37

The waves will not travel forever. Water particles moving against and around each other will have friction, and the friction will cause motion energy to be converted to heat (which will dissipate throughout the water and air). The wave will eventually cease to exist unless energy is added.


22

To answer this, I would appeal to the general principle which we call the 2nd law of thermodynamics. One way of expressing it is that the entropy of an isolated system cannot decrease. This means that in order to keep going for ever, a wave motion would have to involve no entropy increase. But almost all processes involve some increase of entropy, and in the ...


19

I doubt if anyone has come up with a complete explanation, but some laboratory simulations have created similar patterns. They happen if the central and surrounding areas in a flat, circular disk of fluid have different velocities. Emily Lakdawalla at The Planetary Society covers it at this site. She also explains how other patterns (triangles & ...


13

Of course, no. Tsunamis are a series of pressure waves with a longitudinal mode and have much higher wavelengths, speed, and period than the normal ones. Normal ocean waves only involve motion of the uppermost layer of the water, but Tsunami waves involve the movement of the entire water column from surface to seafloor. However, they are still akin to ...


8

A magnetic field configuration corresponds to a knot when for two magnetic field lines given by the parametric curves: $\mathbf{x}_1(s)$ and $\mathbf{x}_2(s)$, the Gauss linking number $$L\{x_1, x_2\} = \int ds_1 ds_2 \frac{d \mathbf{x_1}(s_1)}{ds_1} .\frac{\mathbf{x_2}(s_1) - \mathbf{x_2}(s_2)}{|\mathbf{x_1}(s_1) - \mathbf{x_2}(s_2|^3}\times \frac{d\...


7

No, I believe the Standard Model does not predict monopoles as a result of symmetry breaking. This is because the symmetry breaking $\mathrm{SU(2)} \times \mathrm{U(1)} \rightarrow \mathrm{U(1)}$ does not allow for topological solitons to exist. Edit: $\pi_2(\mathrm{SU(2)} \times \mathrm{U(1)}/\mathrm{U(1))}=\pi_2(S^3)=0$ Source: To be or not to be? ...


7

First let me refer you to Eric Weinberg's book where the instanton moduli space is described in more detail. Principal bundles over 4-dimensional Riemannian manifolds are classified by the second Chern class = Instanton number and the t' Hooft discrete Abelian magnetic fluxes. Please see the following Lecture notes by Måns Henningson. t' Hooft fluxes ...


7

This is a situation where knowing the history of the terminology can be helpful. The QFT/string theory terminology comes from algebraic geometry, where the term moduli space is used for any space whose points correspond to some kind of geometric object. The projective space $\mathbb{P}(V)$, for example, is the moduli space of lines in the vector space $V$. ...


7

A soliton is a localized, non-dispersive solution of a nonlinear theory in Euclidean space. It certainly is a real object: you have a famous story about a certain John Russell who observed soliton-like waves made by a boat on a river (wikipedia knows everything about it!) The so-called morning glory clouds in Australia (http://en.wikipedia.org/wiki/...


7

I'm not sure what intuition you are seeking in similarities of mathematical modeling... It's like intuition about the similar beat of two very different pieces of music? I fear it is all in the math. That is, the KdV being a solvable equation with the prototypical "magical" soliton solution $v(x,t)=-2c \operatorname{sech} ^2 (\sqrt{c}(x-ct))$, this shape ...


6

A particle is not a wavepacket. And there are no particle states for interacting theories. We define particle states in QFT by expanding the free field into its Fourier modes and using these modes as creation/annihilation operators for particle states - the mode of momentum $p$ creates the particle state $\lvert p\rangle$ with momentum $p$. The Hilbert ...


5

The following interpretations are taken from Thorne [2014]. Chapter 17, entitled Miller's Planet, discusses the issue of the large waves on the water planet in the movie Interstellar. There Kip mentions that the waves are due to tidal bore waves with height of ~1.2 km. In the appendix entitled Some Technical Notes, Kip estimates the density of Miller's ...


5

In a linear wave equation, there is nothing to pull a pulse or envelope of running waves apart. But there is nothing to hold it together, either. A minor disturbance such as a small obstacle or some dispersion, will change the waveshape, or break it up, such as losing some of its energy to outward spherical waves from the obstacle. Two or more pulses in ...


5

1) Suppose that you have two configurations (here I've used Coulomb gauge with euclidean time $\tau$): $$ \tag 0 A_{i}(x) = \begin{cases} 0 = U^{(0)}\partial_{i}(U^{(0)})^{-1}, \quad \tau = -\infty \\ U^{(1)}\partial_{i}(U^{(1)})^{-1}, \quad \tau = \infty\end{cases} $$ Such situation describes tunneling between vacua with topological charges $0$ and $1$. ...


5

No, "real" plain waves do not exist in nature and neither does anything "exist" the way a physical theory describes it. That's about as trivial as it is irrelevant. We are not performing experimental mathematics here. In physics we are merely finding approximate explanations to natural observations. My first theory professor said it this way to the entire ...


5

The topological charge which is the space integral of the zeroth component of the topological curent is responsible for the stability of the kink: A configuration with a nonvanishing topological charge cannot evolve into a vacuum solution by means of any Hamiltonian, because Hamiltonian evolution is continuous, thus cannot change the topological charge. ...


5

If we start in the unbroken phase, and if there are multiple degenerate vacua after spontaneously breaking the symmetry, generically we should have domain walls. The reason is that as we pass the phase transition, there will be random fluctuations in the field(s) that will make different values of the order parameter locally more favorable, so that different ...


4

I didn't go through all of your equations. However, if you take (1), differentiate it w.r.t $\lambda$ and set $\lambda = 1$, then since $\lambda=1$ is the stationary point $E'(\lambda)|_{\lambda=1} = 0$. This is equation (2)


4

I) Consider a Yang-Mills type theory with gauge group $G$. In principle we can consider the same theory with its covering group $\tilde{G}$, with $\pi:\tilde{G}\to G$. The covering group is by definition simply connected: $$\tag{1} \pi_1(\tilde{G})~=~\{\bf 1\}.$$ Any representation $\rho$ of $G$ can naturally be viewed as a representation $\rho\circ \pi$ ...


4

A (generalized) 't Hooft-Polyakov monopole and a Dirac monopole with a Dirac string attached are two types of magnetic monopoles, which differ in several ways, as OP and user ACuriousMind correctly state. On one hand, a (generalized) 't Hooft-Polyakov monopole is a regular, soliton-like, finite-energy solution to the classical Euler-Lagrange field ...


4

OK, perhaps the notation in Ref. 1 is a bit confusing. Let us elaborate on Derrick's No-Go theorem: Derrick's No-Go theorem: For the number of spatial dimensions $D>2$, the only time-independent finite-energy solutions are ground states. In a nutshell, the idea of the proof is to derive a necessary condition by a simple 1-parameter scaling argument. (...


4

Quantum fluctuations in the kink sectors of Sin-Gordon and the quartic interaction theory are described by reflectionless Pöschl-Teller-Operators, which form a SUSY-Chain with $N$ elements. The second quantization is then obtained by the spectral decompostion of those operators that are connected through annihilation and creation operators. Sin Gordon ...


4

Plane waves are useful because we can take any physical function of space, e.g. some field, and Fourier transform it to represent it as a sum (well, integral) of plane waves. This is often a very useful way to approach complicated problems. For example Fourier developed the technique as a way of solving the heat equation, and it's the way quantum field ...


4

The usual soliton for the NLS is $$ \psi(x,t)=e^{ikx-i\omega t}\sqrt{\frac{\alpha}{m\lambda}}{\rm sech}(\sqrt{\alpha}(x-Ut) $$ where $m$ is the mass and $\lambda$ is the coeficient of the $|\psi|^2\psi$ term. The parameters $\alpha$ and $U$ are arbitrary. Your book has interchanged the role of $x$ and $t$. I suspect that it deals with optical fibres in ...


4

You don't explain what you mean by "topological charges" but I expect you mean the sum of the Hopf indices of the zeros of a tangent-vector field on a manifold. The resulting Poincare-Hopf theorem says that the sum of these numbers is indeed $\chi$. A discussion and sketch of a proof can be found starting on on page 547 of my book with Paul ...


3

In one dimensional case you have the oscillation theorem: $n$-th level has $n$-zeroes. As a special case ground state has no zeroes. It doesn't generalize on the case of several dimensions however there is still a theorem that the ground state is non-degenerate and has no zeroes. Thus the observation that the Goldstone mode vanishes somewhere means that it ...


3

Type I' string theory is equivalent to M-theory compactified on a line segment times a circle, i.e. M-theory on a cylinder. M-theory on a line segment only is the Hořava-Witten M-theory, a dual description of the $E_8\times E_8$ heterotic string, because every 9+1-dimensional boundary in M-theory has to carry the $E_8$ gauge supermultiplet. The extra ...


3

It's a bit hard to be sure without seeing the whole text, but it looks like they're discussing the problem of of obtaining finite minimum energy solutions of a gauge/Higgs system. In 3 space dimensions, for example, for the Georgi-Glashow model, $$ \mathcal{L}= \frac{1}{2}Tr(F^{\mu\nu}F_{\mu\nu})+Tr(D_{\mu}\phi D^{\mu}\phi)-\frac{\lambda}{4}(|\phi|^2-v^2)^2 ...


3

For the pure e.g. $d=4$ gauge theory instanton and gauge field perturbations around it, there is no negative mode – the counterpart of the bound state. It's the only one among the 3 classes that is absent here. One may see this absence by noticing that the gauge theory may be embedded into a supersymmetric theory with the same gauge-field degrees of freedom,...


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