Task007

acoustics/acousticsOfMirrors.m

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+function acousticsOfMirrors
+	% Speed of longitudinal wave propagation in the considered media.
+	% we take the mean value of all source we found
+	vlWater = mean([1480,1483,1484,1493,1496]);
+	vlCopper = mean([4660,4760,4760,4660,3560]);
+	vlPCB = mean([4726,3070,2100,2460]);
+
+	% Speed of shear wave propagation in the considered media.
+	% we take the mean value of all source we found
+	vsWater = vlWater;
+	vsCopper = mean([2330,2325,2325]);
+	vsPCB = vlPCB / 2; % hypothese
+
+	% critical angles:
+	disp (strcat("critical angle of longitudinal transmission Water->Copper : ", num2str(rad2deg(criticalAngleOfTransmission(vlWater, vlCopper)))));
+	disp (strcat("critical angle of shear transmission Water->Copper : ", num2str(rad2deg(criticalAngleOfTransmission(vlWater, vsCopper)))));
+
+	% hereafter are computed the angles between the propagation direction and
+	% the normal vector of the planar boundary between media
+	% naming convention: ("P"|"S") "_" ("R"|"T") "_" <X> <N> is the angle between the normal vector
+	% of the boundary X and the propagation vector
+	% of the pressure (P) or shear (S) wave reflected (R) or transmitted (T) by the boundary X for the Nth time
+	% example: P_R_A_2 is the pressure wave reflected by the boundary A for the second time
+
+	incident = deg2rad(45);
+
+	% Pressure waves
+	P_T_A_1 = transmittedAngle(vlWater, vlCopper, incident);
+	P_T_A_2 = transmittedAngle(vlCopper, vlWater, P_T_A_1);
+	P_T_B_1 = transmittedAngle(vlCopper, vlPCB, P_T_A_1);
+
+	% Shear waves
+	S_R_A_1 = transmittedAngle(vlWater, vsWater, incident);
+	S_T_A_1 = transmittedAngle(vsWater, vsCopper, incident);
+	S_T_A_2 = transmittedAngle(vsCopper, vsWater, S_T_A_1);
+	S_T_B_1 = transmittedAngle(vsCopper, vsPCB, S_T_A_1);
+
+	% display in octave console
+	disp(strcat("incident : ",num2str(rad2deg(incident))));
+	disp(strcat("P_T_A_1 : ",num2str(rad2deg(P_T_A_1))));
+	disp(strcat("P_T_A_2 : ",num2str(rad2deg(P_T_A_2))));
+	disp(strcat("P_T_B_1 : ",num2str(rad2deg(P_T_B_1))));
+	disp(strcat("S_R_A_1 : ",num2str(rad2deg(S_R_A_1))));
+	disp(strcat("S_T_A_1 : ",num2str(rad2deg(S_T_A_1))));
+	disp(strcat("S_T_A_2 : ",num2str(rad2deg(S_T_A_2))));
+	disp(strcat("S_T_B_1 : ",num2str(rad2deg(S_T_B_1))));
+
+	return;
+endfunction
+
+% Share of the energy which is reflected at the boundary between two 
+% media of acoustic impedences "z1" and "z2"
+function x = reflectionCoeff (z1, z2)
+	x = ((z2-z1)/(z2+z1))^2;
+	return;
+endfunction
+
+% Angle of the transmitted wave meeting the boundary between two 
+% media  of longitudinal wave velocities "v1" and "v2" at an incendent 
+% angle "incident_angle"
+function x = transmittedAngle (v1, v2, incident_angle)
+	x = asin (sin(incident_angle)*v2/v1);
+	return;
+endfunction
+
+% returns the critical angle for longitudinal wave transmission. Any
+% wave meeting the boundary with a greater angle than those returned
+% will be totally reflected
+function x = criticalAngleOfTransmission(v1, v2)
+	x = asin (v1/v2);
+	return;
+endfunction

acoustics/envelope_detection.m

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+function [envelope] = envelope_detection (signal, fech, fmin, fmax,option)
+
+%function to determine the envelope of a the data vector named signal
+%fech is the sampling frequency of the data vector in MHz
+%we add a filtering of the signal between fmin and fmax in MHz
+%option refer to the option of the zero padding process:
+%sigzp is filled with 0 from N to Nzeropad if option = 0
+%sigzp is filled with the mean of sig from N to Nzeropad if option != 0
+%%%%
+%if signal have an offset, the output envelope won't have, but still need to put
+%option != 0 if signal have an offset and want a clean envelope
+
+if fmin<0
+    fmin=0;
+endif
+if fmax>fech/2
+    fmax=fech/2;
+endif
+if fmax<fmin
+    fmax=fmin;
+endif
+
+N=length(signal);
+[Nzeropad, sigzp] = zero_padding (signal, option);
+Sig=fft(sigzp);
+deltaf=fech/Nzeropad;
+
+nmin=floor(fmin/deltaf)+1;
+nmax=floor(fmax/deltaf)+1;
+
+Sig(1:nmin)=0;
+Sig(nmax:Nzeropad)=0;
+tmpenv=abs(ifft(Sig))*2;
+envelope=tmpenv(1:N);
+
+endfunction

acoustics/readme.md

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+# Acoustic implications of using a mirror
+[TOC]
+
+## Problem description
+
+* Problem 1 (solved): define the angles of all reflected and transmitted longitudinal (black vectors) and shear (red vectors) waves.
+* Problem 2 (solved): situate the echos, that is, the distance between the useful reflected wave and the echos
+* Problem 3 (in progress): determine the magnitude of all vectors
+
+![./Problem_description.png](./Problem_description.png)
+
+## Results
+
+* Critical angle of longitudinal transmission from water to copper : 19.4°.
+* Critical angle of shear transmission from water to copper : 39.7°
+* The incident angle (45°) is greater than the critical angles of longitudinal and shear transmission, therefore, there is no transmission of the incident wave into the copper layer, therefore neither longitudinal nor shear echo ([total internal reflection](https://en.wikipedia.org/wiki/Total_internal_reflection)).
+
+*(To reproduce the results, execute in Octave the script “acousticsOfMirrors.m”)*
+
+## Method
+
+* The angles of the longitudinal and shear waves transmitted and reflected in the case of a longitudinal wave hitting a boundary between two media A and B are given by [Snell's law](https://de.wikipedia.org/wiki/Snelliussches_Brechungsgesetz#Akustik):
+```
+sin(a)/vLA = sin(aRS)/vSA = sin(aRL)/vLA = sin(aTS)/vSB = sin(aTL)/vLB
+Where
+  * a is the incident angle
+  * vLA is the velocity of longidudinal waves in the medium A
+  * aRS is the angle of the reflected shear wave in medium A
+  * vSA is velocity of shear waves in the medium A
+  * aRL is the angle og the reflected longitudinal wave in medium A
+  * vLA is the velocity of longitudinal waves in the medium A
+  * aTS is the angle of the transmitted shear wave in medium B
+  * vSB is the velocity of shear waves in the mediom B
+  * aTL is the transmitted longitudinal wave in medium B
+  * vLB is the velocity of longitudinal waves in the B
+```
+
+* The critical angle of wave transmission is the angle above which an incindent wave is totally reflected. It is  [derived from Snell's law](https://en.wikipedia.org/wiki/Snell%27s_law#Total_internal_reflection_and_critical_angle): 
+```
+alpha = asin (vA/vB)
+Where
+  * alpha is the critical angle
+  * vA is the velocity of longitudinal waves in medium A
+  * vB is the velocity of longitudinal waves in medium B
+```
+
+## Variables
+
+Compressional velocity (speed of longitudinal wave propagation, in m/s):
+-   In copper :
+    -   4660 [2]
+    -   4760 (annealed) [6]
+    -   4760 (annealed) [7]
+    -   4660 [10]
+    -   3560 [13]
+-   In air:
+    -   331.45 (dry) [6]
+    -   331.2 (dry at 0°C) [8]
+    -   331 [13]
+-   In composite:
+    -   4726 (Glass fiber-reinforced polyester composite) [9]
+    -   3070 (graphite/epoxy) [10]
+    -   2100 (L385:340 epoxy at 20°C) [11]
+    -   2460 to 3170 (depending on thickness and material ratio) [14]
+-   In water:
+    -   1480 (at 20°C) [10]
+    -   1483 (at 20°C) [12]
+    -   1484 [8]
+    -   1493 [13]
+    -   1496 (distilled) [6]
+Shear velocity (speed of shear wave propagation, in m/s):
+-   In copper:
+    -   2330 [2]
+    -   2325 [6]
+    -   2325 [7]
+-   In composite:
+    -   No data found. As default value we take compression velocity / sqrt(2)
+-   In water and air: no shear wave propagation
+Acoustic impedence (in Ns/m³):
+-   Copper:
+    -   41.6e6 [3]
+-   Air
+    -   413 (at 20°C) [3]
+-   Water
+    -   1.48e6 [3]
+
+## References
+
+[1] https://www.slideshare.net/RakeshSingh125/minor-project-report-28478524, p 9-12
+[2] https://www.slideshare.net/RakeshSingh125/minor-project-report-28478524, p 15
+[3] https://www.slideshare.net/RakeshSingh125/minor-project-report-28478524, p 16
+[4] https://www.slideshare.net/RakeshSingh125/minor-project-report-28478524, p 17
+[5] https://www.slideshare.net/RakeshSingh125/minor-project-report-28478524, p 18
+[6] http://www.rfcafe.com/references/general/velocity-sound-media.htm
+[7] https://en.wikipedia.org/wiki/Speeds_of_sound_of_the_elements_(data_page)
+[8] https://en.wikipedia.org/wiki/Speed_of_sound
+[9] https://arxiv.org/ftp/arxiv/papers/1511/1511.04543.pdf
+[10] https://www.olympus-ims.com/de/ndt-tutorials/thickness-gage/appendices-velocities/
+[11] http://www.ndt.net/article/wcndt2004/pdf/materials_characterization/616_mchugh.pdf
+[12] http://www.ondacorp.com/images/Liquids.pdf
+[13] http://hyperphysics.phy-astr.gsu.edu/hbase/Tables/Soundv.html
+[14] http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.457.3039&rep=rep1&type=pdf
+
+## Furher messy notes
+https://www.nde-ed.org/EducationResources/CommunityCollege/Ultrasonics/Physics/modeconversion.htm
+https://acoustics.byu.edu/content/mode-conversion

acoustics/test_code.m

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+clear all;
+close all;
+
+tlim=2.3;
+Npoint=1025;
+offset=0.5;
+
+t=linspace(-tlim,tlim,Npoint)*10^(-6);
+fech=1/(t(2)-t(1))/10^6;
+f=3.5*10^6;
+om=2*pi*f;
+sig=sin(om*t);
+tau=1/f*1;
+sig=sig.*exp(-(t/tau).^2);
+sig=sig+offset;
+
+[Nzeropad, sigzp] = zero_padding (sig, 0);
+
+figure(1)
+plot(t,sig);
+figure(2)
+plot(sigzp);
+
+[env1] = envelope_detection (sig, fech, 0.5, 8,0);
+[env2] = envelope_detection (sig, fech, 0.5, 8,1);
+
+figure(3)
+plot(t,(sig-offset),t,env1,t,env2);

acoustics/zero_padding.m

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+function [Nzeropad, sigzp] = zero_padding (sig, option)
+
+%this function is made for making a zero padding to a data vector sig
+%zero_padding consiste in changing the number of elements of the the initial vector of size N
+%to a number Nzeropad wich is a power of 2 to enjoy the speed of fft algorithm
+%sigzp is filled with 0 from N to Nzeropad if option = 0
+%sigzp is filled with the mean of sig from N to Nzeropad if option != 0
+
+N=length(sig);
+
+pow=0;
+
+while (2^pow<N)
+    if (2^pow != N)
+        pow++;
+    endif
+ endwhile
+
+ Nzeropad=2^pow;
+ sigzp=zeros(Nzeropad, 1);
+ sigzp(1:N)=sig;
+
+if option!=0
+    mean = sum(sig)/N;
+    sigzp(N+1:Nzeropad)=mean;
+ endif
+
+endfunction