# Pin point sound wave on phonon dispersion

Imagine a sound wave of 1 Mhz is pushed into a material, in order to plot this on a phonon dispersion relation (E-K plot) - should I convert the 1 Mhz into wavelength(lambda) and find the equivalent K(=2*pi/lambda). Then use the intensity of the sound to find the energy and using this energy and K to plot it on to the dispersion curve ?

Probably K is extremely small, is this why speed of sound is estimated at near the gamma point or beginning of the Brillouin zone?

You need to find the $k$ vector in the medium. $$k = \frac{2\pi f}{v_m}$$ where $v_m$ is the speed of sound in the medium, which can be found from the dispersion relationship $$v_m = \frac{1}{\hbar}\frac{\mathrm{d}E}{\mathrm{d}k}$$

The intensity has nothing to do with phonon energy. Intensity provides the total energy incident in a unit area.

Typically the wavelength of a sound wave is many many times the inter-atomic spacing, so indeed the wavevector is small, and the $\Gamma$ point is the appropriate place to be.

But I would not call a 1 MHz disturbance a sound wave, as it is well beyond the range of human hearing. But that's just vocabulary.

First, yes the speed of sound is the limit of the group velocity at small wave vectors. In the limit of small wave vectors (long wavelengths), the material acts like the continuum material we see at a macroscopic scale. (You don't see any atoms with your eye. Macroscopic materials appear to be infinitely divisible.) The speed of these long-wavelength waves is "the speed of sound".

However, be aware that there is not simply a single speed of sound in solids. Solids support many types of waves (longitudinal, transverse, Rayleigh, etc.), and each has its own "speed of sound". In contrast, there is only one type of wave in air (a pressure wave) so there is only one speed of sound. Things can be even more complicated still because materials can be anisotropic; the wave speed is different in different directions.

Now that all that is said, let's examine your question. All you specify is a frequency. You also need to specify a mode and a direction. A 1 MHz transverse wave will have a different speed (and thus a different wavelength) than a 1 MHz longitudinal wave. Likewise, what direction is the wave propagating in? This makes things complicated since the group velocity and the wave vector can be in different directions for an anisotropic material.