Propagation modes of a wave I'm studying Debye's Theory and I am really confused about what really is a propagation mode.
The energy of a $3$-dimensional object is given by $$ \xi_{\alpha , k} = \hbar v_{\alpha} |k|$$

where $\alpha = 1$ refers to the "longitudinal mode" and  $\alpha = 2,3$ the other two transversal modes and $v_{\alpha}$ is the sound velocity.

I'm not sure what those modes really means. My first thought was that the longitudinal mode refers to $k_{x}$ wave number and the other two transversal modes refers to $k_{y},k_{z}\\$.
So, in this line of thinking, if I have a $2$-dimensional object, there are only 2 modes.
But  if this is correct , I'm not sure if the modes have any relation to the wave number of each direction.
Can someone explain to me?
 A: Elastic waves in an isotropic solid are of two types.  Longitudinal in which the particles move backwards and forwards in the direction of propagation and transverse where they move at right angles to the direction of probabgation. The atom at point ${\bf r}$ has displacement
$$
{\boldsymbol \xi}=  {\bf A} \sin({\bf k}\cdot {\bf r}-\omega t)
$$ where ${\bf A}$ is parallel to ${\bf k}$ for longitudinal modes and ${\bf A}$ obeys ${\bf A}\cdot {\bf k}=0$ for transverse modes.  These are the "P" waves and "S" waves of earthquakes.
A: To expand on another answer, a "propagation mode" is just a traveling wave, where the underlying medium is the crystal lattice of atoms in the solid; that is, the waves are really just collective vibrations of the atoms in the solid.  The polarization is just one property that defines a particular wave/mode.  Some details:
In order to completely specify a wave (i.e. a mode), you need these pieces of information:


*

*A wavelength $\lambda$
We instead use the magnitude of the wave vector $k=2\pi/\lambda$, for convenience, because it's easier to write $e^{ikx}$ than it is to write $e^{i2\pi x/\lambda}$.


*A direction of propagation for the wave


We specify this by turning the wave vector into an actual vector, i.e. the direction of $\vec{k}$ is the direction of propagation of the wave, and the magnitude $k$ of $\vec{k}$ is as in bullet point 1 above. Then, a plane wave can be specified as $e^{i\vec{k}\cdot\vec{r}}$.


*A polarization (which specifies the actual motion of each point in the medium relative to the direction of propagation of the wave)


For a wave propagating through the bulk of a three-dimensional solid, there are three different independent possibilities for the motion of the actual atoms in the solid.  (a) The atoms move back and forth along the direction of motion: longitudinal wave.  (b,c) The atoms move perpendicular to the direction of propagation: transverse waves (there are two independent directions perpendicular to the direction of propagation).  (See Wikipedia pages on Longitudinal and Transverse waves for animations.)

My first thought was that the longitudinal mode refers to  wave number and the other two transversal modes refers to ,.

So, this is incorrect!  The different components of $\vec{k}$ combine to characterize the wavelength and direction of propagation and not the polarization.

So, in this line of thinking, if i have a 2 dimensional object, there are only 2 modes.

Sort of!  In 3D, there are three independent modes at every $\vec{k}$ corresponding to the three polarizations.  In 2D, there are two independent modes at every $\vec{k}$, corresponding to one longitudinal and (only) one transverse polarization.


*An amplitude, which is the same as specifying an energy stored in the wave.


In the Debye model, just as in the Einstein model, the energies of these collective vibrations are quantized as $E_{n,\vec{k}} = n\hbar\omega_{\vec{k}}$, and so specifying $n$ is basically specifying the amplitude of the wave.

In general, you also need


*A dispersion relation which relates the time-dependent and spatial-dependent aspects of the wave, i.e. something that gives you the frequency if you know the wavelength.  For the Debye model, we assume that the dispersion relation is linear, i.e. $\omega=vk$, which is the same as $v=f\lambda$ is the speed of sound (wave speed) in the solid.

