Dispersion refers to the frequency dependence of the properties of a wave.

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PDE from dispersion relation?

Suppose I have knowledge of a system's dispersion relation $f(\omega,k)$. Is it possible to recover the underlying PDE describing the system? Can I simply use the replacement $k=-i\nabla$, ...
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Equations of motion of displacement field

We have an action: $$S[\boldsymbol{u}] = \frac{1}{2} \int dt \int d^3x \left\{ \mu (\frac{\partial u_{i}}{\partial t})^{2} - \nu (u_{ii})^{2} - \rho(u_{ij})^{2}\right\} $$ Where $u_{ij} = ...
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Why do dispersive waves get wider?

Consider the two waves $$y_1=Acos(\omega_1 t+k_1 x), \tag{1}$$ $$y_2=Acos(\omega_2 t+k_2 x), \tag{2}$$ where $\omega_i=k_iv(k_i)$ for $i=1,2$ so we have a dispersive medium. Then if we take their ...
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Rayleigh length determination for Laguerre-Gaussian Modes

Recently I have measured the Rayleigh length of a Gaussian electron beam probe in a scanning electron microscope, using the function: $$w(z) = w_0 \times \sqrt{1 + (z/z_r)^2}$$ Where $w$ is beam ...
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How to calculate the dispersion relation for a wave equation with non-constant speed of wave propagation?

Specifically, it is a one-dimensional wave equation for waves on a string with a non-constant cross-section, i. e. $$S(x)=S_1+S_2 \cos{2x}; \qquad c(x)=\sqrt{F/\rho\, S(x)}.$$ Separating the variables ...
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What is the difference between Lehmann-Kallen and Dispersion relation?

I know that the Lehmann-Kallen (LK) form of an operator concerns just that, an operator. But the LK is very similar in form to dispersion relations found in analytic S-matrix theory.
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Third-order dispersion of glass

When working with short laser pulses, it is important to compensate the dispersion of a pulse passing through a material because it changes the pulse's shape. The effect of dispersion can be described ...
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Deriving Graphene energy dispersion in tight binding model

I'm trying to get into graphene, in detail, I try to derive the elec. energy dispersion. Sadly, I am not that familiar with condensed matter QM by now, so I got some basic questions and I hope to find ...
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What phenomenon is responsible for the evolution pattern of waves created by waterfalls?

I have been fascinated lately by the pattern of the waves created by a waterfall in my town. Specifically, the pattern shows a gradual decrease in the density of the waves as they travel away from ...
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Dispersion of prism: Graph of angle of deviation versus wavelength

The figure below is the schematic diagram fro measuring angle of minimum deviation using a collimator. High pressure sodium vapor lamp is used as a light source in this experiment. At position of ...
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Refractive index and electric susceptibility

Suppose we have a complex refractive index $n_{ref}=n+ik$ whose value is given at a precise frequency $\omega_l$ from experimental data. We know that the imaginary part is responsible for the ...
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Dispersion of electrons in conductive medium

Say you have a 1'x 1'x 1' cube of conductive metal. If you placed an electrode on the center of one of the sides, and quickly turned it on and off, how would the electrons disperse? Both initially ...
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Dispersing ultrasound in a tube - patters

I am looking for patterns to efficiently disperse reflected ultrasound in the range of wavelengths 1mm to 4mm within the interior of a narrow tube (I do not want wall reflections). For various reasons ...
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Two-Dimensional Tight-Binding Dispersion Relation

As in my last post, I am doing out a calculation in Giamarchi's Many-Body text: http://dpmc.unige.ch/gr_giamarchi/Solides/Files/many-body.pdf. This time, I am going through the derivation of the ...
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Drawing the wave function for a wave packet

I have the following infotmation: Amplitude-Function: $U(k) = Ae^{-a|k-k_0|}$ Wave Function: $u(x,t) = \frac{A}{\sqrt{2\pi}} \frac{2a}{(x-vt)^2+a^2}e^{ik_0(x-vt)}$ Uncertainty in x: $\Delta x = 1$ ...
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Mass frequency problem

For Dispersion relation , according to Gaussian profile, the author in the equation 3 wrote as $\omega= \left(k^2+\omega_{mass}^2\right)^{1/2}$ My question is what is mass frequency and how it arose ...
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antiferromagnetic spin wave

I have a hamiltonian that is derived from a spin wave energy dispersion calculation for a nearest neighbor interacting cubic antiferromagnet. After a Holstein-Primakoff transformation and making a ...