SaM - Planar Waves or Plane Waves

Remeber:

A plane wave propagates only in a single direction, let’s call it $x_1$, and all the variables which describes the waves are constant on planes perpendicular to $x_1$.

When we perform the derivative on $x_2$ and $x_3$, both will result in a $0$, so rememberIMPORTANTE :$\frac{\partial u_i}{\partial x_2}=\frac{\partial u_i}{\partial x_3}=0$Using the wave equation:$\rho {\ddot{u}}_j=c_{ijkl}\frac{\partial u_k}{\partial x_j\partial x_l}$For a planar wave we find (specifically for isotropic materials, so that we can use the collapsed stifness tensor):IMPORTANTE $\begin{array}{l}\rho {\ddot{u}}_1=c_{11}\frac{{\partial }^2{u}_1}{\partial x_1^2}\\ \rho {\ddot{u}}_2=c_{44}\frac{{\partial }^2{u}_2}{\partial x_1^2}\\ \rho {\ddot{u}}_3=c_{44}\frac{{\partial }^2{u}_3}{\partial x_1^2}\end{array}$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

If we look at the definitions of longitudinal wave and transverse wave, we can say that the planar wave is a combination of the two.NOT_SURE_ABOUT_THIS

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We take into account a plane wave, which propagates along $x_1$.

• What is a Planar Wave?
  • A planar wave, also known as a plane wave, is a type of electromagnetic or acoustic wave characterized by the property that the wavefronts, which are surfaces of constant phase, are flat and planar.
    In other words, a plane wave’s wavefronts are parallel planes, and the wave propagates uniformly in a single direction without spreading out as it travels.
• What is a Planar Wave? (Professor’s Explenation)
  • All the variables which describes the waves are constant on plates perpendicular to $x_1$.
• So the Hooke’s formula is much more compact, since we have to evalutated only for $j=l=1$, where $j$ and $l$ are referning tho this part: $\frac{{\partial }^2{u}_k}{\partial x_j\partial x_l}$ and using the symmetry of $C$, we end up with only 3 formulas (that are not equal to $0$).
• 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.
• NOTE: In the first formula we use $c_{11}$ and in the other two we use $c_{44}$ this is because