├── LICENSE ├── README.md ├── array_simulation.m ├── libraries ├── Gauss_2D.m ├── NPGauss_2D.m ├── T_fluid_solid.m ├── delay_laws2D.m ├── delay_laws2D_int.m ├── delay_laws3D.m ├── delay_laws3Dint.m ├── discrete_windows.m ├── elements.m ├── ferrari2.m ├── fresnel_2D.m ├── fresnel_int.m ├── gauss_c10.m ├── gauss_c15.m ├── init_xi.m ├── init_xi3D.m ├── interface2.m ├── ls_2Dint.m ├── ls_2Dv.m ├── on_axis_foc2D.m ├── ps_3Dint.m ├── ps_3Dv.m ├── pts_2Dintf.m ├── pts_3Dint.m └── rs_2Dv.m └── output └── beam_steering.mp4 /LICENSE: -------------------------------------------------------------------------------- 1 | GNU GENERAL PUBLIC LICENSE 2 | Version 3, 29 June 2007 3 | 4 | Copyright (C) 2007 Free Software Foundation, Inc. 5 | Everyone is permitted to copy and distribute verbatim copies 6 | of this license document, but changing it is not allowed. 7 | 8 | Preamble 9 | 10 | The GNU General Public License is a free, copyleft license for 11 | software and other kinds of works. 12 | 13 | The licenses for most software and other practical works are designed 14 | to take away your freedom to share and change the works. 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See the 645 | GNU General Public License for more details. 646 | 647 | You should have received a copy of the GNU General Public License 648 | along with this program. If not, see . 649 | 650 | Also add information on how to contact you by electronic and paper mail. 651 | 652 | If the program does terminal interaction, make it output a short 653 | notice like this when it starts in an interactive mode: 654 | 655 | {project} Copyright (C) {year} {fullname} 656 | This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'. 657 | This is free software, and you are welcome to redistribute it 658 | under certain conditions; type `show c' for details. 659 | 660 | The hypothetical commands `show w' and `show c' should show the appropriate 661 | parts of the General Public License. Of course, your program's commands 662 | might be different; for a GUI interface, you would use an "about box". 663 | 664 | You should also get your employer (if you work as a programmer) or school, 665 | if any, to sign a "copyright disclaimer" for the program, if necessary. 666 | For more information on this, and how to apply and follow the GNU GPL, see 667 | . 668 | 669 | The GNU General Public License does not permit incorporating your program 670 | into proprietary programs. If your program is a subroutine library, you 671 | may consider it more useful to permit linking proprietary applications with 672 | the library. If this is what you want to do, use the GNU Lesser General 673 | Public License instead of this License. But first, please read 674 | . 675 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # matlab-simulation 2 | Matlab simulation scripts for my ultrasonic phased array. 3 | The libraries used in this simulation are created by Lester W Schmerr Jr, a reference how to used them as well as a explanation of the physics this model is based on can be found in the books "Fundamentals of Ultrasonic Nondestructive Evaluation" and "Fundamentals of Ultrasonic Phased Arrays", both written by Lester W Schmerr Jr. 4 | 5 | # Simulation output 6 | An example can be found in "output", a video showing beam steering of a phased array as used in this project. In the future this simulation might be ported to a more efficient language like numpy or C/C++ to calculate real time beam patterns. 7 | -------------------------------------------------------------------------------- /array_simulation.m: -------------------------------------------------------------------------------- 1 | % This script solves for the normalized pressure wave field of a 2-D 2 | % array of rectangular elements radiating waves in a fluid using the 3 | % MATLAB function ps_3Dv. Both time delay and apodization laws can 4 | % be specified for the array to steer it and focus it. 5 | 6 | steps = 40; %steps for dynamic beam steering 7 | 8 | % For video recording: 9 | %writerObj = VideoWriter('out.avi'); 10 | %writerObj.FrameRate = 30; % How many frames per second. 11 | %open(writerObj); 12 | 13 | for n=0:0.1:steps 14 | % ------------- give input parameters ------------------------- 15 | lx = 5; % element length in x-direction (mm) 16 | ly = 5; % element length in y-direction (mm) 17 | gx=5; % gap length in x-direction (mm) 18 | gy = 5; % gap length in y-direction (mm) 19 | f= 0.04; % frequency (MHz) 20 | c = 340; % wave speed (m/sec) 21 | L1 =19; % number of elements in x-direction 22 | L2 =19; % number of elements in y-direction 23 | theta =n; % steering angle in theta direction (deg) 24 | phi =0; %steering angle in phi direction (deg) 25 | Fl = 200; % focal distance (mm) 26 | %weighting choices are 'rect','cos', 'Han', 'Ham', 'Blk', 'tri' 27 | ampx_type ='rect'; % weighting coeffcients in x-direction 28 | ampy_type ='rect'; % weighting coefficients in y-direction 29 | 30 | % field points (x,y,z)to evaluate 31 | xs= linspace(-150,150, 200); 32 | zs= linspace(1, 300, 200); 33 | y=0; 34 | [x,z]=meshgrid(xs,zs); 35 | 36 | % ---------------- end input parameters ---------------------- 37 | 38 | % calculate array pitches 39 | sx = lx+gx; 40 | sy = ly+gy; 41 | 42 | % compute centroid locations for the elements 43 | Nx = 1:L1; 44 | Ny = 1:L2; 45 | ex =(2*Nx -1-L1)*(sx/2); 46 | ey =(2*Ny -1 -L2)*(sy/2); 47 | 48 | % generate time delays, put in exponential 49 | % and calculate amplitude weights 50 | td =delay_laws3D(L1,L2,sx,sy,theta,phi,Fl,c); 51 | delay = exp(1i.*2.*pi.*f.*td); 52 | 53 | Cx = discrete_windows(L1,ampx_type); 54 | Cy = discrete_windows(L2,ampy_type); 55 | 56 | 57 | % calculate normalized pressure 58 | p=0; 59 | for nn=1:L1 60 | for ll=1:L2 61 | p = p + Cx(nn)*Cy(ll)*delay(nn,ll)... 62 | *ps_3Dv(lx,ly,f,c,ex(nn),ey(ll),x,y,z); 63 | end 64 | end 65 | 66 | % ---------------- outputs -------------------------- 67 | %plot results based on specification of (x,y,z) points 68 | subplot(2,1,1); imagesc(xs,zs,abs(p)); title("2D section view"); ylabel("Z in mm"); 69 | 70 | %heatmap of individual phase delay 71 | phase = angle(delay); 72 | subplot(2,1,2); pcolor(phase); title("Phase of individual transceivers"); 73 | 74 | colormap hot 75 | colorbar 76 | 77 | %Video recording settings: 78 | %set(gca, 'CameraViewAngle',cva); 79 | %set(gca, 'CameraUpVector',cuv); 80 | %set(gca, 'CameraTarget',ct); 81 | %set(gca, 'CameraPosition',cp); 82 | drawnow; 83 | 84 | %frame = getframe(gcf); % 'gcf' can handle if you zoom in to take a movie. 85 | %writeVideo(writerObj, frame); 86 | end 87 | %close(writerObj); % Saves the movie. 88 | -------------------------------------------------------------------------------- /libraries/Gauss_2D.m: -------------------------------------------------------------------------------- 1 | function p=Gauss_2D(b, f, c, x, z) 2 | % p = Gauss_2D(b,f,c,x,z) calculates the normalized pressure at a 3 | % point (x, z) (in mm)in a fluid whose wave speed is c (in m/sec) 4 | % for a 1-D element of length 2b (in mm) radiating at a frequency, f, 5 | % (in MHz). The function uses a paraxial multi-Gaussian beam model and 6 | % 15 Gaussian coefficients developed by Wen and Breazeale that are 7 | % contained in the MATLAB function gauss_c15. 8 | 9 | % retrieve Wen and Breazeale coefficients 10 | [A, B] = gauss_c15; 11 | 12 | % calculate the wave number 13 | kb = 2000*pi*f*b/c; 14 | 15 | %normalize the (x,z) coordinates 16 | xb = x/b; 17 | zb = z/b; 18 | 19 | %initialize the pressure to zero and then superimpose 15 20 | %Gaussian beams to calculate the pressure wave field 21 | p=0; 22 | for nn = 1:15 23 | qb=zb-i*1000*pi*f*b./(B(nn)*c); 24 | qb0 = -i*1000*pi*f*b./(B(nn)*c); 25 | p=p+sqrt(qb0./qb).*A(nn).*exp(i*kb*xb.^2./(2*qb)); 26 | end -------------------------------------------------------------------------------- /libraries/NPGauss_2D.m: -------------------------------------------------------------------------------- 1 | function p = NPGauss_2D(b,f,c,e,x,z) 2 | % p = NPGauss_2D(b,f,c,e,x,z) calculates the normalized pressure 3 | % of an element of length 2b (in mm), at a frequency, f, (in MHz),in a 4 | % fluid whose wave speed is c (in m/sec). The offset of the center of 5 | %the element in the x-direction is e (in mm) and the pressure is 6 | % calculated at a point (x,z) (in mm). The function uses a non-paraxial 7 | % expansion of a cylindrical wave and 10 Gaussians to model piston 8 | % behavior of the element. 9 | 10 | % get the Gaussian coefficients of Wen and Breazeale 11 | [A, B] = gauss_c10; 12 | 13 | %define non-dimensional quantities 14 | xb=x/b; 15 | zb=z/b; 16 | eb=e/b; 17 | Rb=sqrt((xb-eb).^2 +zb.^2); 18 | kb= 2000*pi*f*b/c; 19 | Db = kb/2; 20 | cosp=zb./Rb; 21 | 22 | 23 | %calculate normalized pressure field from 10 Gaussians 24 | p =0; 25 | for nn= 1:10 26 | arg =(cosp.^2 +1i*B(nn).*Rb./Db); 27 | Dn = sqrt(arg); 28 | amp = A(nn).*exp(1i.*kb.*Rb)./Dn; 29 | p = p + amp.*exp(-1i.*kb.*(xb.^2)./(2.*Rb.*arg)); 30 | end -------------------------------------------------------------------------------- /libraries/T_fluid_solid.m: -------------------------------------------------------------------------------- 1 | function [tpp,tps]= T_fluid_solid(d1,cp1,d2,cp2,cs2, theta1) % T_fluid_solid(d1,cp1,d2,cp2,cs2, theta1) computes the P-P (tpp) % and P-S (tps) transmission coefficients based on velocity ratios % for a plane fluid-solid interface. (d1,cp1) are the density and wave %speed of the fluid. (d2,cp2,cs2) are the density, compressional wave %speed and shear wave speed of the solid, and theta1 is the incident angle % (in degrees) % put incident angle in radians iang = (theta1.*pi)./180; %calculate sin(theta) for refracted p- and s-waves sinp = (cp2/cp1)*sin(iang); sins =(cs2/cp1)*sin(iang); %calculate cos(theta) for refracted p- and s-waves % for angles beyond critical, the value of the cosine is % computed for postive frequencies only cosp = 1i*sqrt(sinp.^2 - 1).*(sinp >= 1) + sqrt(1 - sinp.^2).*(sinp < 1); coss = 1i*sqrt(sins.^2 - 1).*(sins >= 1) + sqrt(1 - sins.^2).*(sins < 1) ; %calculate transmission coefficients denom = cosp + (d2/d1)*(cp2/cp1)*sqrt(1-sin(iang).^2).*(4.*((cs2/cp2)^2).*(sins.*coss.*sinp.*cosp) ... + 1 - 4.*(sins.^2).*(coss.^2)); tpp = (2*sqrt(1 - sin(iang).^2).*(1 - 2*(sins.^2)))./denom; tps = -(4*cosp.*sins.*sqrt(1 - sin(iang).^2))./denom; -------------------------------------------------------------------------------- /libraries/delay_laws2D.m: -------------------------------------------------------------------------------- 1 | function td=delay_laws2D(M, s, Phi, F, c) 2 | % td = delay_laws2D(M,s,Phi,F,c) generates the time delay 3 | % td (in microsec) for an array with M elements, pitch s 4 | % (in mm), where we want to steer the beam at the angle Phi 5 | % (in degrees) and focus it at the distance F (in mm) 6 | % in a single medium of wave speed c (in m/sec). For steering 7 | % at an angle Phi only the focal length, F, must be set equal 8 | % to inf. 9 | 10 | Mb=(M-1)/2; 11 | m=1:1:M ; 12 | em =s*((m-1)-Mb); % location of centroids of elements 13 | 14 | switch (F) 15 | % steering only case 16 | case inf 17 | if Phi > 0 18 | td=1000*s*sind(Phi)*(m-1)/c; 19 | else 20 | td=1000*s*sind(abs(Phi))*(M-m)/c; 21 | end 22 | 23 | %steering and focusing case 24 | otherwise, 25 | r1=sqrt(F^2 +(Mb*s)^2 + 2*F*Mb*s*sind(Phi)); 26 | rm = sqrt(F^2+em.^2 - 2*F*em*sind(Phi)); 27 | rM=sqrt(F^2 +(Mb*s)^2 + 2*F*Mb*s*sind(abs(Phi))); 28 | if Phi > 0 29 | td=1000*(r1-rm)/c; 30 | else 31 | td=1000*(rM-rm)/c; 32 | end 33 | end 34 | 35 | -------------------------------------------------------------------------------- /libraries/delay_laws2D_int.m: -------------------------------------------------------------------------------- 1 | function td=delay_laws2D_int( M, s, angt,ang20,DT0,DF, c1, c2, plt) 2 | % td = delay_laws2D_int(M,s,angt, an20, DT0, DF,c1,c2, plt) calculates 3 | % the delay laws for steering and focusing an array of 1-D elements 4 | % through a planar interface between two media in two dimensions. The 5 | % number of elements is M, the pitch is s (in mm), the angle that array 6 | % makes with the interface is ang(in degrees). The height of the 7 | % center of the array above the interface is DT0 (in mm). Steering and 8 | % focusing to a point in the second medium are specified by giving the 9 | % refracted angle, ang20, (in degrees) and the depth in the second 10 | % medium, DF, (in mm). The wave speeds of the first and second media 11 | % are (c1, c2), respectively (in m/sec). The plt argument is a string 12 | % ('y' or 'n')that specifies if a plot of the rays from the centroids 13 | % of the elements to the point in the second medium is wanted ('y') or 14 | % not ('n') 15 | 16 | 17 | 18 | cr = c1/c2; % wave speed ratio 19 | Mb=(M-1)/2; 20 | %compute location of element centroids, e 21 | m=1:1:M; 22 | e =(m-1-Mb)*s; 23 | % computed parameters: 24 | % ang10, incident angle of central ray, deg 25 | % DX0, distance along interface from center of array to focal point, mm 26 | % DT, heights of elements above interface, mm 27 | % DX, distances along interface from elements to focal point, mm 28 | ang10 = asind((c1/c2).*sind(ang20)); 29 | DX0 = DF.*tand(ang20) + DT0.*tand(ang10); 30 | DT = DT0 + e.*sind(angt); 31 | DX =DX0 - e.*cosd(angt); 32 | switch (DF) 33 | % steering only case, use linear law 34 | case inf 35 | if (ang10 -angt)>0 36 | td = 1000*(m-1)*s*sind(ang10-angt)/c1; 37 | else 38 | td = 1000*(M-m)*s*abs(sind(ang10-angt))/c1; 39 | end 40 | % plotting rays option 41 | if strcmp(plt,'y') 42 | for nn = 1:M 43 | xp2(1, nn) = e(nn)*cosd(angt); 44 | yp2(1, nn) = DT(nn); 45 | xp2(2, nn) = e(nn)*cosd(ang10-angt)/cosd(ang10) +DT0*tand(ang10); 46 | dm=e(nn)*cosd(ang10-angt)/cosd(ang10); 47 | if ang20 >0 48 | dM = e(M)*cosd(ang10-angt)/cosd(ang10); 49 | else 50 | dM =e(1)*cosd(ang10-angt)/cosd(ang10) ; 51 | end 52 | yp2(2, nn) = 0; 53 | xp2(3, nn) = xp2(2,nn) + (dM-dm)*sind(ang20)*sind(ang20); 54 | yp2(3, nn) = -(dM-dm)*sind(ang20)*cosd(ang20); 55 | end 56 | plot(xp2, yp2, 'b') 57 | end 58 | % end plotting rays option 59 | 60 | % steering and focusing case 61 | otherwise, 62 | 63 | %solve for ray intersection locations on interface, xi,(in mm) 64 | %and path lengths in medium 1 and medium 2, r1, r2 (mm) 65 | 66 | xi=zeros(1,M); 67 | r1=zeros(1,M); 68 | r2=zeros(1,M); 69 | for mm = 1:M 70 | xi(mm) = ferrari2(cr,DF,DT(mm),DX(mm)); 71 | r1(mm) =sqrt(xi(mm)^2 +(DT0+e(mm)*sind(angt))^2); 72 | r2(mm) =sqrt( (xi(mm) +e(mm)*cosd(angt)-DX0)^2 +DF^2); 73 | end 74 | 75 | % solve for time advances (in microsec), turn into delays, and 76 | % make the delays ,td, positive 77 | t= 1000*r1/c1 +1000*r2/c2; 78 | td=max(t) -t; 79 | % plotting rays option 80 | if strcmp(plt, 'y') 81 | 82 | for nn = 1:M 83 | xp(1, nn) = e(nn)*cos(angt*pi/180); 84 | yp(1, nn) = DT(nn); 85 | xp(2, nn) = e(nn)*cos(angt*pi/180) +xi(nn); 86 | yp(2, nn) = 0; 87 | xp(3, nn) = DX0; 88 | yp(3, nn) = -DF; 89 | end 90 | plot(xp, yp, 'b') 91 | end 92 | %end plotting rays option 93 | 94 | end 95 | end 96 | 97 | -------------------------------------------------------------------------------- /libraries/delay_laws3D.m: -------------------------------------------------------------------------------- 1 | function td= delay_laws3D(M, N, sx, sy, theta, phi, F, c) 2 | % td = delay_laws3D(M,N,sx,sy,theta, phi, F, c) generates the time delays 3 | % td (in microseconds) for a 2-D array of MxN elements in a single medium 4 | % with elements whose pitches are (sx,sy) in the x- and 5 | % y-directions, respectively (in mm). The steering direction is 6 | % specified by the spherical coordinate angles (theta, phi) (both in 7 | % degrees) and the focusing distance is specied by F (in mm). For steering 8 | % only, F = inf. The wavespeed of medium is c (in m/sec). 9 | 10 | % calculate locations of element centroids in x- and y-directions 11 | m=1:M; 12 | n=1:N; 13 | Mb =(M-1)/2; 14 | Nb=(N-1)/2; 15 | exm=(m-1-Mb)*sx; 16 | eyn=(n-1-Nb)*sy; 17 | 18 | %calculate delays (in microseconds) 19 | switch(F) 20 | % if steering only specified, use explicit steering law 21 | case(inf) 22 | for mm=1:M 23 | for nn=1:N 24 | dt(mm,nn)=1000*(exm(mm)*sind(theta)*cosd(phi) + ... 25 | eyn(nn)*sind(theta)*sind(phi))/c; 26 | end 27 | end 28 | % make delays all positive 29 | td = abs(min(min(dt))) + dt; 30 | % otherwise, if steering and focusing specified, use time delays to 31 | % the specified point 32 | otherwise, 33 | for mm=1:M 34 | for nn=1:N 35 | r(mm,nn) = sqrt((F*sind(theta)*cosd(phi) -exm(mm))^2 ... 36 | +(F*sind(theta)*sind(phi)-eyn(nn))^2 +F^2*(cosd(theta))^2); 37 | end 38 | end 39 | td = max(max(1000*r/c)) -1000*r/c; 40 | end 41 | end 42 | -------------------------------------------------------------------------------- /libraries/delay_laws3Dint.m: -------------------------------------------------------------------------------- 1 | function td = delay_laws3Dint(Mx,My,sx,sy,thetat, phi, ... 2 | theta2,DT0, DF, c1,c2, plt) 3 | % td = delay_laws3Dint(Mx,My,sx,sy,thetat,phi, theta2, DT0,DF,c1,c2,plt) 4 | % calculates the delay laws for steering and focusing a 2-D array 5 | % through a planar interface between two media in three dimensions. 6 | % (Mx, My)are the number of elements in the (x', y') directions, (sx, sy) 7 | % are the pitches (in mm), and thetat is the the angle that array 8 | % makes with the interface(in degrees).Steering and focusing to a point in 9 | % the second medium is specified by giving the angles theta2 and 10 | % phi,(both in degrees). The height of the center of the array above 11 | % the interface is DT0 (in mm).The wave speeds of the first and second 12 | % media are (c1, c2), respectively (in m/sec). The plt argument is a string 13 | % ('y' or 'n')that specifies if a plot of the rays from the centroids 14 | % of the elements to the point in the second medium is wanted ('y') or 15 | % not ('n'). Plotting is not done if steering only (DF = inf) is specified. 16 | 17 | % compute wave speed ratio 18 | cr=c1/c2; 19 | 20 | % compute element centroid locations 21 | Mbx=(Mx-1)/2; 22 | Mby=(My-1)/2; 23 | mx=1:1:Mx; 24 | ex=(mx-1-Mbx)*sx; 25 | my=1:1:My; 26 | ey=(my-1-Mby)*sy; 27 | 28 | %initialize variables to be used 29 | t=zeros(Mx,My); 30 | Db=zeros(Mx,My); 31 | De=zeros(1,Mx); 32 | xi=zeros(Mx,My); 33 | 34 | ang1 =asind(c1*sind(theta2)/c2); % ang1e in first medium (in degrees) 35 | 36 | switch(DF) 37 | % steering only case, use linear steering law 38 | case inf 39 | 40 | ux= sind(ang1)*cosd(phi)*cosd(thetat) -cosd(ang1)*sind(thetat); 41 | uy =sind(ang1)*sind(phi); 42 | for m =1:Mx 43 | for n = 1:My 44 | t(m,n)= 1000*(ux*ex(m)+uy*ey(n))/c1; %time in microsec 45 | end 46 | end 47 | td = abs(min(min(t))) +t; % make sure delay is positive 48 | 49 | % steering and focusing case 50 | otherwise 51 | % determine distances De, Db needed in arguments of ferrari2 52 | % function 53 | DQ=DT0*tand(ang1)+DF*tand(theta2); 54 | x=DQ*cosd(phi); 55 | y=DQ*sind(phi); 56 | for m=1:Mx 57 | for n = 1:My 58 | Db(m,n) = sqrt((x-ex(m)*cosd(thetat))^2 +(y-ey(n))^2); 59 | end 60 | end 61 | De = DT0 +ex*sind(thetat); 62 | % use ferrari2 method to determine distance, xi, where a ray from an 63 | % element to the point (x, y, DF)intesects the interface 64 | % in the plane of incidence 65 | for m=1:Mx 66 | for n = 1:My 67 | xi(m,n) = ferrari2(cr,DF,De(m),Db(m,n)); 68 | end 69 | end 70 | % use ray distances to calculate time advances (in microsec) 71 | for m=1:Mx 72 | for n=1:My 73 | t(m,n) = 1000*sqrt(xi(m,n)^2 +De(m)^2)/c1 +... 74 | 1000*sqrt(DF^2+(Db(m,n) -xi(m,n))^2)/c2; 75 | end 76 | end 77 | % turn time advances into delays and make all delays positive 78 | td =max(max(t)) -t; 79 | 80 | % plotting rays option 81 | if strcmp(plt, 'y') 82 | 83 | for m=1:Mx 84 | for n = 1:My 85 | xp(1,1) = ex(m)*cosd(thetat); 86 | zp(1,1)=DT0 +ex(m)*sind(thetat); 87 | yp(1,1) = ey(n); 88 | xp(2,1) = ex(m)*cosd(thetat) + xi(m,n)*(x-ex(m)*cosd(thetat))/Db(m,n); 89 | yp(2,1) = ey(n) + xi(m,n)*(y-ey(n))/Db(m,n); 90 | zp(2,1) =0; 91 | xp(3,1) = x; 92 | yp(3, 1) = y; 93 | zp(3,1) =-DF; 94 | plot3(xp,yp,zp) 95 | hold on 96 | end 97 | end 98 | hold off 99 | end 100 | %end plotting rays option 101 | end 102 | -------------------------------------------------------------------------------- /libraries/discrete_windows.m: -------------------------------------------------------------------------------- 1 | function amp=discrete_windows(M, type) 2 | % amp = discrete_windows(M, type) returns the discrete apodization 3 | % amplitudes for M elements of type 'cos' (cosine), 'Han' (Hanning) 4 | % 'Ham' (Hamming), 'Blk' (Blackman), 'tri' (triangle), 5 | % and 'rect' (a window with all ones, i.e. no apodization) 6 | 7 | m=1:M; 8 | switch type 9 | case 'cos' 10 | amp = sin(pi*(m-1)/(M-1)); 11 | case 'Han' 12 | amp =(sin(pi*(m-1)/(M-1))).^2; 13 | case 'Ham' 14 | amp= 0.54 -0.46*cos(2*pi*(m-1)/(M-1)); 15 | case 'Blk' 16 | amp=0.42 -0.5*cos(2*pi*(m-1)/(M-1)) + ... 17 | 0.08*cos(4*pi*(m-1)/(M-1)); 18 | case 'tri' 19 | amp =1 - abs(2*(m-1)/(M-1) -1); 20 | case 'rect' 21 | amp = ones(1,M); 22 | otherwise 23 | disp(' Wrong type. Choices are ''cos'', ''Han'', ''Ham'', ''Blk'', ''tri'', ''rect'' ') 24 | 25 | end 26 | -------------------------------------------------------------------------------- /libraries/elements.m: -------------------------------------------------------------------------------- 1 | function [ A, d, g, xc]=elements(f, c, dl, gd, N) 2 | % [A,d,g,xc]=elements(f,c,dl,gd,N) calculates the 3 | % total length of an array,A (in mm),the element size,d=2b, 4 | % (in mm), the gap size, g, (in mm) and the location of the 5 | % centroids of the array elements, xc, (in mm) for an array 6 | % with N elements. The imputs are the frequency, f, (in MHz) 7 | % the wave speed, c, (in m/sec), the element length divided 8 | % by the wavelength, dl, the gap size divided by the element 9 | % length,gd, and the number of elements, N. 10 | 11 | % dl is the element diameter, d, divided by the 12 | % wavelength,l, i.e. dl =d/l. 13 | d=dl.*c./(1000*f); 14 | %gd is the gap size, g, between elements as a fraction of the 15 | %element size, i.e. gd =g/d 16 | g=gd.*d; 17 | % A is the total aperture size of the array 18 | A = N*d + (N-1)*g; 19 | % x= xc is the location of the centroid of each element 20 | % where x = 0 is at the center of the array 21 | for nn = 1:N 22 | xc(nn) = (g+d)*((2*nn -1)/2 - N/2); 23 | end 24 | -------------------------------------------------------------------------------- /libraries/ferrari2.m: -------------------------------------------------------------------------------- 1 | function xi= ferrari2(cr,DF,DT,DX) 2 | % xi = ferrari2(cr, DF, DT, DX) solves for the intersection point, xi, on 3 | % a plane interface along a Snells law ray path from a point located a 4 | % distance DT (in mm)above the interface to a point located a 5 | % distance DF (in mm)below the interface. 6 | % Both DT and DF must be positive. DX (in mm)is the separation 7 | % distance between the points along the plane interface and can be positive 8 | % or negative. cr = c1/c2 is the ratio of the wave speed in medium 1 to 9 | % that of the wavespeed in medium 2. 10 | % The intersection point,xi, is obtained by writing Snells law as a quartic 11 | % equation in xi and solving the quartic with Ferrari's method. Of the 12 | % four roots, two will be complex, one will be the wanted real solution 13 | % in the interval [0,DX] and one will be real but outside that interval. 14 | % reference: http://exampleproblems.com/wiki/index.php/Quartic_equation 15 | % If the root returned by Ferrari's method lies inside the permissable 16 | % interval, [0, DX], and is essentially real(set by a tolerance value 17 | % in line 76), the solution obtained by Ferrari's method is used. 18 | % Otherwise, the MATLAB function fzero is used instead to find the 19 | % intersection point. 20 | 21 | % if two media are identical, use explicit solution for the interface point 22 | % along a straight ray 23 | if abs(cr-1) < 10^(-6) 24 | xi = DX*DT/(DF+DT); 25 | %otherwise, use Ferrari's method 26 | else 27 | cri=1/cr; % cri = c2/c1 28 | %define coefficients of quartic Ax^4 +Bx^3 +Cx^2 + Dx + E =0 29 | A = 1-cri^2; 30 | B = (2*(cri)^2*DX -2*DX)/DT; 31 | C = (DX^2 +DT^2 -(cri)^2*(DX^2 +DF^2))/(DT^2); 32 | D = -2*DX*DT^2/(DT^3); 33 | E= DX^2*DT^2/(DT^4); 34 | % begin Ferrari's solution 35 | alpha = -3*B^2/(8*A^2) + C/A; 36 | beta = B^3/(8*A^3) - B*C/(2*A^2) + D/A; 37 | gamma = -3*B^4/(256*A^4) + C*B^2/(16*A^3) - B*D/(4*A^2) + E/A; 38 | % if beta =0 the quartic is a bi-quadratic whose solution is easier 39 | if(beta == 0) 40 | x(1) = -B/(4*A) + sqrt( (-alpha + sqrt(alpha^2-4*gamma))/2); 41 | x(2) = -B/(4*A) + sqrt( (-alpha - sqrt(alpha^2-4*gamma))/2); 42 | x(3) = -B/(4*A) - sqrt( (-alpha + sqrt(alpha^2-4*gamma))/2); 43 | x(4) = -B/(4*A) - sqrt( (-alpha - sqrt(alpha^2-4*gamma))/2); 44 | % otherwise, proceed with Ferrari's method 45 | else 46 | 47 | P= -alpha^2/12 - gamma; 48 | Q= -alpha^3/108 + alpha*gamma/3 - beta^2/8; 49 | % 50 | 51 | Rm= Q/2 - sqrt(Q^2/4 + P^3/27); 52 | % 53 | U=Rm^(1/3); 54 | % 55 | if(U == 0) 56 | y=-5/6*alpha - U; 57 | else 58 | y=-5/6*alpha - U + P/(3*U); 59 | end 60 | % 61 | W=sqrt(alpha + 2*y ); 62 | % 63 | x(1) = -B/(4*A) + 0.5*( + W + sqrt(-(3*alpha + 2*y + 2*beta/W ))); 64 | x(2) = -B/(4*A) + 0.5*( - W + sqrt(-(3*alpha + 2*y - 2*beta/W ))); 65 | x(3) = -B/(4*A) + 0.5*( + W - sqrt(-(3*alpha + 2*y + 2*beta/W ))); 66 | x(4) = -B/(4*A) + 0.5*( - W - sqrt(-(3*alpha + 2*y - 2*beta/W ))); 67 | end 68 | % end of bi-quadratic solution or ferrari method with four roots 69 | 70 | % find root that is real,lies in the interval [0, DX] 71 | flag =0; 72 | for nn=1:4 73 | xr=real(x(nn)); 74 | axi= DT*abs(imag(x(nn))); 75 | xt=xr*DT; 76 | tol = 10^(-6); 77 | % f is a function which should be zero if Snell's law 78 | % is satisfied and can also be used to check the 79 | % accuracy of Ferrari's solution. Currently not used. 80 | % f =(DX-xt)*sqrt(xt^2+DT^2)-cri*xt*sqrt((DX-xt)^2+DF^2); 81 | 82 | if DX >=0 && (xt >=0 && xt<= DX) && axi < tol 83 | xi = xr*DT; 84 | flag =1; 85 | 86 | elseif DX <0 && (xt <=0 && xt >= DX) && axi < tol 87 | 88 | xi = xr*DT; 89 | flag =1; 90 | 91 | end 92 | end 93 | if flag == 0 94 | 95 | % if interface intersection value returned by Ferrari's 96 | % method lies outside the permissable region or the 97 | % tolerance on being real is not met, use fzero instead 98 | 99 | xi=fzero(@interface2,[0,DX], [], cr, DF, DT, DX); 100 | 101 | end 102 | 103 | end 104 | end -------------------------------------------------------------------------------- /libraries/fresnel_2D.m: -------------------------------------------------------------------------------- 1 | function p = fresnel_2D(b, f, c, x, z) 2 | % p = fresnel_2D(b, f, c, x, z) calculates the normalized pressure 3 | % field at a point (x, z), (in mm), of a 1-D element of 4 | % length 2b(in mm), at a frequency, f,(in MHz)radiating 5 | % into a fluid with wave speed, c, (in m/sec). This function uses the 6 | % fresnel_int function to calculate the Fresnel integral numerically. 7 | 8 | % calculate wave number 9 | kb =2000*pi*f*b/c; 10 | 11 | % put (x, z) coordinates in normalized form 12 | xb=x/b; 13 | zb=z/b; 14 | % calculate term in Fresnel integral argument 15 | arg = sqrt(kb./(pi*zb)); 16 | 17 | %calculate normalized pressure 18 | p=sqrt(1/(2*i)).*exp(i*kb*zb).*(fresnel_int(arg.*(xb+1))... 19 | -fresnel_int(arg.*(xb -1))); -------------------------------------------------------------------------------- /libraries/fresnel_int.m: -------------------------------------------------------------------------------- 1 | function y=fresnel_int(x) 2 | % y = fresnel_int(x) computes the Fresnel integral defined as the integral 3 | % from t = 0 to t = x of the function exp(i*pi*t^2/2). Uses the approximate 4 | % expressions given by Abramowitz and Stegun, Handbook of Mathematical 5 | % Functions, Dover Publications, 1965, pp. 301-302. 6 | 7 | 8 | %separate arguments into positive and negative values, change sign 9 | %of the negative values 10 | xn =-x(x<0); 11 | xp=x(x >=0); 12 | 13 | %compute cosine and sine integrals of the negative values, using the 14 | %oddness property of the cosine and sign integrals 15 | 16 | [cn,sn] =cs_int(xn); 17 | cn= -cn; 18 | sn = -sn; 19 | 20 | %compute cosine and sine integrals of the positive values 21 | 22 | [cp, sp]=cs_int(xp); 23 | 24 | %combine cosine and sine integrals for positive and negative 25 | %values and return the complex Fresnel integral 26 | ct =[cn cp]; 27 | st =[sn sp]; 28 | y=ct+i*st; 29 | 30 | %cs_int(xi) calculates approximations of the cosine and sine integrals 31 | %for positive values of xi only(see Abramowitz and Stegun reference above) 32 | function [c, s]=cs_int(xi) 33 | f =(1+0.926.*xi)./(2+1.792.*xi +3.104.*xi.^2); % f function (see ref.) 34 | g=1./(2+4.142.*xi+3.492.*xi.^2+6.67.*xi.^3); % g function (see ref.) 35 | c=0.5 +f.*sin(pi.*xi.^2./2) -g.*cos(pi.*xi.^2./2); % cos integral approx. 36 | s = 0.5 -f.*cos(pi.*xi.^2./2)-g.*sin(pi.*xi.^2./2); % sin integral approx. 37 | -------------------------------------------------------------------------------- /libraries/gauss_c10.m: -------------------------------------------------------------------------------- 1 | function [a, b] = gauss_c10 % [a, b] = gauss_c10 returns the ten Wen and Breazeale % coefficients for a multi-Gaussian beam model a = zeros(10,1); b = zeros(10,1); % enter Wen and Breazeale Coefficients a(1) = 11.428 + 0.95175*i; a(2) = 0.06002 - 0.08013*i; a(3) = -4.2743 - 8.5562*i; a(4) = 1.6576 +2.7015*i; a(5) = -5.0418 + 3.2488*i; a(6) = 1.1227 - 0.68854*i; a(7) = -1.0106 - 0.26955*i; a(8) = -2.5974 + 3.2202*i; a(9) = -0.14840 -0.31193*i; a(10) = -0.20850 - 0.23851*i; b(1) = 4.0697 + 0.22726*i; b(2) = 1.1531 - 20.933*i; b(3) = 4.4608 + 5.1268*i; b(4) = 4.3521 +14.997*i; b(5) = 4.5443 + 10.003*i; b(6) = 3.8478 + 20.078*i; b(7) = 2.5280 -10.310*i; b(8) = 3.3197 - 4.8008*i; b(9) = 1.9002 - 15.820*i; b(10) = 2.6340 + 25.009*i; -------------------------------------------------------------------------------- /libraries/gauss_c15.m: -------------------------------------------------------------------------------- 1 | function [a, b] = gauss_c15 % [a,b] = gauss_c15 returns the 15 "optimized" coefficients % obtained by Wen and Breazeale to simulate the wave field % of a circular planar piston transducer radiating into a fluid. % Reference: % Wen, J.J. and M. A. Breazeale," Computer optimization of the % Gaussian beam description of an ultrasonic field," Computational % Acoustics, Vol.2, D. Lee, A. Cakmak, R. Vichnevetsky, Eds. % Elsevier Science Publishers, Amsterdam, 1990, pp. 181-196. a = zeros(15,1); b = zeros(15,1); a(1) = -2.9716 + 8.6187*i; a(2) = -3.4811 + 0.9687*i; a(3) = -1.3982 - 0.8128*i; a(4) = 0.0773 - 0.3303*i; a(5) = 2.8798 + 1.6109*i; a(6) = 0.1259 - 0.0957*i; a(7) = -0.2641 - 0.6723*i; a(8) = 18.019 + 7.8291*i; a(9) = 0.0518 + 0.0182*i; a(10) = -16.9438 - 9.9384*i; a(11) = 0.3708 + 5.4522*i; a(12) = -6.6929 + 4.0722*i; a(13) = -9.3638 - 4.9998*i; a(14) = 1.5872 - 15.4212*i; a(15) = 19.0024 + 3.6850*i; b(1) = 4.1869 - 5.1560*i; b(2) = 3.8398 - 10.8004*i; b(3) = 3.4355 - 16.3582*i; b(4) = 2.4618 - 27.7134*i; b(5) = 5.4699 + 28.6319*i; b(6) = 1.9833 - 33.2885*i; b(7) = 2.9335 - 22.0151*i; b(8) = 6.3036 + 36.7772*i; b(9) = 1.3046 - 38.4650*i; b(10) = 6.5889 + 37.0680*i; b(11) = 5.5518 + 22.4255*i; b(12) = 5.4013 + 16.7326*i; b(13) = 5.1498 + 11.1249*i; b(14) = 4.9665 + 5.6855*i; b(15) = 4.6296 + 0.3055*i; -------------------------------------------------------------------------------- /libraries/init_xi.m: -------------------------------------------------------------------------------- 1 | function [xi, P, Q] = init_xi(x,z) 2 | % [xi,P,Q] =init_xi(x,z) examines the points(x,z), where x can be a row 3 | % or column vector and z a scalar, or z a row or column vector and x a 4 | % scalar, or both x and z can be equal sized scalars, vectors or matrices. 5 | % The dimensions (P,Q) of xi are chosen accordingly so that calls to 6 | % functions of (x, z, xi) can be made transparently and consistently 7 | % if one is evaluating that function along an axis, a line, or over a 2-D 8 | % array of points. An empty xi matrix of dimensions PxQ is returned, along 9 | % with the dimensions P and Q. 10 | 11 | % get sizes of x and z variables 12 | [nrx, ncx] =size(x); 13 | [nrz, ncz] = size(z); 14 | 15 | % if x and z are equal sized matrices,vectors,or scalars, xi is of the 16 | % same size 17 | if nrx == nrz && ncx ==ncz 18 | xi=zeros(nrx, ncx); 19 | P=nrx; 20 | Q=ncx; 21 | % if x is a column vector and z a scalar, xi is the same size column vector 22 | elseif nrx > 1 && ncx ==1 && nrz ==1 && ncz ==1 23 | xi=zeros(nrx,1); 24 | P=nrx; 25 | Q=1; 26 | % if z is a column vector and x a scalar, xi is the same size column vector 27 | elseif nrz >1 && ncz == 1 && nrx ==1 && ncx ==1 28 | xi =zeros(nrz,1); 29 | P=nrz; 30 | Q=1; 31 | % if x is a row vector and z a scalar, xi is the same size row vector 32 | elseif ncx > 1 && nrx ==1 && nrz ==1 && ncz ==1 33 | xi=zeros(1, ncx); 34 | P=1; 35 | Q=ncx; 36 | % if z is a row vector and x a scalar, xi is the same size row vector 37 | elseif ncz > 1 && nrz ==1 && nrx ==1 && ncx ==1 38 | xi=zeros(1,ncz); 39 | P=1; 40 | Q=ncz; 41 | % other combinations are not supported 42 | else error('(x,z) must be (vector,scalar) pairs or equal matrices') 43 | end 44 | -------------------------------------------------------------------------------- /libraries/init_xi3D.m: -------------------------------------------------------------------------------- 1 | function [xi, P, Q] = init_xi3D(x,y,z) 2 | % [xi, P, Q] = init_xi3D(x,y z) examines the sizes of the (x,y,z) variables 3 | % (which specify points in the second medium, across a plane interface, 4 | % to which a ray must travel from an array element or element segment) 5 | % and returns a PxQ array of zero values to hold the distances xi 6 | % at which the ray intersects an interface, as well as the values (P,Q). 7 | % Eleven different combinations of sizes for (x,y,z) are 8 | % allowed, which permits (x,y,z) to represent values in planes parallel 9 | % to the x-,y-,or z-axes (three cases), or values along lines parallel to 10 | % the x-, y-,or z-axes (six cases since the line could be represented 11 | % as row or column vectors),or values along an inclined line 12 | % in 3-D (two cases since the line could be represented as row or column 13 | % vectors). 14 | 15 | % get sizes of (x,y,z) 16 | [nrx,ncx ]= size(x); 17 | [nry,ncy]=size(y); 18 | [nrz,ncz] = size(z); 19 | 20 | % if x,z are equal size [nrx,ncx] matrices and y is single value, make 21 | % xi a [nrx, ncx] matrix 22 | if nrx == nrz && ncx == ncz && nry ==1 && ncy ==1 23 | xi = zeros(nrx,ncx); 24 | P = nrx; 25 | Q = ncx; 26 | % if x, y are equal size [nrx, ncx] matrices and z is a single value, make 27 | % xi a [ nrx, ncx] matrix 28 | elseif nrx == nry && ncx == ncy && nrz ==1 && ncz ==1 29 | xi = zeros(nrx,ncx); 30 | P = nrx; 31 | Q = ncx; 32 | % if y, z are equal size [nry,ncy] matrices and x is a single value, make 33 | % xi a [nry, ncy] matrix 34 | elseif nry ==nrz && ncy == ncz && nrx ==1 && ncx ==1 35 | xi=zeros(nry, ncy); 36 | P = nry; 37 | Q = ncy; 38 | % if z is a [1,ncz] vector and x and y are single values, make 39 | % xi a [1,ncz] vector 40 | elseif nrz ==1 && ncz >1 && nrx ==1 && ncx ==1 && nry == 1 && ncy ==1 41 | xi =zeros(1, ncz); 42 | P = 1; 43 | Q = ncz; 44 | % if z is a [nrz, 1] vector and x and y are single values, make 45 | % xi a [nrz,1] vector 46 | elseif ncz ==1 && nrz >1 && nrx ==1 && ncx == 1 && nry == 1 && ncy == 1 47 | xi =zeros(nrz,1); 48 | P= nrz; 49 | Q =1; 50 | % if x is a [1,ncx] vector and y and z are single values, make 51 | % xi a [1,ncx] vector 52 | elseif nrx ==1 && ncx >1 && nry ==1 && ncy == 1 && nrz == 1 && ncz == 1 53 | xi =zeros(1,ncx); 54 | P= 1; 55 | Q = ncx; 56 | % if x is a [nrx, 1] vector and y and z are single values, make 57 | % xi a [nrx, 1] vector 58 | elseif ncx == 1 && nrx >1 && nry ==1 && ncy ==1 && nrz == 1 && ncz == 1 59 | xi = zeros(nrx, 1); 60 | P= nrx; 61 | Q =1; 62 | % if y is a [1, ncy] vector and x and z are single values, make 63 | % xi a [1, ncy] vector 64 | elseif nry ==1 && ncy >1 && nrx ==1 && ncx == 1 && nrz == 1 && ncz == 1 65 | xi=zeros(1, ncy); 66 | P =1; 67 | Q = ncy; 68 | % if y is a [nry, 1] vector and x and z are single values, mke 69 | % xi a [nry,1] vector 70 | elseif nry >1 && ncy ==1 && nrx ==1 && ncx ==1 && nrz ==1 && ncz ==1 71 | xi=zeros(nry, 1); 72 | P = nry; 73 | Q =1; 74 | % if x, y, z are equal size [1, ncx] vectors, make 75 | % xi a [ 1,ncx] vector 76 | elseif nrx ==nry && ncx == ncy && nrz == nrx && ncz == ncx && nrx ==1 77 | xi =zeros(1, ncx); 78 | P= 1; 79 | Q = ncx; 80 | % if x, y, z are equal size [nrx,1] vectors, make 81 | % xi a [nrx,1] vector 82 | elseif nrx ==nry && ncx == ncy && nrz == nrx && ncz == ncx && ncx ==1 83 | xi =zeros(nrx,1); 84 | P = nrx; 85 | Q = 1; 86 | else error(' (x,y,z) combination given is not supported') 87 | end 88 | 89 | -------------------------------------------------------------------------------- /libraries/interface2.m: -------------------------------------------------------------------------------- 1 | function y =interface2(x, cr, df, dp, dpf) 2 | % y = interface2(x, cr, df, dp, dpf) outputs the value of a function, y, 3 | % which is zero if the input argument,x,is the location along an interface 4 | % where Snell's law is satisfied. The input parameter cr =c1/c2, where c1 5 | % is the wave speed in medium one, and c2 is the wave speed in medium 2, 6 | % The other input parameters (df, dp, dpf) define a ray which goes from 7 | % a point in medium one to the interface and then to a point in medium 8 | % two, where df = DF is the depth of the point in medium two, 9 | % dp = DT is the height of the point in medium one, and dpf = DX is the 10 | % separation distance between the points in medium one and two 11 | % (see Fig 5.4 in the text). The function y used here is c1 times the 12 | % function defined in Eq.(5.2.6) in the text. 13 | 14 | % the function,y, 15 | 16 | 17 | y =x./sqrt(x.^2+dp^2)-cr*(dpf-x)./sqrt((dpf-x).^2 +df^2); 18 | -------------------------------------------------------------------------------- /libraries/ls_2Dint.m: -------------------------------------------------------------------------------- 1 | function p = ls_2Dint(b, f, mat,e, angt, Dt0, x, z, varargin) 2 | % p= ls_2Dint(b, f, mat, e, angt, Dt0, x, z, Nopt)computes the normalized 3 | % pressure, p, for an element in a 1-D array radiating waves across 4 | % a plane fluid/fluid interface where p is calculated 5 | % at a location (x, z) (in mm)in the second fluid for a 6 | % source of length 2b (in mm) at a frequency, f, (in MHz). 7 | % The vector mat = [d1, c1, d2, c2] where d1 is the density in the first 8 | % medium (in gm/cm^3), c1 is the wave speed in the first medium 9 | % (in m/sec)and similarly d2 is the density in the second medium (in 10 | % gm/cm^3)and c2 is the wave speed in the second medium (in m/sec). 11 | % The distance e (in mm) is the offset of the center of the element from 12 | % the center of the array. The parameter angt(in degrees) 13 | % specifies the angle of the array with respect to the x-axis 14 | % and Dt0 (in mm) is the distance of the center of the array from the 15 | % interface. The assumed harmonic time dependency is exp(-2i*pi*f*t). 16 | % The model used is a Rayleigh-Sommerfeld type of integral for a 17 | % piston source where ray theory has been used to propagate the cylindrical 18 | % waves generated by the element across the interface. 19 | % Nopt gives the number of segments to use. If Nopt is not 20 | % specified as an input argument the function uses one segment 21 | % per wavelength, based on the input frequency, f, which must 22 | % be a scalar when Nopt is not given. 23 | 24 | % extract material parameters 25 | d1 =mat(1) ; 26 | c1 = mat(2); 27 | d2 = mat(3) ; 28 | c2 = mat(4); 29 | % compute wave numbers 30 | k1b = 2000*pi*b*f/c1 ; 31 | k2b=2000*pi*b*f/c2; 32 | 33 | % if number of segments is specified, use 34 | 35 | if nargin == 9 36 | N = varargin{1}; 37 | else 38 | % else choose number of segments so that the size of each segment 39 | % is a wavelength 40 | N = round((2000)*f*b/c1); 41 | if N < 1 42 | N=1; 43 | end 44 | end 45 | 46 | % compute centroid locations for the segments 47 | xc =zeros(1,N); 48 | for jj=1:N 49 | xc(jj) = b*(-1 + 2*(jj-0.5)/N); 50 | end 51 | 52 | % calculate normalized pressure as a sum over all the segments 53 | 54 | p=0; 55 | for nn= 1:N 56 | % find the distance, xi, where the ray from the center of a segment 57 | % to point(x,z)intersects the interface 58 | xi = pts_2Dintf(e, xc(nn), angt, Dt0, c1,c2, x, z); 59 | % compute distances and angles needed in the model 60 | Dtn=Dt0+(e+xc(nn)).*sin(angt*pi/180); 61 | Dxn = x-(e+xc(nn)).*cos(angt*pi/180); 62 | r1 = sqrt(xi.^2.+ Dtn.^2)./b; 63 | r2 = sqrt((Dxn -xi).^2 +z.^2)./b; 64 | ang1 = asin(xi./(b*r1)); 65 | ang2 =asin((Dxn-xi)./(b*r2)); 66 | ang = angt*pi/180 -ang1; 67 | ang = ang + eps.*( ang == 0); 68 | % form up the segment directivity 69 | dir =sin(k1b.*sin(ang)/N)./(k1b.*sin(ang)/N); 70 | % compute plane wave transmission coefficient(based on pressure ratio) 71 | Tp = 2*d2*c2.*cos(ang1)./(d1.*c1.*cos(ang2) +d2.*c2.*cos(ang1)); 72 | % compute phase term and denominator 73 | ph =exp(1i*k1b.*r1 + 1i*k2b.*r2); 74 | den =r1+(c2/c1).*r2.*((cos(ang1)).^2)./(cos(ang2)).^2; 75 | % put terms together for pressure due to each segment 76 | p= p + Tp.*dir.*ph./sqrt(den); 77 | 78 | end 79 | p = p.*(sqrt(2*k1b./(1i*pi)))/N; % include external factor 80 | 81 | -------------------------------------------------------------------------------- /libraries/ls_2Dv.m: -------------------------------------------------------------------------------- 1 | function p = ls_2Dv(b, f, c, e, x, z, varargin) 2 | % p= ls_2Dv(b, f, c, e, x, z, Nopt)computes the normalized 3 | % pressure, p, at a location (x, z) (in mm) in a fluid 4 | % for a two-dimensional source of length 5 | % 2b (in mm) along the x-axis at a frequency, f, (in MHz) 6 | % and for a wave speed, c, (in m/sec) of the fluid. This 7 | % function can used to describe an element in an array by 8 | % specifying a non-zero value for e (in mm), which is the offset 9 | % of the center of the element along the x-axis. 10 | % The assumed harmonic time dependency is exp(-2i*pi*f*t)and 11 | % the 2-D version of the Rayleigh-Sommerfeld integral for a 12 | % piston source is used as the model where the Hankel function 13 | % is approximated by its asymptotic cylindrical wave form for 14 | % large wave numbers. 15 | % Nopt gives the number of segments to use. If Nopt is not 16 | % given as an input argument the function uses 1 segment 17 | % per wavelength, based on the input frequency, f, which must 18 | % be a scalar in the case where Nopt is not given. 19 | 20 | % compute wave number 21 | kb = 2000*pi*b*f/c ; 22 | 23 | % if number of segments is specified, use 24 | 25 | if nargin == 7 26 | N = varargin{1}; 27 | else 28 | % else choose number of terms so that the size of each segment 29 | % is a wavelength 30 | N = round((2000)*f*b/c); 31 | if N < 1 32 | N=1; 33 | end 34 | end 35 | % use normalized positions in the fluid 36 | xb = x/b; 37 | zb = z/b; 38 | eb=e/b; 39 | % compute normalized centroid locations for the segments 40 | xc =zeros(1,N); 41 | for jj=1:N 42 | xc(jj) = -1 + 2*(jj-0.5)/N; 43 | end 44 | % calculate normalized pressure as a sum over all the 45 | % segments as an approximation of the Rayleigh-Sommerfeld 46 | % type of integral 47 | p=0; 48 | for kk = 1:N 49 | ang =atan((xb-xc(kk) -eb)./zb); 50 | ang = ang + eps.*( ang == 0); 51 | dir =sin(kb.*sin(ang)/N)./(kb.*sin(ang)/N); 52 | rb = sqrt((xb-xc(kk)- eb).^2 + zb.^2); 53 | ph = exp(1i*kb.*rb); 54 | p= p + dir.*exp(i*kb.*rb)./sqrt(rb); 55 | 56 | end 57 | p = p.*(sqrt(2*kb./(i*pi)))/N; % include external factor 58 | 59 | -------------------------------------------------------------------------------- /libraries/on_axis_foc2D.m: -------------------------------------------------------------------------------- 1 | function p = on_axis_foc2D(b, R, f, c, z) 2 | % p = on_axis_foc2D(b,R, f,c,z) computes the on-axis normalized 3 | % pressure for a 1-D focused piston element of length 2b 4 | % and focal length R (in mm). 5 | % The frequency is f (in MHz), b is the transducer half-length 6 | % (in mm), c is the wave speed of the surrounding fluid 7 | % (in m/sec),and z is the on-axis distance (in mm). The 8 | % paraxial approximation is used to write the pressure field in terms 9 | % of a Fresnel integral. Note: the propagation term exp(ikz) is removed 10 | % from the wave field calculation. 11 | 12 | % ensure no division by zero at z =0 13 | z = z +eps*(z == 0); 14 | 15 | % define transducer wave number 16 | kb = 2000*pi*f*b/c; 17 | 18 | % define u and prevent division by zero 19 | u =(1-z/R); 20 | u = u + eps*( u == 0); 21 | 22 | % argument of the Fresnel integral and denominator in on-axis pressure 23 | % equation 24 | x = sqrt((u.*kb.*b)./(pi.*z)).*( z <= R)+... 25 | sqrt((-u.*kb.*b)./(pi.*z)).*(z > R); 26 | denom = sqrt(u).*(z <= R) + sqrt(-u).*( z > R); 27 | Fr = fresnel_int(x).*( z <= R) + conj(fresnel_int(x)).*(z >R); 28 | 29 | % compute normalized on-axis pressure (p/rho*c*v0) with 30 | % the propagation phase term exp(ikz) removed. Use analytical 31 | % values near the focus and the numerical Fresnel integral values 32 | % away from the focus 33 | p=(sqrt(2/i).*sqrt((b/R).*kb/pi)).*( abs(u) <= .005) + ... 34 | (sqrt(2/i).*Fr./denom).*(abs(u) > .005); 35 | 36 | 37 | 38 | 39 | -------------------------------------------------------------------------------- /libraries/ps_3Dint.m: -------------------------------------------------------------------------------- 1 | function [vx,vy,vz] = ps_3Dint(lx,ly,f,mat,ex,ey,angt, Dt0,x,y,z, varargin ) 2 | % [vx,vy,vz] = ps_3Dint(lx,ly,f,mat,ex,ey,angt, Dt0, x,y,z,Ropt, Qopt) 3 | % calculates the normalized velocity components (vx,vy,vz) of a rectangular 4 | % array element radiating waves through a planar fluid/solid interface. The 5 | % parameters (lx, ly) are the lengths of the element in the x'- and y'- 6 | % directions, respectively (in mm), f is the frequency (in MHz), and mat is 7 | % a vector mat = [d1, cp1, d2, cp2, cs2, type] where (d1, cp1) are the 8 | % density (in gm/cm^3) and compressional wave speed (in m/sec) for the 9 | % fluid and (d2, cp2, cs2) are similarly the density, P-wave speed, and 10 | % S-wave speed for the solid, and type ='p' or 's' for a P-wave or 11 | % S-wave, respectively, in the solid. The distances (ex, ey) are the 12 | % x'- and y'- coordinates of the centroid of the element relative to the 13 | % center of the array (in mm). The parameters angt is the angle 14 | % (in degrees) the array makes with respect to the interface, and Dt0 15 | % is the distance of the center of the array above the interface (in mm). 16 | % The parameters (x,y,z) specify the point(s) in the second medium at 17 | % which the fields are to be calculated (in mm), where x- and y- are 18 | % parallel to the interface and z is normal to the interface, pointing 19 | % into the second medium. 20 | % Ropt and Qopt are optional arguments. Ropt specifies the number of 21 | % segments to use in the x'-direction while Qopt specifies the number of 22 | % segments in the y'-direction . If either Ropt or Qopt are not 23 | % given as input arguments for a given direction then the function uses 24 | % one segmentper wavelength in that direction, based on the input 25 | % frequency, f, which must be a scalar when either Ropt or Qopt 26 | % are not given. 27 | 28 | 29 | %extract material densities, wave speeds, and the type of wave in the 30 | %second medium from mat vector 31 | d1 =mat(1); 32 | cp1=mat(2); 33 | d2 =mat(3); 34 | cp2=mat(4); 35 | cs2 =mat(5); 36 | type =mat(6); 37 | 38 | % wave speed in the first medium (a fluid) is for compressional waves 39 | c1 =cp1; 40 | % decide which wave speed to use in second medium for specified wave type 41 | if strcmp(type, 'p') 42 | c2 =cp2; 43 | elseif strcmp(type,'s') 44 | c2=cs2; 45 | else error(' type must be ''p'' or ''s'' ') 46 | end 47 | 48 | %compute wave numbers for waves in first and second medium 49 | k1=2000*pi*f/c1; 50 | k2 =2000*pi*f/c2; 51 | 52 | %if number of x-segments is specified then use 53 | if nargin > 11 54 | R = varargin{1}; 55 | 56 | % else choose number of terms so each segment 57 | % is a wave length or less 58 | else 59 | R=ceil(1000*f*lx/c1); 60 | if R < 1 61 | R=1; 62 | end 63 | end 64 | 65 | % if number of y-segments is specified then use 66 | if nargin >12 67 | Q = varargin{2}; 68 | 69 | % else choose number of terms so that each segment 70 | % is a wave length or less 71 | else 72 | Q=ceil(1000*f*ly/c1); 73 | if Q < 1 74 | Q=1; 75 | end 76 | end 77 | 78 | % compute centroid locations of segments in x'- and y'-directions 79 | % relative to the element centroid 80 | xc=zeros(1,R); 81 | yc=zeros(1,Q); 82 | for rr=1:R 83 | xc(rr) = -lx/2 +(lx/R)*(rr-0.5); 84 | end 85 | for qq=1:Q 86 | yc(qq) = -ly/2 +(ly/Q)*(qq-0.5); 87 | end 88 | 89 | % calculate normalized velocity components as a sum over all the 90 | % segments as an approximation of the Rayleigh-Sommerfeld 91 | % integral 92 | vx=0; 93 | vy=0; 94 | vz=0; 95 | 96 | for rr = 1:R 97 | for qq = 1:Q 98 | % calculate distance xi along the interface for a ray from a 99 | %segment to the specified point in the second medium 100 | Db = sqrt((x-(ex+xc(rr)).*cosd(angt)).^2 +(y-(ey+yc(qq))).^2); 101 | Ds = Dt0 + (ex +xc(rr)).*sind(angt); 102 | xi = pts_3Dint(ex,ey,xc(rr),yc(qq),angt,Dt0,c1,c2,x,y,z); 103 | 104 | % calculate incident and refracted angles along the ray, 105 | % including the special case when ray is at normal incidence 106 | if Db ==0 107 | ang1 =0; 108 | else 109 | ang1 = atand(xi./Ds); 110 | end 111 | 112 | if ang1 == 0 113 | ang2 =0; 114 | else 115 | ang2=atand((Db-xi)./z); 116 | end 117 | % calculate ray path lengths in each medium 118 | r1 =sqrt(Ds.^2 +xi.^2); 119 | r2=sqrt((Db-xi).^2 +z.^2); 120 | % calculate segment sizes in x'- and y'- directions 121 | dx=lx/R; 122 | dy =ly/Q; 123 | 124 | % calculate (x', y')components of unit vector along the ray in the 125 | % first medium 126 | if Db ==0 127 | uxt =-sind(angt); 128 | uyt = 0; 129 | else 130 | uxt=xi.*(x-(ex+xc(rr)).*cosd(angt)).*cosd(angt)./(Db.*r1) ... 131 | -Ds.*sind(angt)./r1; 132 | uyt = xi.*(y - (ey+yc(qq)))./(Db.*r1); 133 | end 134 | 135 | % calculate polarization components for P- and S-waves in the 136 | % second medium, including special case of normal incidence 137 | if Db == 0 138 | dpx =0; 139 | dpy=0; 140 | dpz=1; 141 | dsx =1; 142 | dsy =0; 143 | dsz=0; 144 | else 145 | dpx = (1-xi./Db).*(x-(ex+xc(rr)).*cosd(angt))./r2; 146 | dpy = (1 -xi./Db).*(y-(ey+yc(qq)))./r2; 147 | dpz=z./r2; 148 | dsx = sqrt(dpy.^2 +dpz.^2); 149 | dsy= -dpx.*dpy./dsx; 150 | dsz = -dpx.*dpz./dsx; 151 | end 152 | % choose polarization components to use based on wave type in the 153 | % second medium 154 | if strcmp(type, 'p' ) 155 | px=dpx; 156 | py=dpy; 157 | pz =dpz; 158 | elseif strcmp(type, 's') 159 | px = dsx; 160 | py = dsy; 161 | pz =dsz; 162 | else error('wrong type') 163 | end 164 | % calculate transmission coefficients (based on velocity ratios) 165 | % for P- and S-waves and choose appropriate coefficient for the 166 | % specified wave type 167 | [tpp,tps]= T_fluid_solid(d1,cp1,d2,cp2,cs2, ang1); 168 | 169 | if strcmp(type,'p') 170 | T=tpp; 171 | elseif strcmp(type, 's') 172 | T = tps; 173 | end 174 | % form up the directivity term 175 | argx = k1.*uxt.*dx/2; 176 | argx =argx +eps.*(argx == 0); 177 | argy = k1.*uyt.*dy/2; 178 | argy = argy + eps.*( argy == 0); 179 | dir = (sin(argx)./argx).*(sin(argy)./argy); 180 | % form up the denominator term 181 | D1 = r1 + r2.*(c2/c1).*(cosd(ang1)./cosd(ang2)).^2; 182 | D2 = r1 + r2.*(c2/c1); 183 | % put transmission coefficient, polarization, directivity, phase 184 | % term and denominator together to calculate velocity components. 185 | vx = vx + T.*px.*dir.*exp(1i.*k1.*r1 +1i.*k2.*r2)./sqrt(D1.*D2); 186 | vy = vy + T.*py.*dir.*exp(1i.*k1.*r1 +1i.*k2.*r2)./sqrt(D1.*D2); 187 | vz = vz + T.*pz.*dir.*exp(1i.*k1.*r1 +1i.*k2.*r2)./sqrt(D1.*D2); 188 | end 189 | end 190 | % include external factor for these components 191 | vx = vx.*(-1i*k1*dx*dy)/(2*pi); 192 | vy = vy.*(-1i*k1*dx*dy)/(2*pi); 193 | vz = vz.*(-1i*k1*dx*dy)/(2*pi); 194 | 195 | 196 | 197 | 198 | 199 | -------------------------------------------------------------------------------- /libraries/ps_3Dv.m: -------------------------------------------------------------------------------- 1 | function p = ps_3Dv(lx,ly,f,c,ex,ey,x,y,z, varargin ) 2 | % p =ps_3Dv(lx, ly, f, c, ex, ey, x,y,z,Popt,Qopt) computes the normalized 3 | % pressure, p, at a location (x,y,z) (in mm) in a fluid 4 | % for a rectangular element of lengths (lx, ly) 5 | % (in mm) along the x- and y-axes, respectively,at a frequency, f, (in MHz) 6 | % ,and for a wave speed, c, (in m/sec) of the fluid. This 7 | % function can used to describe an element in an array by 8 | % specifying non-zero values for (ex,ey) (in mm), which are the offsets 9 | % of the center of the element along the x- and y-axes, respectively. 10 | % The assumed harmonic time dependency is exp(-2i*pi*f*t)and 11 | % the Rayleigh-Sommerfeld integral for a piston source is used 12 | % as the beam model. 13 | % Popt and Qopt are optional arguments. Popt specifies the number of 14 | % segments to use in the x-direction while Qopt specifies the number of 15 | % segments in the y-direction . If either Popt or Qopt are not 16 | % given as input arguments for a given direction the function uses 17 | % one segment per wavelength in that direction, based on the input 18 | % frequency, f, which must be a scalar when either Popt or 19 | % Qopt are not given. 20 | 21 | 22 | %compute wave number 23 | k=2000*pi*f/c; 24 | 25 | %if number of x-segments is specified then use 26 | if nargin > 9 27 | P = varargin{1}; 28 | 29 | % else choose number of terms so each segment 30 | % length is at most a wave length 31 | else 32 | P=ceil(1000*f*lx/c); 33 | if P < 1 34 | P=1; 35 | end 36 | end 37 | 38 | % if number of y-segments is specified then use 39 | if nargin >10 40 | Q = varargin{2}; 41 | 42 | % else choose number of terms so that each segment 43 | %is a wave length or less 44 | else 45 | Q=ceil(1000*f*ly/c); 46 | if Q < 1 47 | Q=1; 48 | end 49 | end 50 | 51 | %compute centroid locations of segments in x- and y-directions 52 | xc=zeros(1,P); 53 | yc=zeros(1,Q); 54 | for pp=1:P 55 | xc(pp) = -lx/2 +(lx/P)*(pp-0.5); 56 | end 57 | for qq=1:Q 58 | yc(qq) = -ly/2 +(ly/Q)*(qq-0.5); 59 | end 60 | 61 | % calculate normalized pressure as a sum over all the 62 | % segments as an approximation of the Rayleigh-Sommerfeld 63 | % integral 64 | p=0; 65 | for pp = 1:P 66 | for qq = 1:Q 67 | rpq=sqrt((x-xc(pp) -ex).^2 +(y-yc(qq)-ey).^2 +z.^2); 68 | ux= (x -xc(pp)-ex)./rpq; 69 | uy = (y-yc(qq)-ey)./rpq; 70 | ux =ux+eps*(ux == 0); 71 | uy =uy+eps*(uy == 0); 72 | dirx = sin(k.*ux.*lx/(2*P))./(k.*ux.*lx/(2*P)); 73 | diry =sin(k.*uy.*ly/(2*Q))./(k.*uy.*ly/(2*Q)); 74 | p=p + dirx.*diry.*exp(1i*k.*rpq)./rpq; 75 | end 76 | end 77 | p = p.*(-1i*k*(lx/P)*(ly/Q))/(2*pi); % include external factor 78 | 79 | 80 | 81 | 82 | 83 | 84 | -------------------------------------------------------------------------------- /libraries/pts_2Dintf.m: -------------------------------------------------------------------------------- 1 | function xi = pts_2Dintf( e, xn, angt, Dt0, c1,c2, x, z) 2 | % xi = pts_2Dintf(e, xn, angt, Dt0, c1, c2, x, z) calculates the 3 | % intersection of a ray from the center of a segment of an array element in 4 | % one fluid to a point (x, z) (in mm) in a second fluid across a plane 5 | % interface, where e is the offset of the element from the center of the 6 | % array and xn is the offset of the segment from the center of the element. 7 | % (both in mm). The parameter angt is the angle (in degrees) that the array 8 | % makes with respect to the x-axis (the interface) and Dt0 is the distance 9 | % of the center of the array above the interface (in mm). The parameters 10 | % c1, c2 are the wave speeds in the first and second medium, respectively, 11 | % (both in m/sec). This function uses the function init_xi(x,z) to examine 12 | % the sizes of the (x,z) variables to decide on the corresponding number 13 | % of rows and columns needed to calculate the locations xi (in mm) at 14 | % which rays from the center of a segment to the points (x,z) intersect the 15 | % interface. The function ferrari2 is then used with the appropriate input 16 | % arguments to calculate the xi values (in mm). 17 | 18 | 19 | % calculate wave speed ratio 20 | cr =c1/c2; 21 | 22 | % based on sizes of (x, z), determine corresponding number of rows and 23 | %columns (P,Q) needed for xi calculations and initialize xi as zeros. 24 | [xi,P,Q]=init_xi(x,z); 25 | 26 | % obtain sizes of (x,z) so appropriate arguments can be found in the calls 27 | % to the function ferrari2 when making the xi calculations 28 | [nrx, ncx] =size(x); 29 | [nrz,ncz]=size(z); 30 | 31 | % calculate xi locations using ferrari's method 32 | for pp=1:P 33 | for qq=1:Q 34 | Dtn=Dt0+(e+xn)*sin(angt*pi/180); 35 | % if x is a point,and z is a row or column vector 36 | if nrx ==1 && ncx == 1 37 | Dxn= x -(e+xn)*cos(angt*pi/180); 38 | xi(pp,qq)=ferrari2(cr, z(pp,qq), Dtn,Dxn); 39 | % if z is a point, and x is a row or column vector 40 | elseif nrz ==1 && ncz ==1 41 | Dxn = x(pp,qq) -(e+xn)*cos(angt*pi/180); 42 | xi(pp,qq)=ferrari2(cr, z, Dtn,Dxn); 43 | % if x and z are equal size PxQ matrices 44 | else 45 | Dxn = x(pp,qq) -(e+xn)*cos(angt*pi/180); 46 | xi(pp,qq)=ferrari2(cr, z(pp,qq), Dtn,Dxn); 47 | end 48 | 49 | end 50 | end 51 | -------------------------------------------------------------------------------- /libraries/pts_3Dint.m: -------------------------------------------------------------------------------- 1 | function xi = pts_3Dint(ex, ey, xn, yn, angt, Dt0, c1, c2, x, y, z) 2 | % xi = pts3Dint(ex, ey, xn,yn,angt,Dt0, c1,c2,x,y,z) calculates the 3 | % distance, xi, (in mm) along the interface in the plane of incidence, 4 | % at which a ray from the center of an array element segment to a point 5 | % in the second medium intersects the interface. The parameters 6 | % (ex, ey) are the element offsets (in mm)from the center of the 7 | % entire array to the center of the element in the x'- and y'-directions, 8 | % respectively, and (xn,yn)are similarly the offsets as measured 9 | % to the center of the element segment from the center of the 10 | % element in the x'- and y'-directions (in mm). The parameter angt, is the 11 | % angle of the array (in degrees) from the interface, and Dt0 is the 12 | % distance (in mm) of the center of the array from the interface. (c1,c2) 13 | % are the wave speeds in the first and second medium (in m/sec) and 14 | % (x,y,z) are the coordinates of end point of the ray in the second medium 15 | % (all in mm). 16 | 17 | %calculate wave speed ratio 18 | cr=c1/c2; 19 | % determine size of array needed for xi calculations based on the sizes of 20 | % the (x,y,z) variables) and also determine those sizes 21 | [xi, P, Q ] = init_xi3D(x,y,z); 22 | 23 | [nrx,ncx] =size(x); 24 | [nry, ncy] =size(y); 25 | [nrz,ncz] =size(z); 26 | 27 | 28 | % call ferrari2 function to compute xi with the arguments of that function 29 | % determined by the sizes of the (x,y,z) variables. 30 | De = Dt0 +(ex + xn)*sind(angt); 31 | for pp=1:P 32 | for qq = 1:Q 33 | 34 | % x and y are points, z is a row or column vector 35 | if nrx ==1 && ncx ==1 && nry ==1 && ncy ==1 36 | Db=sqrt((x-(ex +xn)*cosd(angt)).^2 +(y-(ey+yn)).^2); 37 | xi(pp,qq) =ferrari2(cr, z(pp,qq), De, Db); 38 | % y and z are points, x is a row or column vector 39 | elseif nry == 1 && ncy ==1 && nrz ==1 && ncz ==1 40 | Db=sqrt((x(pp,qq)-(ex +xn)*cosd(angt)).^2 +(y-(ey+yn)).^2) ; 41 | xi(pp,qq) =ferrari2(cr, z, De, Db); 42 | % x and z are points, y is a row or column vector 43 | elseif nrx ==1 && ncx ==1 && nrz ==1 && ncz ==1 44 | Db=sqrt((x-(ex +xn)*cosd(angt)).^2 +(y(pp,qq)-(ey+yn)).^2); 45 | xi(pp,qq) =ferrari2(cr, z, De, Db); 46 | % y is a point, x and z are equal size PxQ matrices 47 | elseif nry ==1 && ncy ==1 && nrx == nrz && ncx == ncz 48 | Db=sqrt((x(pp,qq)-(ex +xn)*cosd(angt)).^2 +(y-(ey+yn)).^2); 49 | xi(pp,qq) = ferrari2(cr, z(pp,qq), De, Db); 50 | 51 | 52 | % z is a point, x and y are equal size PxQ matrices 53 | elseif nrz == 1 && ncz ==1 && nrx == nry && ncx == ncy 54 | Db=sqrt((x(pp,qq)-(ex +xn)*cosd(angt)).^2 +(y(pp,qq)-(ey+yn)).^2); 55 | xi(pp,qq) = ferrari2(cr, z, De, Db); 56 | % x is a point, y and z are equal size PxQ matrices 57 | elseif nrx ==1 && ncx ==1 && nry == nrz && ncy == ncz 58 | Db=sqrt((x-(ex +xn)*cosd(angt)).^2 +(y(pp,qq)-(ey+yn)).^2); 59 | xi(pp,qq) = ferrari2(cr, z(pp,qq), De, Db); 60 | % x, y, z are all equal size row or column vectors 61 | else 62 | Db=sqrt((x(pp,qq)-(ex +xn)*cosd(angt)).^2 +(y(pp,qq)-(ey+yn)).^2); 63 | xi(pp,qq) = ferrari2(cr, z(pp,qq),De, Db); 64 | end 65 | end 66 | end -------------------------------------------------------------------------------- /libraries/rs_2Dv.m: -------------------------------------------------------------------------------- 1 | function p = rs_2Dv(b, f, c, e, x, z, varargin) 2 | % p= rs_2Dv(b, f, c, e, x, z, Nopt)computes the normalized 3 | % pressure, p, at a location (x, z) (in mm) 4 | % in a fluid for a 1-D element of length 5 | % 2b (in mm) along the x-axis at a frequency, f,(in MHz). 6 | % and for a wave speed, c, (in m/sec) of the fluid. This 7 | % function can used to describe an element in an array by 8 | % specifying a non-zero value for e (in mm), which is the offset 9 | % of the center of the element along the x-axis. 10 | % The assumed harmonic time dependency is exp(-2i*pi*f*t)and 11 | % the 2-D version of the Rayleigh-Sommerfeld integral for a 12 | % piston source is used as the model. 13 | % Nopt gives the number of segments to use. If Nopt is not 14 | % given as an input argument the function use 10 segments 15 | % per wavelength, based on the input frequency, f, which must 16 | % be a scalar when Nopt is not given. 17 | 18 | 19 | % compute wave number 20 | kb = 2000*pi.*b.*f./c ; 21 | % if number of segments is specified, use 22 | if nargin == 7 23 | N = varargin{1}; 24 | else 25 | % else choose number of segments so that the size of each segment 26 | % is one-tenth a wavelength 27 | N = round((20000)*f*b/c); 28 | if N <= 1 29 | N = 1; 30 | end 31 | end 32 | % use normalized positions in the fluid 33 | xb = x./b; 34 | zb = z./b; 35 | eb = e./b; 36 | % compute normalized centroid locations for the segments 37 | xc =zeros(1,N); 38 | for jj=1:N 39 | xc(jj) = -1 + 2*(jj-0.5)/N; 40 | end 41 | % calculate normalized pressure as a sum over all the 42 | % segments as an approximation of the Rayleigh-Sommerfeld 43 | % type of integral 44 | p=0; 45 | for kk = 1:N 46 | rb = sqrt((xb-xc(kk)-eb).^2 + zb.^2); 47 | p= p + besselh(0, 1,kb.*rb); 48 | 49 | end 50 | p = p.*(kb./N); % include external factor 51 | 52 | -------------------------------------------------------------------------------- /output/beam_steering.mp4: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/ultrasonic-phased-array/matlab-simulation/859b7cf2b645380d598cf50a53f418bac27e08c0/output/beam_steering.mp4 --------------------------------------------------------------------------------