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Questions tagged [boundary-conditions]

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Boundary conditions for $\mathbf D$ and $\mathbf H$

I understand the derivation for the boundary conditions for $\mathbf B$ and $\mathbf E$ as it was explained to me in Griffiths, but Griffiths states the following: $$H_{\text{above}}^{\bot} - H_{\...
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Robin Boundary Conditions in Electrostatics

Are robin boundary conditions ever used in electrostatics? I can find three references on the internet that say they are: https://en.wikipedia.org/wiki/Uniqueness_theorem_for_Poisson%27s_equation "...
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1answer
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Deriving the path integral for periodic boundary conditions

I'm thinking about path integrals with the Euclidean time formalism, where I have partition function $Z=\operatorname{Tr} e^{-\beta \hat H}$. I'm used to the following derivation of the path integral: ...
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0answers
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Realistic vacuum boundary conditions in fluid mechanics?

What are some realistic boundary conditions between a fluid and vacuum? Is there an interface or does the fluid kind of spray out into the vacuum?
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1answer
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Deriving the Electrostatic boundary conditions

When deriving the electrostatic boundary conditions for any charge distribution (to my knowledge at least), Griffiths in his textbook references this illustration: So, when considering the boundary ...
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Variational formulation of Maxwell equations with interface/boundary conditions

Consider $\Omega = \Omega_1 \cup \Omega_2$, where $\Omega _1$ and $\Omega_2$ are two different media with conductivity and permeability \begin{equation} \sigma= \begin{cases} \sigma _1 & \text{in ...
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5answers
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Is tangential component of $\mathbf{B}$ undefined at the boundary of two media?

Tangential component of $\mathbf{B}$ is discontinuous at the boundary of two media. Does this mean that tangential component of $\mathbf{B}$ is undefined at the boundary of two media? If yes, then: $...
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1answer
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Uniqueness Theorem and the 1D Infinite Square Well

Consider the 1D infinite square well problem: $$\frac{d^2\psi (x)}{dx^2} = -k^2\psi (x)\tag{1}$$ along with the boundary conditions $\psi (0) = \psi (L) = 0$. This seems to be a well posed problem ...
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1answer
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Why is the $k$-space in multiples of $2\pi/L$?

So when you find the solution to the Schrödinger equation you get that the wave function can have $k=n\pi/L$, $n=1, 2,3 \dots $ The problem I have is that when calculating the density of states of a ...
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0answers
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Boundary conditions of spun string

Problem: Consider a string with mass per unit length $\rho$ and length $L$. It is spun about one end, with angular velocity $\omega$ , such that the motion is in a plane (we neglect gravity). Let $x$...
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0answers
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Polarization depending phase shift of EM waves on reflection off denser medium

I've seen this video: https://www.youtube.com/watch?v=JjGep0W8ZHI, There it is explained that an electromagnetic (here radio) wave has a phase shift if it was radiated in horizontal polarization, but ...
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1answer
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Dealing with the electrostatic boundary condition

In Griffiths, it is noted that there is a discontinuity in the electric field for a material with a surface charge density. What is the significance of this boundary condition in practicality when ...
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1answer
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Electric potential and field due to a continuous charge distribution

(1) The electric potential due to a continuous charge distribution is: $$\psi=\int_V \dfrac{\rho}{r}\ dV$$ To calculate this integral $\rho$ must be continuous over $V$. But $\rho$ is discontinuous ...
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1answer
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Reflection of transverse wave from free end?

I have been using David Morin' drafts on waves along with French's wave book and Fox Smith's book for my undergrad wave course and one thing I don't understand is the physical intuition behind ...
3
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1answer
68 views

Diffusion equation with time-dependent boundary condition

I was trying to solve this 1D diffusion problem \begin{equation} \dfrac{\partial^2 T}{\partial \xi^2} = \dfrac{1}{\kappa_S}\dfrac{\partial T}{\partial t}\, , \label{eq_diff_xi} \end{equation} with ...
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Is the electrostatic potential also undetermined by a constant in 2d periodic boundary conditions?

In 3D periodic boundary conditions (PBC), the electrostatic potential is underdetermined by a constant. Is this also true for any other periodicity as 2D or 1D?
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Linear elasticity boundary conditions

I came across this post from the computational science board: https://scicomp.stackexchange.com/questions/26495/well-posedness-of-elasticity-boundary-conditions I agree with the posted answer, but I ...
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1answer
41 views

Changes in boundaries with the application of Faraday's law

Reviewing Faraday's law of an induced electric field due to a changing magnetic field $$ \nabla \times E = -\frac{\partial B}{\partial t}$$ In integral form via application of Stokes theorem: $$ \...
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Strain field and periodic boundary conditions

Let's say I have a lattice, and I impose periodic boundary conditions. I want to construct a tight-binding model on a strained lattice, and I can determine the change in the hopping parameter based on ...
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0answers
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How to choose the boundary condition for Maxwell's equations in the vacuum?

I need to solve the Maxwell's equations with sources in the vacuum numerically. The simplified problem is as following. A charged particle moving along the $z$ direction with speed $v_z$. Then, it ...
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1answer
42 views

Why normal component of particle velocity must be continuous at boundary?

I have problems for understanding the following: Source: https://mycourses.aalto.fi/pluginfile.php/393850/mod_resource/content/1/Lecture7.pdf Why there would be a vacuum at the boundary? I dont see ...
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0answers
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Would Bekenstein bound disappear in some holographic models?

In Holographic principle models there's a limit to the information that the system can store known as the "Bekenstein bound". In physics, the Bekenstein bound is an upper limit on the entropy S, or ...
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1answer
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Pridictions and Observational evidences of No Boundary Condition of S.Hawking

Reference: http://www.hawking.org.uk/the-beginning-of-time.html Predictions of No Boundary Condition proposal: 1) Irregularities in the current universe same as the Big Bang theory predicts and it ...
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1answer
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Boundary terms and Symmetries

Consider Maxwell-Chern-Simons theory in 2+1 dimension, with Lagrangian $$L = -(1/4)F_{\mu v}F^{\mu v} + (m^2/4) \epsilon_{\mu v \rho}A^\mu F^{v \rho},$$ when I make a gauge transformation $A_\mu \to ...
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2answers
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Wave simulation without reflection on the boundaries [duplicate]

I would like to numerically simulate a wave (let's say in a string) with different boundary conditions: Fixed endpoints Periodic Boundless $\varphi(x, t)$ is the value of the wave (vertical ...
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1answer
40 views

Solving Lagrangian given initial and final coordinate

Consider a Lagrangian $$L=L\left(q, \dot{q}\right)$$ I can use the Euler-Lagrange equation to find an expression $$\ddot{q}=A\left(q,\dot{q}\right).$$ Let's assume that the equation can be ...
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2answers
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How does Hamilton's Principle give us the path taken?

We defined the action as: $$\mathcal{S}(t)=\int_{t_1}^{t_2}\mathcal{L}(q_i,\dot{q_i},t) dt$$ where $q_i(t_1)$ and $q_i(t_2)$ are known and fixed. Hamilton's principle states that the path that is ...
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0answers
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Current density on Perfect Magnetic Conductor

I know that PMC boundary condition requires tangential magnetic fields to be 0. I also learned that PMC condition requires tangential current density to be 0. Is this condition a result of 0 ...
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0answers
28 views

Deriving Canonical Transformation from Generating Function using Principle of Stationary Action

In Hamill's "A Student's Guide to Lagrangians and Hamiltonians", section 5.2, the equations for a canonical transformation $(q,p) \to (Q,P)$, induced by the generating function $F(q,Q,t)$ are derived ...
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0answers
24 views

SOUND WAVES :: organ pipes [duplicate]

Why doesn't sound wave escape in a open end pipe, why does it reflect again at open end of organ pipe when it can just move outside.
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0answers
41 views

What is happening to a wave at a media boundary?

Say we have a light wave going from air to plastic, refractive indexes 1 and 1.5 respectively. What exactly is happening to the properties of the wave and why? Taking wavespeed as v, frequency as f, ...
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1answer
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Confusion regarding a basic boundary value problem

Consider a rectangular area, defined by the region $x=0,x=a,y=0,y=b$. Now, there is a potential $\phi(x,y)$ defined in this region, which satisfies, $\nabla^2 \phi=0$, and the following boundary ...
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3answers
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Why, in an open or half-open pipe, must an open end of a standing sound wave have a pressure of zero?

I believe this question was asked in some form before, but I'm not clear on the answer. If a sound wave must equal air pressure when it exits a tube, why is it possible that at many points after the ...
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0answers
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How can we predict how a system evolves using the stationary action principle even though we need to specify the final state? [duplicate]

The stationary action principle states that a system evolves between a fixed initial and fixed final configuration in such a way that the action is stationary. But isn't the final configuration what ...
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0answers
51 views

Green's function for infinite square well

The Green's function can be given in terms of left and right solutions. $G(x,x';k) = \frac{1}{W}\left(\Psi_{L}(x_{<})\Psi_{R}(x_{>})\right)$ But I don't understand how to determine these left ...
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0answers
23 views

Intuition for construction of a wave reflected from a general corner reflector

Consider a corner reflector with angle $\alpha$ between its semi-planes: Let a plane wave come from the bottom into this reflector (possible at an angle). The objective is to find the total wave ...
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0answers
46 views

How is Poisson's Equation solved numerically?

This question is of pure interest. I would like to know, how a mixed boundary value problem like the following can be solved numerically: Lets say I have two conducting plates (not necessarily ...
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1answer
89 views

Schrödinger Equation for a freely falling body near the surface of Earth

Near Earth's surface the Schrödinger equation of a freely falling particle takes the form, $$ \frac {-\hbar^2}{2m} \frac {d^2 \psi (y)}{dy^2} + mgy\psi (y) = E \psi (y). $$ Putting $k=\frac {\sqrt {...
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1answer
39 views

Heat equation volume source vs. heat flux boundary condition

I want to solve the heat equation in the 3D unit sphere $B$ with a general heat flux boundary condition, no volume sources and some given constant initial temperature: $$ \rho c_p\partial_t T - \...
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0answers
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Couette Flow encountering an airfoil obstruction

Im interested in what would happen to an airfoil place within a Couette type fluid flow bounded between a fixed and moving boundary plate If we say the plate is infinite to establish a steady state ...
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0answers
51 views

Commutation of differential operators with boundary conditions

First post ever. Let's see how this goes... My question concerns the commutation of differential operators in the presence of boundary conditions. If it is of any help, this is relevant to me in the ...
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0answers
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Phase difference of waves [duplicate]

Why do light and sound waves under change in phase on reflection and is the change in phase for displacement and pressure wave the same in case of sound waves??
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1answer
62 views

Mass formula for open string with mixed boundary conditions

I want to give an expression for the mass formula of an open string with has Neumannn condtion in $m$ directions and Dirichlet in $n$ directions $X^{i}(\sigma ^{1}=0)=x_{0}^{i}, X^{i}(\sigma ^{1}=\pi)=...
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1answer
50 views

Question about Mode expansion of free compact boson

$(1+1)$-Dim free compact boson, Lagrangian is $$\mathcal{L}= \frac{1}{2}(\partial_\mu\phi(\sigma,t))^2$$ with $\phi(x,t)\sim\phi(x,t)+2\pi r$ and periodic boundary condition along $x$, i.e. $\phi(\...
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0answers
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Differentiating D3 brane worldvolume theories with NS5 brane and NS5 antibrane boundary conditions

In 'Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory', Gaiotto and Witten derive boundary conditions for the worldvolume theory of the D3 brane. In particular the boundary conditions (...
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0answers
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Covariantly constant Lie algebra-valued field with Dirichlet boundary condition

I have a question about a statement in Witten's paper 'Analytic Continuation of Chern-Simons Theory' (https://arxiv.org/abs/1001.2933). On page 66, below equation 4.13, he discusses a Lie algebra-...
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0answers
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EM Induction in non-uniformly conducting ohmic toroid, thought experiment

Assume that there's a conducting toroid with radius of revolution $R$ and an own radius of $D$ so that the cross section of the toroid is given by $\pi D^2$ Assume that there's a circular region of ...
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0answers
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Fluid dynamics - boundary later equation $f'''+ff''-f'^2+\theta = 0$ and $\theta''+Prf\theta' = 0$

Our lecturer gave us a system of boundary layer equations: $$\begin{align}f'''+ff''-f'^2+\theta &= 0 \\ \theta''+Prf\theta'&=0 \end{align}$$ subject to boundary conditions: $$f=f'=0, \...
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2answers
95 views

Energy dependence on boundary conditions for particle in a box

I am taking a course in solid state physics, and I have some trouble with the "hard wall" and the periodic boundary conditions for a particle in a box. The thing is that we obtain, for a box of ...
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1answer
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What is meant when we say “any solution is *the* solution” due to the uniqueness theorem?

I understand the proofs for the uniqueness theorems in electrostatics, but I'm having trouble understanding their application to problem solving. A classic example is a system of concentric shells of ...