SaM - Standing Waves

Remeber:

The reason ultrasonic transducer vibrates is beacuse a standing wave resonates inside of the piezoelectric cristal. ~ For example let’s take a slice of PZT, in contact with air (which as a high acustic impedance). We can imagine a standing wave like so:

• A standing wave is a resonance.
• We can represent it with a formula, and we have that the forward part of the wave ($U_F$) and backward part ($U_B$) assume:$U=U_F+U_B=A\sin \left(\frac{2\pi x_1}{{\lambda }_w}-\omega t\right)+A\sin \left(\frac{2\pi x_1}{{\lambda }_w}+\omega t\right)$Where:
  • ${\lambda }_w$ is the wavelength.
  • $t$ is the time, in the picture there is also another $t$ there it means thickness.
  • $x_1$ is the direction of propagation of the wave: we have limited it between $[-\frac{t}{2}÷\frac{t}{2}]$, here $t$ means thickness.

~Ex: Here a real world example of a standing wave:
You can see how it is described by the formula:$U=A\sin \left(\frac{2\pi x_1}{{\lambda }_w}-\omega t\right)+A\sin \left(\frac{2\pi x_1}{{\lambda }_w}+\omega t\right)$

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And what happens actually is that we have in the PZT a standing wave is created, and the frequency of the standing wave is corresponding to the resonance that we have found in our simple analysis:

• The vibration of the surface is given by this standing wave.
• The standing wave comes from waves which propagate in the piezoelectric material, and it is the reusult of their superimposition.
• If you consider to have air as the surrounding material, the stress transmitted to the air is approximately $0$.
• I have also written the formula for a standing wave $U_W$ in case of a planar sine wave.
• What I see from outside is that this surface vibrating in time.
• So our coarse approximation can be used in order to understand what happens but taking to account that we have actually this small complex situation inside we don’t have a slice which simply moves something which is related to a field inside it.
• What is a Standing Wave?
  • A standing wave is a type of wave pattern that occurs when two waves of the same frequency and amplitude traveling in opposite directions superpose (combine) in such a way that they interfere with each other, creating a stationary or “standing” pattern.
• Example of Standing Wave:
  • Here is a simple demonstration of a standing wave formed at different specific frequencies, here we have the 2nd 3rd and 4th armonics:
    
    
  • Source: Youtube