SaM - Acustic Impedance • Ultrasonic Lumped Parameter System

Remeber:

For ultrasonic waves and ultrasonic sensors or actuators we have not defined a lumped parameter system, we have just defined the two quanitites for flow and effort and the generated impedance that binds the two. So for ultrasonic systems we have:

• Flow quantity: ${\dot{u}}_1$
• Effort quantity: $T_{11}$ (stress) for solids, $p$ (preassure) for liquids.
• ==Acustic Impendance== $\left(\frac{\text{effort quantity}}{\text{flow quantity}}\right)$:IMPORTANTE $\begin{array}{lc}\text{fluids:}& Z=\frac{p}{{\dot{u}}_1}\\ \text{solids:}& Z=\frac{T_{11}}{{\dot{u}}_1}\end{array}$==We can also define the acustic impedance in another, more useful way==:IMPORTANTE $Z=\rho \cdot v_l$==This formula is valid for both solids and liquids==.
  Here we have:
  • $\rho$ : density of the material, measured in $\left[\frac{\text{Kg}}{{\text{m}}^3}\right]$.
  • $v_l$ : longitudinal velocity of a wave inside a certain material $\left(v_l=\sqrt{\frac{c_{11}}{\rho }}\right)$.
  • Remeber also that $u_1$ and $v_l$ are reciprocal:${\dot{u}}_1=-\frac{T_{11}}{\rho }\frac{1}{v_l}$Where:
    • ${\dot{u}}_1$ represents “==how fast the wave changes form==”.
    • $v_l$ is the longitudinal propagation velocity.
    • Both are velocities, and as such are expressed in $\left[\frac{\text{m}}{\text{s}}\right]$.
• Power: (effort quantity $\cdot$ flow quantity): $\begin{array}{lc}\text{fluids:}& P=p\cdot \dot{u}\\ \text{solids:}& P=T_{11}\cdot {\dot{u}}_1\end{array}$

IMPORTANTE Here’s the value of the acustic impedance for the PZT:$Z_{\text{PZT}}=32.5\text{Rayl}$The unit of measure for the acustic impedance is $\left[\text{Rayl}=\frac{\text{kg}}{{\text{m}}^2\text{s}}\right]$

Formulas that we need to arrive at this conclusion:

1. From the “Elastic Waves and Acustic Waves” lecture:${\lim }_{\Delta V\to 0}\rho \ddot{u_1}=\frac{\partial T_{11}}{\partial x_1}+\frac{\partial T_{12}}{\partial x_2}+\frac{\partial T_{13}}{\partial x_3}$And from the same lecture we have the wave equation:$\rho {\ddot{u}}_j=c_{ijkl}\frac{\partial u_k}{\partial x_j\partial x_l}$
2. We can simplifiy it if we consider it for planar waves in isotropic material:$\rho {\ddot{u}}_1=c_{11}\frac{\partial u_1}{{\partial }^2{x}_1}$Since for planar waves we have that: $\frac{\partial T_{12}}{\partial x_2}=\frac{\partial T_{13}}{\partial x_3}=0$
3. From the “Generic Ultrasonic Wave Function” lecture, we use a a simple longitudinal wave function: $U_1(t,x_1)=U_{1F}\left(t-\frac{x_1}{v_l}\right)$
4. From the “Generic Ultrasonic Wave Function” lecture:$v_l=\sqrt{\frac{c_{11}}{\rho }}$

Here are the passages: (NOTE: These are my calcuation, they can be wrongNOT_SURE_ABOUT_THIS )

1. From the “Elastic Waves and Acustic Waves” lecture:${\lim }_{\Delta V\to 0}\rho \ddot{u_1}=\frac{\partial T_{11}}{\partial x_1}+\frac{\partial T_{12}}{\partial x_2}+\frac{\partial T_{13}}{\partial x_3}$
2. We assume a planar waves that propagates in isotropic materials, so we can use a simplified version of the hooke’s law: $\rho {\ddot{u}}_1=c_{11}\frac{\partial u_1}{{\partial }^2{x}_1}$
3. Since we are talking about a planar wave we will have that:$\frac{\partial T_{12}}{\partial x_2}=\frac{\partial T_{13}}{\partial x_3}=0$So we can rewrite $(1.)$ as: $c_{11}\frac{{\partial }^2{u}_1}{{\partial }^2{x}_1}=\frac{\partial T_{11}}{\partial x_1}$
4. We now define as we have done in the “Generic Ultrasonic Wave Function” lecture a simple longitudinal wave function:$U_1(t,x_1)=U_{1F}\left(t-\frac{x_1}{v_l}\right)$This time we don’t consider the backward part of the formula, we assume that this wave only propagates forward.
5. If we derive $U_1(t,x_1)$ over $t$ we obtain: ${\dot{U}}_1(t,x_1)=V_{1F}\left(t-\frac{x_1}{v_l}\right)$Where $V_{1F}={\dot{U}}_{1F}$.
   And if we derive it over $x_1$ we obtain:$\frac{\partial U_1(t,x_1)}{\partial x_1}=-\frac{1}{v_l}\cdot V_{1F}\left(t-\frac{x_1}{v_l}\right)$Since $U_{1F}$ depends on $\left(t-\frac{x_1}{v_l}\right)$.
6. So we can say:$\frac{\partial U_1(t,x_1)}{\partial x_1}=-\frac{1}{v_l}\cdot {\dot{U}}_1(t,x_1)$
7. Similarly using $(3.)$ we can say: $c_{11}\frac{{\partial }^2{U}_1(t,x_1)}{\partial x_1^2}=\frac{\partial T_{11}}{\partial x_1}$Or, if we integrate from both parts over $x_1$: $c_{11}\frac{\partial U_1(t,x_1)}{\partial x_1}=T_{11}$
8. So using $(7.)$: $T_{11}=c_{11}\cdot \left[-\frac{1}{v_l}\cdot {\dot{U}}_1(t,x_1)\right]$
9. We know form the “Generic Ultrasonic Wave Function” lecture that:$v_l=\sqrt{\frac{c_{11}}{\rho }}$And if we calculate:$\frac{c_{11}}{v_l}=\rho \cdot v_l$
10. So we can rewrite $(8.)$ as:$T_{11}=-\rho \cdot v_l\cdot {\dot{U}}_1(t,x_1)$For simplicity i write ${\dot{U}}_1(t,x_1)$ as ${\dot{u}}_1$, since we have defined the impedance as: $Z=\frac{T_{11}}{{\dot{u}}_1}$Now we know that it is equal to, and we define it as a positive value:$Z=\rho \cdot v_l$We have demonstrated it starting from a solid, but the same formula is true also for liquids.

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• So acoustic impedance is like a generalized impedance.
• The flow quantity (represented as a current) is the speed of the particles, which are vibrating and give the propagation of the wave.
• Whereas the effort quantity (represented as a voltage) is the stress ($T$), or in fluids the pressure ($P$).
• So the power ($P$) of the wave would be for us the product of this two quantity.

Let’s take for example now a simple plane wave, defined using an ultrasonic wave function:

• So you will have uniform speed, preassure and stress.
• NOTE: $V_{1F}$ represents “how fast the wave changes form”, the propagation speed is still $v_l$.

So we can say, using Hooke’s law:

• We change the derivative since integating $T_{11}$ over time or space is similar, except it is scaled.

So the impedance is equal to:

• $\rho$ is the density as usual.
• So the impedance can be written in this two ways.
• $B=c_{11}=\frac{P}{\frac{\Delta V}{V}}$ : Bulk modulus ($V$ in the formula is volume)