SaM - MEMORIZE (Ultrasonic Waves)

List of things to memorize:

SaM - Ultrasonic Waves

• Elastic Waves and Acustic Waves
• Planar Waves
• Longitudinal Waves
• Transverse Waves
• Generic Ultrasonic Wave Function
• ~Ex.: Sine Plane Wave
• Reflection
• Scattering and Diffuse Reflection
• Matching Layer
• Standing Waves
• ~ Ex.: Propagation of an Ultrasonic Wave in Different Materials
  • Wavelength

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SaM - Elastic Waves and Acustic Waves • Calculation of the Wave Equation

• Elastic waves: propagates in solids.
  These are also called pressure wave or stress wave and are related to the elastic behavior of solids.
• Acoustic waves : propagates in fluids.
  Air and human tissues are considered fluids.
• How we defined a simple wave funtion:
  1. Take a small cube:
     
  2. Define:
     • Small volume of the cube: $\Delta V=d{x}_1\cdot d{x}_2\cdot d{x}_3$
     • Stress: $T=\frac{F}{A}$
     • Strain: ${\epsilon }_{kl}=\frac{1}{2}\left(\frac{\partial u_k}{\partial x_l}+\frac{\partial u_l}{\partial x_k}\right)$
     • Second Newton’s law: $F=m\cdot a$.
       We will consider $F=⟨F_1,0,0⟩$ for simplicity.
     • Hooke’s law: $T_{ij}={\sum }_{k=1}^3{\sum }_{l=1}^3{c}_{ijkl}\cdot {\epsilon }_{kl}$
  3. The force $(F_1)$ can be found in two ways: from stress $(F=T\cdot A)$ : $\begin{array}{lcc}{F}_1& =& T_{11}(x_1+d{x}_1,x_2,x_3)\cdot d{x}_2\cdot d{x}_3-T_{11}(x_1,x_2,x_3)\cdot d{x}_2\cdot d{x}_3+\\ & +& T_{12}(x_1,x_2+d{x}_2,x_3)\cdot d{x}_1\cdot d{x}_3-T_{12}(x_1,x_2,x_3)\cdot d{x}_1\cdot d{x}_3+\\ & +& T_{13}(x_1,x_2,x_3+d{x}_3)\cdot d{x}_1\cdot d{x}_2-T_{13}(x_1,x_2,x_3)\cdot d{x}_1\cdot d{x}_2\end{array}$And from the Second Newton’s law $(F=m\cdot a)$ :$\begin{array}{lcc}{F}_1& =& {\ddot{u}}_1\cdot (\rho \cdot \Delta V)\\ & =& \rho {\ddot{u}}_{1}d{x}_{1}d{x}_{2}d{x}_3\end{array}$
  4. Calculate the wave equation:$\rho {\ddot{u}}_j=c_{ijkl}\frac{\partial u_k}{\partial x_j\partial x_l}$
• Terminology:
  • $x_i$ is the $i$-the axis on which the waves propagates.NOT_SURE_ABOUT_THIS
  • While $\vec{u}$ defines the direction where the particles that compose the wave move.NOT_SURE_ABOUT_THIS

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SaM - Planar Waves or Plane Waves

• A plane wave propagates only in a single direction:
  
• Formula:$\frac{\partial u_i}{\partial x_2}=\frac{\partial u_i}{\partial x_3}=0$
• A planar wave in an isotropic material (so that we can use the collapsed stifness tensor):$\begin{array}{l}\rho {\ddot{u}}_1=c_{11}\frac{{\partial }^2{u}_1}{\partial x_1^2}\\ \rho {\ddot{u}}_2=c_{44}\frac{{\partial }^2{u}_2}{\partial x_1^2}\\ \rho {\ddot{u}}_3=c_{44}\frac{{\partial }^2{u}_3}{\partial x_1^2}\end{array}$
• Terminology:
  • $x_i$ is the $i$-the axis on which the waves propagates.NOT_SURE_ABOUT_THIS
  • While $\vec{u}$ defines the direction where the particles that compose the wave move.NOT_SURE_ABOUT_THIS
  • If we look at the definitions of longitudinal wave and transverse wave, we can say that the planar wave is a combination of the two.

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SaM - Longitudinal Waves or Compressive Waves

• In a longitudinal wave the displacement of the particles are parallel to the propagation of the wave
  So: $\vec{u}\parallel \vec{x}$ :
  
• Formulas (in an isotropic material):$\begin{array}{l}\rho {\ddot{u}}_1=c_{11}\frac{{\partial }^2{u}_1}{\partial x_1^2}\\ \rho {\ddot{u}}_2=0\\ \rho {\ddot{u}}_3=0\end{array}$
• Terminology:
  • $x_i$ is the $i$-the axis on which the waves propagates.NOT_SURE_ABOUT_THIS
  • While $\vec{u}$ defines the direction where the particles that compose the wave move.NOT_SURE_ABOUT_THIS

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SaM - Transverse Waves or Shear Waves

• Pure transverse waves are possible only in solids.
• In a shear wave the displacement of the particles are perpendicular to the propagation of the wave
  So: $\vec{u}\perp \vec{x}$ :
  
• Formulas (in an isotropic material):$\begin{array}{l}\rho {\ddot{u}}_1=0\\ \rho {\ddot{u}}_2=c_{44}\frac{{\partial }^2{u}_2}{\partial x_1^2}\\ \rho {\ddot{u}}_3=c_{44}\frac{{\partial }^2{u}_3}{\partial x_1^2}\end{array}$
• Terminology:
  • $x_i$ is the $i$-the axis on which the waves propagates.NOT_SURE_ABOUT_THIS
  • While $\vec{u}$ defines the direction where the particles that compose the wave move.NOT_SURE_ABOUT_THIS

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SaM - Generic Ultrasonic Wave Function • Longitudinal and Transvers Propagation Velocities of an Ultrasonic Wave

• Formula:$U_1(t,x_1)=U_{1F}\left(t-\sqrt{\frac{\rho }{c_{11}}}\cdot x_1\right)+U_{1B}\left(t+\sqrt{\frac{\rho }{c_{11}}}\cdot x_1\right)$
• In the same media we can have two velocity for a wave:$\begin{array}{lc}\text{longitudinal velocity:}& v_l=\sqrt{\frac{c_{11}}{\rho }}\\ \text{transverse velocity:}& v_t=\sqrt{\frac{c_{44}}{\rho }}\end{array}$
• We can define the “components” of a generic planar wave ($U_1,U_2,U_3$, primitives of $\vec{\ddot{u}}$):$\begin{array}{lc}\text{longitudinal:}& U_1(t,x_1)=U_{1F}\left(t-\frac{x_1}{v_l}\right)+U_{!B}\left(t+\frac{x_1}{v_l}\right)\\ \text{transverse:}& U_2(t,x_1)=U_{2F}\left(t-\frac{x_1}{v_t}\right)+U_{2B}\left(t+\frac{x_1}{v_t}\right)\end{array}$
• Sine plane wave example:$U_1(t,x_1)=A\sin \left(2\pi f_0\left(t-\frac{x_1}{v_l}\right)\right)$
• Wavelength:${\lambda }_w=\frac{v_l}{f_0}$
• Definition of wavelength in a sine plane wave:
  
• Real World Measures:
  • Usually: $v_l\approx 2v_t$
  • For Air: $v_l\sim 340\frac{\text{m}}{\text{s}}$.
  • For Water: $v_l\sim 1500\frac{\text{m}}{\text{s}}$.
  • For Steel: $v_l\sim 6000\frac{\text{m}}{\text{s}}$ and $v_t\sim 3000\frac{\text{m}}{\text{s}}$.
• Terminology
  • $c_{11}$ and $c_{44}$ come from the collapsed stifness tensor for isotropic materials.
  • $\rho$ is the density of the material.
  • This two velocities relate to the longitudinal wave and transverse wave.
  • $U_1$ and $U_2$ are functions of any shape.
    • $U_{1F}$ and $U_{2F}$ : function of any shape that propagates Forward.
    • $U_{1B}$ and $U_{2B}$ : function of any shape that propagates Backward.

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SaM ~ Example of an Ultrasonic Wave Function • Sine Plane Wave

• Sine plane wave example:$U_1(t,x_1)=A\sin \left(2\pi f_0\left(t-\frac{x_1}{v_l}\right)\right)$
• Wavelength:${\lambda }_w=\frac{v_l}{f_0}$
• Graph:
  
• Real World Measures:
  • Air: $v_l\sim 340\frac{\text{m}}{\text{s}}$, $f_{\text{RANGE}}\in \left[50\text{HKz}÷1\text{MKz}\right]$, ${\lambda }_w\in \left[7\text{mm}÷340\mu \text{m}\right]$.
  • Water: $v_l\sim 1500\frac{\text{m}}{\text{s}}$, $f_{\text{RANGE}}\in \left[1\text{MKz}÷10\text{MKz}\right]$, ${\lambda }_w\in \left[1.5\text{mm}÷340\mu \text{m}\right]$.
  • Steel: $v_l\sim 6000\frac{\text{m}}{\text{s}}$, $f_{\text{RANGE}}\in \left[1\text{MKz}÷50\text{MKz}\right]$, ${\lambda }_w\in \left[6\text{mm}÷1.2\text{mm}\right]$.

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SaM - Reflection of an Ultrasonic Wave

• Acustic impedance: $Z=\rho \cdot v_l$
• Reflection:
  
• We define:$\begin{array}{lc}{v}_i& \text{(intial velocity of the wave)}\\ v_R={\Gamma }_V\cdot v_i& \text{(velocity of the reflected wave)}\\ v_T=T_V\cdot v_i& \text{(velocity of the transmitted wave)}\\ T_{11_i}& \text{(intial stress over the i-th dimension)}\\ T_{11_R}={\Gamma }_T\cdot T_{11_i}& \text{(reflected wave’s stress)}\\ T_{11_T}=T_T\cdot T_{11_i}& \text{(transmitted wave’s stress)}\end{array}$
• Formulas:$\begin{array}{lc}{T}_V=\frac{2Z_1}{Z_1+Z_2}& \text{(transmitted coefficent for the velocity)}\\ T_T=\frac{Z_1-Z_2}{Z_1+Z_2}& \text{(transmitted coefficent for the stress)}\\ {\Gamma }_V=\frac{2Z_2}{Z_1+Z_2}& \text{(reflected coefficent for the velocity)}\\ \Gamma T=\frac{Z_2-Z_1}{Z_1+Z_2}& \text{(reflected coefficent for the stress)}\end{array}$
• Tilted reflection:
  
• Formula:$\begin{array}{l}{\theta }_2={\theta }_1\\ \frac{\sin {\theta }_3}{Z_2}=\frac{\sin {\theta }_1}{Z_1}\end{array}$

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SaM - Scattering and Diffuse Reflection of an Ultrasonic Wave

• Scattering (if $D\ll {\lambda }_w$):
  
• Diffuse reflection (if $D\gg {\lambda }_w$, but the incident plane is not smooth):
  
• Terminology:
  • $D$ : diameter of the obstacle.
  • ${\lambda }_w$ : wavelength.

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SaM - Matching Layer for Acustic Impedance

• Before applying the matching layer:
  • $Z_{\text{PZT}}=32.5\frac{\text{kg}}{{\text{m}}^2\text{s}}$.
  • $Z_{\text{?}}=430\frac{\text{kg}}{{\text{m}}^2\text{s}}$.
• Resulting transmitted wave (pre-mathcing layer):$\frac{v_T}{v_i}=\frac{2Z_{\text{PZT}}}{Z_{\text{PZT}}+Z_{\text{?}}}$
• Strucutre:
  
• Resulting transmitted wave (post-mathcing layer):$\frac{v_T}{v_i}=\frac{2Z_{\text{PZT}}}{Z_{\text{PZT}}+Z_{\text{A}}}\cdot \frac{2Z_{\text{A}}}{Z_{\text{A}}+Z_{\text{?}}}$
• Real world values:
  • $Z_{\text{PZT}}=32.5\frac{\text{kg}}{{\text{m}}^2\text{s}}$.
  • $Z_{\text{?}}=430\frac{\text{kg}}{{\text{m}}^2\text{s}}$.
  • ~Ex.: matching layer : $Z_A=200\text{Rayl}$.
    • After adding the mathcing layer, the ratio $\left(\frac{v_T}{v_i}\right)$ becomes $126.33\%$ of the intial value, so we have increased it.
  • As a rule of thumb we can consider a single acustic impedance that incorporates the material+matching layer: $Z_M=\sqrt{Z_{\text{A}}{Z}_{\text{PZT}}}$.
  • Given the thickness of the PZT layer: suppose $t_{\text{PZT}}=\frac{{\lambda }_{\omega }}{2}$, we usually take the thickness of the matching layer as $t_A=\frac{{\lambda }_{\omega }}{4}$.NOT_SURE_ABOUT_THIS
• Terminology:
  • The unit of measure $\frac{\text{kg}}{{\text{m}}^2\text{s}}$ is called a $\text{Rayl}$.

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SaM - Standing Waves

• The reason ultrasonic transducer vibrates is beacuse a standing wave resonates inside of the piezoelectric cristal.
• Standing wave:
  
• Formula:$\begin{array}{lcc}U& =& U_F+U_B\\ & =& A\sin \left(2\pi \frac{x_1}{{\lambda }_w}-\omega t\right)+A\sin \left(2\pi \frac{x_1}{{\lambda }_w}+\omega t\right)\end{array}$
• Real world example of a standing wave:
  

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SaM ~ Real World Example • Propagation of an Ultrasonic Wave in Different Materials

• Higher frequency: more special resolution (clearer images) but less penetration depth.
• Real World Measures:
  • Air: $f<1\text{MHz}$, usually: $f\in [30\text{KHz}÷50\text{KHz}]$.
  • Liquids (biomedical field): $f\in [1\text{MHz}÷10\text{MHz}]$.
  • Solids (NDT - Non Destructive Testing): $f\in [1\text{MHz}÷50\text{MHz}]$.