2
votes
1answer
76 views

Linear vs. quadratic dispersion relation

In wave mechanics the dispersion relation between frequency $\omega$ and wave number $k$ is linear: $$\omega_n=c k_n$$ But in quantum mechanics, based on Schrödinger's equation, one can show that we ...
0
votes
0answers
38 views

Wave Packets, Group velocity, and Phase velocity [duplicate]

Mathematically, you find that the wave function of a particle $\Psi (x,t)$ moves with the same velocity as the velocity of the particle ($v_{particle} = v_{group}$). Is there a reason why the ...
2
votes
0answers
45 views

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 ...
2
votes
2answers
84 views

Interpretation of dispersion relation

In my research, I found that my system has the following dissipation relation: $$\omega^2=k^2+k_0^2\ , $$ where $k_0^{-1}$ is an intrinsic lengthscale of the system and the units are chosen so that ...
0
votes
0answers
40 views

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: $\Delta x = 1$ ...
1
vote
1answer
220 views

Dispersion relation for TE and TM waves in general anisotropic medium

I want to calculate the dispersion relation (the relation between $\bf k$ and permittivity and permeability tensors and $\omega$) for a TE and a TM wave with wave vector $\mathbf k=k_x\mathbf {\hat ...
1
vote
1answer
110 views

Why linear wave equation does not have solitonic solutions?

As many people define solitary waves they are localized pulses that propagate without changing the shape. As far as I know the same pulses exist in ordinary wave equation ! why should we look for ...
3
votes
1answer
130 views

More extensions of the wave equation for dispersion

The Phys.SE question Minimal Extension of Wave Equation to Include Dispersion extended the wave equation for only a very simple form of dispersion. However, what about more complex dispersion ...
7
votes
1answer
122 views

Minimal Extension of Wave Equation to Include Dispersion

Let's say you are modeling some process with the wave equation $\frac{1}{c^{2}}\frac{\partial^{2}\psi}{\partial t^{2}} = \nabla^{2}\psi$. You wish to improve your model by including dispersive ...
0
votes
0answers
69 views

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 ...
1
vote
1answer
141 views

Dispersion-less media

As far as I know, vacuum is the only dispersion free medium for electromagnetic waves. This makes me wonder if there are any other dispersion free media for these waves? (Experimentally established or ...
21
votes
3answers
3k views

Why do prisms work (why is refraction frequency dependent)?

It is well known that a prism can "split light" by separating different frequencies of light: Many sources state that the reason this happens is that the index of refraction is different for ...
1
vote
1answer
233 views

Definition of energy

What is the definition of energy $E$ given a dispersion relation $\omega=\omega(k)$ where $k=|\vec k|$ and $\omega$ is not necessarily linearly proportional to $k$? What about momentum $\vec p$? This ...
1
vote
0answers
56 views

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 ...