Physics - Points, points, point everywhere

This exercise reviews the basic relations in ultrasonic wave propagation and echo ranging, with points.

Related materials:

by Alfonso Rodriguez-Molares (alfonso.r.molares@ntnu.no) 17.07.2016

Contents

A point source

A point source is immersed in a fluid: an infinite, isotropic, and homogenous domain with unknown speed of sound. We know however that the fluid density $\rho$ =1052 kg/m3 and that its bulk modulus $K$= 2.6196e9 Pascals.

rho=1052;       % domain density [kg/m3]
K=2.6196e9;     % domain coefficient of stiffness [Pa]
c0=sqrt(K/rho)  % domain speed of sound [m/s]
c0 =

   1.5780e+03

f=10e6;         % central frequency [Hz]
lambda=c0/f     % wavelength [m]
lambda =

   1.5780e-04

T=1/f     % period [s]
T =

   1.0000e-07

A point reflector

Let's consider the location of the source as the origin of coordinates and let us place a point reflector at $(x,z)=(10,30)~\textmd{mm}$,

x=10e-3;        % x-position [m]
z=30e-3;        % z-position [m]

% % create figure
% figure1 = figure('Name','background','Color',[1 1 1]);
% hold on; grid on;
% plot(0,0,'ro','linewidth',2,'markersize',5); axis equal;
% plot(x*1e3,z*1e3,'bo','linewidth',2,'markersize',5);
% set(gca, 'ydir', 'reverse');
% box('on');
% xlabel('x [mm]');
% ylabel('z [mm]')
% legend('Point source','Point reflector');
% title({'Point Source and Reflector'});
% set(figure1,'InvertHardcopy','off');

Notice in the previous figure that we refer to the ordinates as z-axis, not y-axis. Also you may notice that the z-axis is reversed (it increases as we get down). That is the common way we represent space in ultrasound imaging. You better get used to it!

r=sqrt(x^2+z^2);    % distance [m]
TOF=2*r/c0          % time of flight [s]
TOF =

   4.0079e-05

rho1=2028;          % particle's density [kg/m3]
c1=3190;            % particle's speed of sound [m/s]
z1=rho1*c1;         % particle's acoustic impedance [kg/m2/s]
z0=rho*c0;          % domain acoustic impedance [kg/m2/s]
Gamma=(z1-z0)/(z1+z0) % Reflection coefficient [unitless]
Gamma =

    0.5916

% Nope. The position $(x,z)$ of the point reflector can not be retrieved, but
% we can calculate the distance the point reflector is located at:

TOF=30e-6;      % time of flight [s]
r=TOF/2*c0      % distance [m]
r =

    0.0237