SaM - Reflection of an Ultrasonic Wave

Remeber:

==If a wave travels in a homogeneous medium, nothing happens==, let’s see what happens if the medium in not homogeneus. First we need the definition we have given to the acustic impedance:IMPORTANTE $Z=\rho \cdot v_l$

Suppose we have two materials represented by two different acustic impedances:

• Here you can see the phenomena of reflection, part of the wave bounces back, with a different propagation velocity ($v_r$)
• So we define:$\begin{array}{lc}{v}_i& \text{(intial velocity of the wave)}\\ v_R={\Gamma }_V\cdot v_i& \text{(velocity of the reflected wave)}\\ v_T=T_V\cdot v_i& \text{(velocity of the transmitted wave)}\\ T_{11_i}& \text{(intial stress over the i-th dimension)}\\ T_{11_R}={\Gamma }_T\cdot T_{11_i}& \text{(reflected wave’s stress)}\\ T_{11_T}=T_T\cdot T_{11_i}& \text{(transmitted wave’s stress)}\end{array}$And:$\begin{array}{lc}{T}_V=\frac{2Z_1}{Z_1+Z_2}& \text{(transmitted coefficent for the velocity)}\\ T_T=\frac{Z_1-Z_2}{Z_1+Z_2}& \text{(transmitted coefficent for the stress)}\\ {\Gamma }_V=\frac{2Z_2}{Z_1+Z_2}& \text{(reflected coefficent for the velocity)}\\ \Gamma T=\frac{Z_2-Z_1}{Z_1+Z_2}& \text{(reflected coefficent for the stress)}\end{array}$
• Keep in mind that ==we have considered many idealities (as always) and an infinite plane, and completely smooth surface of the plane==

Up till now we have supposed:

• An infinite smooth plane on which the wave reflects.
• A planar wave, with direction $x_1$ perpendicular to the smooth plane.

Suppose now the wave $(v_i,T_{11_i})$ hits the plane with an angle ${\theta }_1$, called “incident angle”, then we have that the resulting transmitted wave $(v_T,T_{11_T})$ and reflected wave $(v_R,T_{11_R})$ are “tilted” like so
Where:$\begin{array}{l}{\theta }_2={\theta }_1\\ \frac{\sin {\theta }_3}{Z_2}=\frac{\sin {\theta }_1}{Z_1}\end{array}$

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Memory Card

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==If a wave travels in a homogeneous medium, nothing happens==. Instead when the wave encounters a surface which separates an homogeneous medium and another homogeneous medium (with different characteristics), then that could be a reflection. Let’s see how to quantify the phenomena of reflection:

• To do this, we have to introduce the acoustic impedance.
• 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.

So we refer to this simple plane wave:

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

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)

So suppose we have two materials represented by two different acustic impedances:

• Here you can see the phenomena of reflection, part of the wave bounces back, with a different propagation velocity ($v_r$)
• It is important that the propagation of the wave is perpendicular to the surface separating $Z_1$ and $Z_2$.
• $T_T$ : transmission coefficient for velocity; the ratio between the starting preassure and the pressure of the wave passed through $Z_2$.
  (It’s a pure number)
• $T_v$ : similar to $T_T$ is a ratio but this time between the propagation velocity.
• ${\Gamma }_v$ is the equivalent of $T_v$ but for the reflected wave, same for ${\Gamma }_T$ and $T_T$.

We define these coefficient, so that we can describe the “transmitted” and “reflected” part of the wave in terms of $v_i$ and $T_{11_i}$.

• $v$ in this formulas are proagation velocities.
• ${\Gamma }_v$ and ${\Gamma }_T$ : reflection coefficients for propagating velocity and stress (indicated by the pedicies ${}_v$ and ${}_T$, $\Gamma$ stands for “reflected”.
• $T_v$ and $T_T$ : transmitted coefficients for propagating velocity and stress (indicated by the pedicies ${}_v$ and ${}_T$, the variable name $T$ stand for “transmitted”

We have considered many idealities (as always) and an infinite plane, completely smooth:

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But I have a different incident angle, which means that I have this is the incident wave:

• So in this case here, if I call these angles: ${\theta }_1$, ${\theta }_2$, ${\theta }_3$.
  We know that ${\theta }_1={\theta }_2$
• While:$\frac{\sin {\theta }_3}{Z_2}=\frac{\sin {\theta }_1}{Z_1}$