Questions tagged [boundary-conditions]
This tag is for questions regarding to the boundary conditions (b.c.) which expresses the behaviour of a function on the boundary (border) of its area of definition. The choice of the b.c. is fundamental for the resolution of the computational problem: a bad imposition of b.c. may lead to the divergence of the solution or to the convergence to a wrong solution.
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Charge accumulation at the boundary of two dielectrics in a capacitor
Say we have a capacitor filled with two dielectrics of different constants. The two charged plates of the capacitor are +V and -V. Does charge accumulate at the boundary of the dielectrics? In my ...
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Why is the potential due to induced charges constant?
Over the past few days, I have taken the time to read this interesting paper, where an unexpected counterexample is given that shows how the electrostatic force between a neutral metal conductor and a ...
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Boundary conditions and field quantization in AdS
While studying the AdS/CFT correspondence, one encounters very early the example of a scalar field in AdS. The general solution to the Klein-Gordon equation in the limit $z\rightarrow 0$ may be ...
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Need for boundary conditions in AdS space?
I am referencing to a passage on wikipedia's page about AdS space:
Because the conformal infinity of AdS is timelike, specifying the initial data on a spacelike hypersurface would not determine the ...
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How, in induction of electric charges, the field developed due to separation of charges exactly cancels out the external field?
I have been learning electrostatics, and I didn't quite understand why charges reside only on the surface. The charges are moved until they reach the surface after which they cannot move further and ...
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How is symmetry boundary condition related to reflection of a plane wave from a rigid flat surface?
My question is in in the context of "method of images" when applied to an incoming incident acoustic pr plane wave.
For a flat rigid surface in $XY$ system, the surface (positioned at $Y=0$) ...
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Boundary condition for density of state in momentum space
I am working on fermions in metal, where you need
$\frac{2}{V}\int_{0}^{P_{f}}\frac{L^{3}}{(2\pi\hbar)^{3}}dP$
to get the total number of occupied states, where the coefficient $\frac{L^{3}}{(2\pi\...
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What is the equation describing the boundary of a 2D charge density?
Consider a charge density $\rho(x,y)$ distributed on the 2D plane. The charge density follows the Poisson equation:
$$\nabla^2 \mathbf{\phi}=\mathbf{ \nabla\cdot E}=-4\pi\rho(x,y),$$
where $\phi(x,y)$ ...
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Acoustic wave incident to pipe wall
I would like to consider the sound incident to a water filled pipe wall. I think the pipe wall is typically considered as a rigid wall boundary, it means all the incident wave is reflected.
Is this ...
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Is Kirchhoff's scalar theory of diffraction mathematically inconsistent?
I've heard that Kirchhoff's scalar diffraction theory is mathematically inconsistent. Is this true? If so, where in the formulation does this inconsistency arise and are there ways to remedy it?
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Added mass and damping of a vertical cylinder
why the velocity potential can be written as this term:
$\Phi(r,\theta ,z,t)=\bar{h}Re[-i\sigma \phi (r,z)\bar{\xi}_3 e^{-i\sigma t}]$
where $\sigma$ is the angular frequency, $\bar{\xi}_3$ the heave ...
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Linear wave theory free surface condition [closed]
How does this formula come about, why it is free boundary conditions?
$$\sigma^2\phi-g\frac{\partial \phi}{\partial z}=0.$$
$\sigma$is the angle frequency, $\phi$ is the potential flow.
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Connectivity of random geometric graph with open boundary conditions
I have a question regarding the existence of a closed-form solution of the connectivity in terms of the radius of vertices (disks) in a two-dimensional ($d=2$) random geometric graph (RGG) with open ...
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Phase jump at medium border with EM wave
I have been reading a book about electrodynamics and I have stumbled upon the following matter which is, to me , contradictory. When the electromagnetic wave changes medium, it is subject to certain ...
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Variation of scalar action on a finite volume, boundary terms
My scalar action in $3d$ Euclidean space reads
$$
\mathcal S[\phi,\psi] = \int_V d^3x\, \psi(\vec x)\, \nabla^2\phi(\vec x).
$$
How do you correctly compute its variation with respect to the field $\...
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Does the Weyl tensor need to be continuous across a membrane with stress-energy that separates two different space-times?
Imagine that I have two $N$+1 dimensional space-times separated by a co-dimension 1 boundary. At the boundary there is a membrane containing stress-energy. The stress-energy tensor of that membrane ...
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Must open string endpoints lie on a D-brane?
In the second edition of professor Zwiebach's book "A First Course in String Theory", on page 331, on the 3rd line of section 15.1, it says
"In the presence of a D-brane, the endpoints ...
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Precise use of Neumann B.C.s in the Nambu-Goto action
Sometime ago I posted this: Logical consistence in Neumann BC in the Nambu-Goto action, whose answer was not helpful to me. Basically in classical string theory with the Nambu-Goto action, we have ...
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Spheroidal eigenvalues with shifted boundary conditions
I was studying the spheroidal differential equation in relation to calculating solutions for fields in a general Kerr background metric and, as far as I can tell, the eigenvalues $\lambda$ that enter ...
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Why does the surface integral over the $B$-field in a Stokesian loop tend to zero as the surface tends to zero (boundary conditions)?
I am confused by the standard argument used for deriving the boundary conditions at the interface of two media as told by Jackson, e.g. see here.
My question concerns the fact that Jackson says that ...
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How to use the boundary conditions of electromagnetic waves to derive the refraction law of light?
In my book it says we can use the boundary conditions of electromagnetic waves to derive the refraction law of light. How to derive it?
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Wave equation of a string fixed on both the ends
I am trying to understand the wave equation of a string fixed on both the ends, which looks like this:
$$ \frac{\partial^2 y}{\partial x^2} - \frac1{c^2}\frac{\partial^2y}{\partial t^2} - \gamma\frac{\...
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No-slip condition tangential and normal component decomposition
No-slip condition on a corrugated surface (modelled by a sinusoidal function $b(x)$))
$\vec{ u} (x,b(x)) =u \vec{i}+ w \vec{k} = 0 \vec{i} + 0 \vec{k}$
expressing in terms of the stream function :
$$\...
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Why can static solutions be found only in problems with Dirichlet boundary conditions in non-relativistic point particle mechanics?
Qmechanic in this answer says that only Dirichlet b.c.s are consistent with static solutions of point particle mechanics. Why is this so?
E.g. The standard classical harmonic oscillator problem with b....
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Least action principle and uniform motion
I'm trying to apply the principle of least action to the case of a uniform motion under no potential. Assume the object starts with initial velocity $v_0$, moving from point $A$ to point $B$. We know ...
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Does a rigid body rotates in an irrotational flow?
Consider a given flow field such that vorticity $\mathbf{\omega} = \text{Curl} ~\mathbf{u} = 0$. In this case, we can consider an arbitrary shape fluid element and look at how it evolves in short ...
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Infinite wire with alternating current and conducting half-plane
I'm kinda confused about the following. Suppose I have a harmonic current-density of
$$j=j_0 e^{i\omega t}=I_0 e^{i\omega t} \, \delta(x)\delta(y-h) \, e_z$$
parallel to the $z$-axis (at $x=0$ and $y=...
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Fermionic path integral with boundary
Given a path integral:
$$K(\eta,\xi) = \int\limits_{\psi(0)=\eta}^{\psi(1)=\xi} e^{\int_0^1\dot{\psi}(t)\psi(t) dt} D\psi\tag{1}$$
where $\psi(t)$ are a real Grassmann fields.
I get two answers ...
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Extending Geodesic Across Boundary in Discontinuous Spacetime
Imagine a metric space to be discontinuous: a Schwarzschild outer metric that changes to a flat FLRW metric once inside the boundary of a $2$-sphere of fixed $r$. If an object (or ray) just hitting ...
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Aharonov-Bohm effect and periodic boundary conditions for particle on a ring
For a particle on a ring, we have the periodic boundary conditions $\psi(\phi+2\pi)=\psi(\phi)$. If we also have a magnetic field penetrating perpendicularly the ring, then when the particle goes ...
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How to add walls or other solid obejcts to Eulers equations? [closed]
I've recently started learning about the physics of fluids and I've found that euler's equations exist and that I can use them to compute the flow of fluids with zero viscosity. But I have a problem ...
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Polchinski's doubling trick for extending open string theory to the whole complex plane
Open string theory can be described on the upper-half complex plane. To simplify the description of open string theory, Polchinski asserts (eq. 2.6.28 in his Vol. I String Theory book) that it is ...
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On obtaining the standard boundary condition for the radial equation
The solution of spherically-symmetric one-particle problems is often facilitated by looking for solutions of the form
$$\psi(r,\theta,\phi) = R(r)Y_l^m(\theta,\phi)$$
where of course the $Y_l^m$ are ...
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Schroedinger equation with derivative boundary conditions [duplicate]
We had the following exam question in our quantum theory undergrads course:
Solve the time independent Schroedinger equation for the following Hamiltonian:
$\hat{H} = \frac{\hat{p}^2}{2m}$ for $x \in [...
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Boundary condition for finite material
Consider a solid state material in some domain $[-L,L]^2$ described by a Hamiltonian
$$ \widehat{H} = \frac{\widehat{p}^2}{2m}+V$$
I wonder what kind of boundary condition I have to impose at the ...
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Why is there a pressure node at the open end?
The pressure at the open end is equal to that of the atmospheric pressure. So how can it be a pressure node when the pressure is not 0? It is the atmospheric pressure.
Edit: Is there an intuitive ...
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Why must there be an antinode at an open end? [duplicate]
Consider the above diagram.
I am currently learning about standing waves in an open and closed tube. To me, it is trivial why at a closed end, there must be a node, as particles are not free to move ...
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How to calculate a momentum space of a semi-finite lattice?
If we have a 2D square lattice of lattice constant a whose $x$ axis has only $N_x$ cells each with one atom and no with spin degeneracy, and periodic boundary conditions on $y$ with $N_y$ cells along ...
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Alternate interpretation of a free rotating string in 2 spatial dimensions
In Zwiebach's A First Course in String Theory, (page 140, 2nd ed.), the case of a straight open string rotating about a fixed point is analyzed.
During the analysis, the condition of periodicity (...
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How do we apply the electrostatic boundary conditions in the method of images for an infinite grounded conducting sheet?
Consider the figure 3.10. How do we apply the electrostatic boundary conditions here in order to get the expression for induced surface charge density. Comparing with figure 2.36, what would be E(...
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For an electric field passing through a charged sheet, what would be the boundary conditions if E(above) and E(below) were both pointing downwards?
Consider the diagram given in Griffith's figure 2.36. we can see that boundary condition is E(above) - E(below) = $\sigma/ \epsilon_0$. According to Griffith's the direction of n is always from "...
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Gauss's Law over two non-symmetric surfaces
Let's say I have a sphere conductor with an electric charge $q$ and radius $R$. The sphere is inserted between two spaces: one material with relative permissivity $\epsilon$ (below) and the other is a ...
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How many Green's functions exist for a given differential operator?
Let's say we have a scalar field $\psi(x)$ that satisfies a field equation
\begin{equation}
\nabla^\mu \nabla _\mu \psi (x) + V(x) \psi (x)=\rho(x)
\end{equation}
where $V(x)$ is some potential and $\...
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Confusion on variation of $\dot{q}$ while applying Hamilton Principle to Lagrangian Mechanics
We restrict that $$\delta q\mid _{t_{1}}= \delta q\mid _{t_{2}}=0$$ while applying Hamilton Principle ($\delta\int_{t_{1}}^{t_{2}}Ldt=0$) to get Euler-Lagrange’s Equations. Hence adding a $$\frac{d}{...
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Existence of Green's functions with specific causality conditions
This is a follow-up to my previous question here: Green's identity for arbitrary differential operators.
Given some scalar field theory with field $\psi$ and source $\rho$ with field equations
\...
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Green's identity for arbitrary differential operators
If we have a scalar field $\psi$ that satisfies an equation $\nabla^\mu \nabla_\mu \psi = \rho$ where $\rho$ is some known source we can use Green's identity to express it as
\begin{equation}
\psi (x)...
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About the usage of the Green's function
This question is very basic, but I'm a little confused about this topic (I "learned" it during my bachelor's degree but never used it before).
I have an electrostatic problem described by ...
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Diffuse boundary condition for fermigas/bosongas
I'm doing math, usually working only in the classical framework, but my background in quantum physics is close to zero so please forgive me if the question does not make sense.
I was looking into ...
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Boundary conditions required to get a unique solution to Schrodinger Equation
If we are considering the scattering of an electron plane wave from a crystal specimen, we could start with the time-independent Schrodinger equation (TISE) as our governing equation:
$$ \{\nabla^2 + ...
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Killing vectors for a boundary condition?
So here's something I was wondering about. Let's say I have to have boundary conditions in flrw metric which does not leave the universe isotropic and homogeneous. This would go against observations. ...
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