SaM - Transverse Waves or Shear Waves

Remeber:

==Pure transverse waves are possible only in solids. If the displacement of the particles are perpendicular to the propagation of the wave==, then it is called a transverse wave:
Using the definition Hooke’s law, we have found previously:$\rho {\ddot{u}}_j=c_{ijkl}\frac{\partial u_k}{\partial x_j\partial x_l}$We have that a transverse wave has:$\begin{array}{l}\rho {\ddot{u}}_1=0\\ \rho {\ddot{u}}_2=c_{44}\frac{{\partial }^2{u}_2}{\partial x_1^2}\\ \rho {\ddot{u}}_3=c_{44}\frac{{\partial }^2{u}_3}{\partial x_1^2}\end{array}$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 transverse waves we have $\vec{u}\perp \vec{x}$==.

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Memory Card

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While if the displacement of the particles and wave move in perpendicular directions, we call it a shear or transfers wave.

• This waves are present only in solid.
• 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.
• Like longitudinal waves they are defined only for $j=l=1$, where $j$ and $l$ are referning tho this part: $\frac{{\partial }^2{u}_k}{\partial x_j\partial x_l}$, however they do not have a $u_1$ component, we find acceleation only for $u_2$ and $u_3$, as we can see by the formula $\rho {\ddot{u}}_{2,3}=c_{44}\frac{{\partial }^2{u}_{2,3}}{\partial x_1^2}$Again for the derivative of $u_2$ and $u_3$ over $x_1$ we multiply it by the coefficent $c_{44}$.
  ⇒ So the only difference between longitudinal waves and transverse waves (looking at the formulas) is that: ==longitudinal waves have a $\rho {\ddot{u}}_1=c_{11}\frac{{\partial }^2{u}_1}{\partial x_1^2}$ component while for transvere waves $\rho {\ddot{u}}_1=0$==.
• NOTE:NOT_SURE_ABOUT_THIS
  • $c_{11}$: represents the elastic constant associated with the longitudinal (compression or dilatation) deformation of the material.
    It characterizes the material’s resistance to changes in volume due to compression or expansion along the same direction as the applied stress.
  • $c_{44}$: represents the elastic constant associated with shear deformation.
    It characterizes the material’s resistance to shearing forces that cause changes in the shape of the material without changing its volume.
• What are Transverse Waves? (ChatGPT)
  • A transverse wave is a type of mechanical wave where the particles of the medium move perpendicular to the direction of the energy transfer.
  • In other words, the oscillations or vibrations of the particles occur in a direction that is perpendicular to the wave’s propagation.