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

Remember

We take for instance a sine plane wave, and we take only a forward-propagating wave, so we define $U_{1F}$ that we said was a generic function as:$U_1(t,x_1)=U_{1F}\left(t-\frac{x_1}{v_l}\right)+0=A\sin \left(2\pi f_0\left(t-\frac{x_1}{v_l}\right)+\phi \right)$Where: $f_0$ is decided by the freqency at which the piezoelectric ultrasonic transducer vibrates.

Here’s a plot of the wave:

• “TX” is the ultrasonic trasnducer.
  It vibrates at a frequency $f_0$, related to the ratio $\sqrt{\frac{K}{m}}$.
• So I generate the wave in the medium with a wavelenght ${\lambda }_w$ well known.
• Remember the wavelenght is defined as:${\lambda }_w=\frac{v_l}{f_0}$The wavelenght is defined using the longitudinal velocity of a wave.
  For transverse waves, we have approximately half the wavelength, since $v_t\simeq \frac{1}{2}{v}_l$.

We can look at some standard values of the frequency range and resulting wavelenght range, used for different materials:IMPORTANTE

• Ultrasonic transducers use different frequency to perform ultrasonic analysis in different media.
• ==For transverse wave, we have approximately half the wavelength, since $v_t\simeq \frac{1}{2}{v}_l$==.
• The wavelength is an important parameter of the wave field.
• Depending on the wavelength, different interaction with obstacle occurs.
  We can have: reflection, scattering or diffuse reflection of the wave, depending on the size difference between the obstacle and the wave.

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We take for instance a sine plane wave, and we take only a forward-propagating wave, so we define $U_{1F}$ that we said was a generic function as:$U_1(t,x_1)=U_{1F}\left(t-\frac{x_1}{v_l}\right)+0=A\sin \left(2\pi f_0\left(t-\frac{x_1}{v_l}\right)+\phi \right)$

• We can define the wavelenght $\lambda$, related to both speed and frequency.
• $v_l$ : wave speed.
• At a fixed frequency $f$ ⇒ $\lambda$ depends linearly on the $v_l$, so the larger the speed the larger the wavelength.
• If we think about a small transduer TX, that vibrates at the frequency $f_0$ related to the ratio $\sqrt{\frac{K}{m}}$, so I generate the wave in the medium with a wavelenght ${\lambda }_w$ well known.
• The plot you see is a “photograph” of the wave, so the wave at a fixed time $t_0$, and you see how it propagates along $x_1$, and how the wavelength ${\lambda }_w$ is defined.
• We can use the wavelength or speed to find some difects in a material, since for example the wave travels in air 20 times slower than in steel, if a block of steel presents a pocket of air an ultrasonic sensor can find it.

• We consider it only going forward.
• We can also define the wave number: $K_w$, and the wave vector: ${\vec{K}}_w$

We can look at some values: (this is considered the longitudinal transverse speed)

⇒ Therefore, ultrasonic transducers use different frequency to perform ultrasonic analysis in different media.

==For transverse wave, we have approximately half the wavelength, since $v_t\simeq \frac{1}{2}{v}_l$==.

The wavelength is an important parameter of the wave field. Sepending on the wavelength, different interaction with obstacle occurs. So we can have a reflection, scattering, diffraction, depending on the ratio between the obstacle and the wave.