SaM - Longitudinal Waves or Compressive Waves

Remeber:

If the ==displacement of the particles are parallel to the propagation of the wave==, then it is called a longitudinal wave:
Using the definition wave formula, we have found previously:$\rho {\ddot{u}}_j=c_{ijkl}\frac{\partial u_k}{\partial x_j\partial x_l}$We have that a logitudinal wave has:$\begin{array}{l}\rho {\ddot{u}}_1=c_{11}\frac{{\partial }^2{u}_1}{\partial x_1^2}\\ \rho {\ddot{u}}_2=0\\ \rho {\ddot{u}}_3=0\end{array}$Where:

• $x_1$ direction in which the wave propagates.
• $\vec{u}$ direction in which the material or particles that form the wave move.NOT_SURE_ABOUT_THIS
  ==For longitudinal waves we have $\vec{u}\parallel \vec{x}$==.

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Memory Card

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If the displacement of the particles are parallel to the propagation of the wave, then it is called a longitudinal wave, and we can write it in the collapsed notation:

• So the collapsed notation takes $j=k=l$, so the condition for a longitudial wave, and writes it as a single index.
• NOTE: The complete extended notation would be $\rho {\ddot{u}}_j={\sum }_i{c}_{ij}\frac{{\partial }^2{u}_j}{{\partial }^2{x}_j}$NOT_SURE_ABOUT_THIS
• So this means that you have a wave, which travels/propagates along $x_1$ and all the particles are displaced along this direction.
• So these are longitudinal waves or compressive waves or waves of pressure because what changes is a pressure because you have molecules which pushes other on contact.
• And this kind of wave can propagate through solid and liquids and fluids in general (air is considered a fluid).
• Remeber to not look, or try to make sense of the indecies for the compressed notation for $c$, since for example $c_{44}$ does not refer to 4th dimension
• What is a Longitudinal Wave? (ChatGPT):
  • A longitudinal wave is a type of mechanical wave in which the particles of the medium vibrate back and forth in the direction parallel to the direction of energy transfer.
  • This is in contrast to transverse waves, where the particles of the medium move perpendicular to the direction of energy transfer