Wave Equation: FD Output ); axis tight soundsc(out,sr); A.5 The 1D wave equation: digital waveguide synthesis % matlab script waveeq1ddw.m % digital waveguide method for the 1D wave equation % fixed boundary conditions % raised cosine initial conditions f0 = 441; % fundamental frequency (Hz)

6 396 APPENDIX A TF = 1; % duration of simulation (s) ctr = 0.7; wid = 0.1; % center location/width of excitation u0 = 1; v0 = 0; % maximum initial displacement/velocity rp = 0.3; % position of readout (0-1) %%%%%% global parameters % begin derived parameters NF = floor(sr*tf); N = floor(0.5*sr/f0); rp_int = 1+floor(N*rp); rp_frac = 1+rp*N-rp_int; % length of delay lines % rounded grid index for readout % fractional part of readout location % initialize delay lines and output wleft = zeros(n,1); wright = zeros(n,1); out = zeros(nf,1); % create raised cosine and integral xax = ([1:N] -1/2)/N; ind = sign(max(-(xax-ctr-wid/2).*(xax-ctr+wid/2),0)); rc = 0.5*ind.*(1+cos(2*pi*(xax-ctr)/wid)); rcint = zeros(n,1); for qq=2:n rcint(qq) = rcint(qq-1)+rc(qq)/n; % set initial conditions wleft = 0.5*(u0*rc+v0*rcint/(2*f0)); wright = 0.5*(u0*rc-v0*rcint/(2*f0)); temp1 = wright(n); temp2 = wleft(1); wright(2:n) = wright(1:n-1); wleft(1:n-1) = wleft(2:n); wright(1) = -temp2; wleft(n) = -temp1; % readout out(n) = (1-rp_frac)*(wleft(rp_int)+wright(rp_int))... +rp_frac*(wleft(rp_int+1)+wright(rp_int+1)); %%%%%% main loop plot([0:nf-1]*k, out, k ); xlabel( t ); ylabel( u ); title( 1D Wave Equation: Digital Waveguide Synthesis Output ); axis tight soundsc(out,sr);

7 APPENDIX A 397 A.6 The 1D wave equation: modal synthesis % matlab script waveeq1dmod.m % modal synthesis method for the 1D wave equation % fixed boundary conditions % raised cosine initial conditions f0 = 441; % fundamental frequency (Hz) TF = 1; % duration of simulation (s) ctr = 0.7; wid = 0.1; % center location/width of excitation u0 = 1; v0 = 0; % maximum initial displacement/velocity rp = 0.3; % position of readout (0-1) %%%%%% global parameters % begin derived parameters/temporary storage NF = floor(sr*tf); N = floor(0.5*sr/f0); % number of modes temp = 2*pi*[1:N] *f0/sr; coeff = 2*cos(temp); outexp = sin([1:n]*pi*rp); % initialize grid functions and output U = zeros(n,1); U1 = zeros(n,1); U2 = zeros(n,1); out2 = zeros(nf,1); % create raised cosine and find Fourier coefficients xax = [0:N-1] /N; ind = sign(max(-(xax-ctr-wid/2).*(xax-ctr+wid/2),0)); rc = 0.5*ind.*(1+cos(2*pi*(xax-ctr)/wid)); rcfs = -imag(fft([rc; zeros(n,1)])); rcfs = 2*rcfs(2:N+1)/N; % set initial conditions U2(1:N) = u0*rcfs; U1(1:N) = (u0*cos(temp)+v0*sin(temp)./(2*pi*[1:n] *f0)).*rcfs; U = -U2+coeff.*U1; out(n) = outexp*u; U2 = U1; U1 = U; % scheme calculation % readout % update of modal weights %%%%%% main loop plot([0:nf-1]*k, out, k ); xlabel( t ); ylabel( u ); title( 1D Wave Equation: Modal Synthesis Output ); axis tight

8 398 APPENDIX A soundsc(out,sr); A.7 The ideal bar % matlab script idealbarfd.m % finite difference scheme for the ideal bar equation % clamped/pivoting boundary conditions % raised cosine initial conditions K = 10; % stiffness parameter TF = 1; % duration of simulation (s) ctr = 0.7; wid = 0.1; % center location/width of excitation u0 = 1; v0 = 0; % maximum initial displacement/velocity mu = 0.5; % scheme free parameter rp = 0.85; % position of readout (0-1) bc = [2 2]; % boundary condition type, [left right] with % 1: clamped, 2: pivoting %%%%%% global parameters % begin derived parameters NF = floor(sr*tf); % stability condition/scheme parameters h = sqrt(k*k/mu); N = floor(1/h); h = 1/N; mu = K*k/h^2; s0 = 2*(1-3*mu^2); s1 = 4*mu^2; s2 = -mu^2; % readout interpolation parameters rp_int = 1+floor(N*rp); rp_frac = 1+rp/h-rp_int; % rounded grid index for readout % fractional part of readout location % create raised cosine xax = [0:N] *h; ind = sign(max(-(xax-ctr-wid/2).*(xax-ctr+wid/2),0)); rc = 0.5*ind.*(1+cos(2*pi*(xax-ctr)/wid)); % initialize grid functions and output u2 = u0*rc; u1 = (u0+k*v0)*rc; u = zeros(n+1,1); out = zeros(nf,1); % scheme calculation (interior) u(3:n-1) = -u2(3:n-1)+s0*u1(3:n-1)+s1*(u1(2:n-2)+u1(4:n))...

9 APPENDIX A 399 +s2*(u1(1:n-3)+u1(5:n+1)); % calculations at boundary points if(bc(1)==2) u(2) = -u2(2)+(s0-s2)*u1(2)+s1*u1(3)+s2*u1(4); if(bc(2)==2) u(n) = -u2(n)+(s0-s2)*u1(n)+s1*u1(n-1)+s2*u1(n-2); out(n) = (1-rp_frac)*u(rp_int)+rp_frac*u(rp_int+1); u2 = u1; u1 = u; % readout % update %%%%% main loop plot([0:nf-1]*k, out, k ); xlabel( t ); ylabel( u ); title( Ideal Bar Equation: FD Output ); axis tight soundsc(out,sr); A.8 The stiff string % matlab script ssfd.m % finite difference scheme for the stiff string % clamped boundary conditions % raised cosine initial conditions % stereo output % implicit scheme: matrix update form % two-parameter frequency-depent loss % sample rate(hz) B = 0.001; % inharmonicity parameter (>0) f0 = 100; % fundamental(hz) TF = 2; % duration of simulation(s) ctr = 0.1; wid = 0.05; % center location/width of excitation u0 = 1; v0 = 0; % maximum initial displacement/velocity rp = [ ]; % positions of readout(0-1) loss = [100, 10; 1000, 8]; % loss [freq.(hz), T60(s), freq.(hz), T60(s)] theta = 1.0; % implicit scheme free parameter (>0.5) %%%%%% global parameters % begin derived parameters NF = floor(sr*tf); gamma = 2*f0; K = sqrt(b)*(gamma/pi); % set parameters % stability conditions

10 400 APPENDIX A h = sqrt((gamma^2*k^2+sqrt(gamma^4*k^4+16*k^2*k^2*(2*theta-1)))/ (2*(2*theta-1))); N = floor(1/h); h = 1/N; mu = K*k/h^2; lambda = gamma*k/h; % readout interpolation parameters rp_int = 1+floor(N*rp); rp_frac = 1+rp/h-rp_int; % rounded grid index for readout % fractional part of readout location % set scheme loss parameters zeta1 = (-gamma^2+sqrt(gamma^4+4*k^2*(2*pi*loss(1,1))^2))/(2*k^2); zeta2 = (-gamma^2+sqrt(gamma^4+4*k^2*(2*pi*loss(2,1))^2))/(2*k^2); sig0 = 6*log(10)*(-zeta2/loss(1,2)+zeta1/loss(2,2))/(zeta1-zeta2); sig1 = 6*log(10)*(1/loss(1,2)-1/loss(2,2))/(zeta1-zeta2); % create update matrices M = sparse(toeplitz([theta (1-theta)/2 zeros(1,n-3)])); A = M+sparse(toeplitz([sig1*k/(h^2)+sig0*k/2 -sig1*k/(2*h^2) zeros(1,n-3)])); C = M+sparse(toeplitz([-sig1*k/(h^2)-sig0*k/2 sig1*k/(2*h^2) zeros(1,n-3)])); B = 2*M+sparse(toeplitz([-2*lambda^2-6*mu^2 lambda^2+4*mu^2 -mu^2... zeros(1,n-4)])); % create raised cosine xax = [1:N-1] *h; ind = sign(max(-(xax-ctr-wid/2).*(xax-ctr+wid/2),0)); rc = 0.5*ind.*(1+cos(2*pi*(xax-ctr)/wid)); % set initial conditions u2 = u0*rc; u1 = (u0+k*v0)*rc; u = zeros(n+1,1); out = zeros(nf,2); u = A\(B*u1-C*u2); out(n,:) = (1-rp_frac).*u(rp_int) +rp_frac.*u(rp_int+1) ; % readout u2 = u1; u1 = u; % update %%%%%% main loop subplot(2,1,1); plot([0:nf-1]*k, out(:,1), k ); xlabel( t ); ylabel( u ); title( Stiff String Equation: FD Output (left) ); subplot(2,1,2); plot([0:nf-1]*k, out(:,2), k ); xlabel( t ); ylabel( u ); title( Stiff String Equation: FD Output (right) ); axis tight soundsc(out,sr);

11 APPENDIX A 401 A.9 The Kirchhoff Carrier equation % matlab script kcfd.m % finite difference scheme for the Kirchhoff-Carrier equation % fixed boundary conditions % triangular initial conditions f0 = 200; % fundamental frequency (Hz) alpha = 10; % nonlinear string parameter TF = 0.03; % duration of simulation (s) ctr = 0.5; % center location of excitation (0-1) u0 = 0.05; % maximum initial displacement rp = 0.5; % position of readout (0-1) lambda = 0.7; % Courant number %%%%%% global parameters % begin derived parameters gamma = 2*f0; NF = floor(sr*tf); % wave equation free parameter % stability condition h = gamma*k/lambda; N = floor(1/h); h = 1/N; lambda = gamma*k/h; % readout interpolation parameters rp_int = 1+floor(N*rp); rp_frac = 1+rp/h-rp_int; % rounded grid index for readout % fractional part of readout location % create triangular function xax = [0:N] *h; tri = min(xax/ctr-1,0)+1+min((1-xax)/(1-ctr)-1,0); % initialize grid functions and output u2 = u0*tri; u1 = u2; u = zeros(n+1,1); out = zeros(nf,1); for n=1:nf % calculate nonlinearity g u1x = (u1(2:n+1)-u1(1:n))/h; u1xx = (u1x(2:n)-u1x(1:n-1))/h; g = (1+0.5*alpha^2*h*sum(u1x.*u1x))/... (1+0.25*alpha^2*k^2*gamma^2*h*sum(u1xx.*u1xx)); % scheme update u(2:n) = 2*u1(2:N)-u2(2:N)+g*gamma^2*k^2*u1xx(1:N-1); % calculation out(n) = (1-rp_frac)*u(rp_int)+rp_frac*u(rp_int+1); % readout

12 402 APPENDIX A u2 = u1; u1 = u; % update %%%%% main loop plot([0:nf-1]*k, out, k ); xlabel( t ); ylabel( u ); title( Kirchhoff-Carrier Equation: FD Output ); axis tight A.10 Vocal synthesis % matlab script vocalfd.m % finite difference vocal tract simulation % radiation loss included % simple glottal source waveform % static pitch L = 0.17; % tract length (m) S0 = ; % vocal tract surface area, left (m^2) c = 340; % wave speed (m/s) f0 = 120; % fundamental frequency (Hz) TF = 1; % simulation duration (s) % vocal tract profile, non-dimensional [pos S] pairs % /E/ S = [0 1; ; ; ; ; ; ; ; ; ; ; ;0.74 1; ; ;0.88 2;0.91 2; ;1 3.2]; % /A/ % S = [0 1; ; ; ; ; ; ;... % ;0.47 1; ;0.59 2; ; ;0.94 2;1 2]; %%%%% global parameters % begin derived parameters NF = floor(tf*sr); gamma = c/l; % sample duration % stability condition/scheme parameters h = gamma*k; N = floor(1/h); h = 1/N; lambda = gamma*k/h; S = interp1(s(:,1),s(:,2),[0:h:1]) ; % interpolate vocal tract profile alf = *L*sqrt(1/(S0*S(N+1))); % radiation parameter bet = /gamma; % radiation parameter Sav = [S(1); 0.25*(S(3:N+1)+2*S(2:N)+S(1:N-1)); S(N+1)]; Sr = 1.5*S(N+1)-0.5*S(N); sr = 0.5*lambda^2*((S(2:N)+S(3:N+1))./Sav(2:N)); sl = 0.5*lambda^2*((S(2:N)+S(1:N-1))./Sav(2:N)); s0 = 2*(1-lambda^2);

13 APPENDIX A 403 q1 = alf*gamma^2*k^2*sr/(sav(n+1)*h); q2 = bet*gamma^2*k*sr/(sav(n+1)*h); r1 = 2*lambda^2/(1+q1+q2); r2 = -(1+q1-q2)/(1+q1+q2); g1 = -(k^2*gamma^2/h/s(1))*(3*s(1)-s(2)); % initialize grid functions and output, generate glottal waveform Psi = zeros(n+1,1); Psi1 = zeros(n+1,1); Psi2 = zeros(n+1,1); uin = sin(2*pi*[0:nf-1]*k*f0); uin = 0.5*(uin+abs(uin)); out = zeros(nf,1); %%%%%% begin main loop for n=1:nf; Psi(2:N) = s0*psi1(2:n)+sl.*psi1(1:n-1)+sr.*psi1(3:n+1)-psi2(2:n); Psi(N+1) = r1*psi1(n)+r2*psi2(n+1); Psi(1) = s0*psi1(1)+2*lambda^2*psi1(2)-psi2(1)+g1*uin(n); out(n) = SR*(Psi(N+1)-Psi1(N+1)); Psi2 = Psi1; Psi1 = Psi; %%%%%% main loop % plot vocal tract profile and output spectrum subplot(2,1,1); plot([0:h:1], sqrt(s), k, [0:h:1], -sqrt(s), k ) title( Vocal Tract Profile ); xlabel( x ); ylabel( sqrt(s) ); subplot(2,1,2); plot([0:nf-1]*sr/nf, 10*log10(abs(fft(out))), k ); title( Output Spectrum ); xlabel( f );ylabel( pressure (db) ); soundsc(out, SR); A.11 The 2D wave equation % matlab script waveeq2dloss.m % finite difference scheme for the 2D wave equation with loss % fixed boundary conditions % raised cosine initial conditions % bilinear interpolation SR = 16000; gamma = 200 T60 = 8; epsilon = 1.3; TF = 2; ctr = [ ]; wid = 0.15; u0 = 0; v0 = 1; rp = [ ]; lambda = 1/sqrt(2); % sample rate(hz) % wave speed (1/s) % 60 db decay time (s) % domain aspect ratio % duration of simulation(s) % center location/width of excitation % maximum initial displacement/velocity % position of readout([0-1,0-1]) % Courant number %%%%%% global parameters

14 404 APPENDIX A % begin derived parameters NF = floor(sr*tf); sig0 = 6*log(10)/T60; % loss parameter % stability condition/scheme parameters h = gamma*k/lambda; Nx = floor(sqrt(epsilon)/h); Ny = floor(1/(sqrt(epsilon)*h)); h = sqrt(epsilon)/nx; lambda = gamma*k/h; s0 = (2-4*lambda^2)/(1+sig0*k); s1 = lambda^2/(1+sig0*k); t0 = -(1-sig0*k)/(1+sig0*k); % readout interpolation parameters rp_int = 1+floor([Nx Ny].*rp); rp_frac = 1+rp/h-rp_int; % create 2D raised cosine % find grid spacing % number of x-subdivisions of spatial domain % number of y-subdivisions of spatial domain % reset Courant number [X, Y] = meshgrid([0:nx]*h, [0:Ny]*h); dist = sqrt((x-ctr(1)).^2 +(Y-ctr(2)).^2); ind = sign(max(-dist+wid/2,0)); rc = 0.5*ind.*(1+cos(2*pi*dist /wid)); % set initial conditions u2 = u0*rc; u1 = (u0+k*v0)*rc; u = zeros(nx+1,ny+1); out = zeros(nf,2); u(2:nx,2:ny) = s1*(u1(3:nx+1,2:ny)+u1(1:nx-1,2:ny)+u1(2:nx,3:ny+1)+... u1(2:nx,1:ny-1))+s0*u1(2:nx,2:ny)+t0*u2(2:nx,2:ny); out(n,:) = (1-rp_frac(1))*(1-rp_frac(2))*u(rp_int(1),rp_int(2))+... (1-rp_frac(1))*rp_frac(2)*u(rp_int(1),rp_int(2)+1)+... rp_frac(1)*(1-rp_frac(2))*u(rp_int(1)+1,rp_int(2))+... rp_frac(1)*rp_frac(2)*u(rp_int(1)+1,rp_int(2)+1); u2 = u1; u1 = u; %%%%% main loop plot([0:nf-1]*k, out, k ); xlabel( t ); ylabel( u ); title( 2D Wave Equation with Loss: FD Output ); axis tight soundsc(out,sr);

15 APPENDIX A 405 A.12 Thin plate % matlab script plateloss.m % finite difference scheme for the thin plate equation with loss % simply supported boundary conditions % raised cosine initial conditions % vector/matrix update form % zeroth-order interpolation K = 20; T60 = 8; epsilon = 1.2; TF = 2; ctr = [ ]; wid = 0.3; u0 = 0; v0 = 1; rp = [ ]; mu = 0.25; % sample rate(hz) % plate stiffness parameter (1/s) % 60 db decay time (s) % domain aspect ratio % duration of simulation(s) % center location/width of excitation % maximum initial displacement/velocity % position of readout([0-1,0-1]) % scheme free parameter %%%%%% global parameters % begin derived parameters NF = floor(sr*tf); sig0 = 6*log(10)/T60; % loss parameter % stability condition/scheme parameters h = sqrt(k*k/mu); % find grid spacing Nx = floor(sqrt(epsilon)/h); % number of x-subdivisions of spatial domain Ny = floor(1/(sqrt(epsilon)*h)); % number of y-subdivisions of spatial domain h = sqrt(epsilon)/nx; ss = (Nx-1)*(Ny-1); % total grid size % generate difference matrix/scheme matrices Dxx = sparse(toeplitz([-2/h^2;1/h^2;zeros(nx-3,1)])); Dyy = sparse(toeplitz([-2/h^2;1/h^2;zeros(ny-3,1)])); D = kron(eye(nx-1), Dyy)+kron(Dxx, eye(ny-1)); DD = D*D; B = sparse((2*eye(ss)-k^2*k^2*dd)/(1+sig0*k)); C = ((1-sig0*k)/(1+sig0*k))*sparse(eye(ss)); % readout interpolation parameters rp_index = (Ny-1)*floor(rp(1)*Nx)+floor(rp(2)*Ny); % create 2D raised cosine [X, Y] = meshgrid([1:nx-1]*h, [1:Ny-1]*h); dist = sqrt((x-ctr(1)*sqrt(epsilon)).^2+(y-ctr(2)/sqrt(epsilon)).^2); ind = sign(max(-dist+wid/2,0)); rc = 0.5*ind.*(1+cos(2*pi*dist/wid)); rc = reshape(rc, ss,1);

16 406 APPENDIX A % set initial conditions/initialize output u2 = u0*rc; u1 = (u0+k*v0)*rc; u = zeros(ss,1);out = zeros(nf,1); u = B*u1-C*u2; u2 = u1; u1 = u; out(n) = u(rp_index); %%%%%% main loop plot([0:nf-1]*k, out, k ); xlabel( t ); ylabel( u ); title( Thin Plate Equation with Loss: FD Output ); axis tight soundsc(out,sr);