Questions tagged [boundary-conditions]

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Boundary condition for a bulk-surface and bulk-bulk diffusion reaction system

Consider this simple example below and the corresponding geometries. I simplified these equations from the real system. Geometry 1 The first geometry is a sphere. Inside this sphere a species $b(t,...
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44 views

Neumann Boundary Conditions for the Open string and the energy momentum tensor

I read in Polchinski's book, "String Theory", page 56, that for the open string the energy momentum tensor satisfies equation (2.6.26) at a boundary $$ T_{ab}n^a t^b=0 \,, $$ where $n^a$ and $t^a$ are ...
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2answers
35 views

How are standing wave patterns created in tubes?

I get how standing wave patterns are created in strings: At a certain resonance/natural frequency of the string, a standing wave is created, increasing the amplitude of the sound, and standing waves ...
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19 views

Why pressures on interface between two liquids are the same

Assume we have a cylindrical container that has a lower part filled with some liquid, say, water, and an upper part filled with another liquid, say, air. Assume that the interface between them is a ...
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37 views

Numerically solving unbounded Fokker-Planck equation

I am wanting to solve a 3D Fokker-Planck equation of the form: $$\partial_{t}p(\mathbf{x}, t) = -\nabla \cdot \mathbf{J}$$ where $\mathbf{J} = \mathbf{v}(\mathbf{x})p(\mathbf{x}, t) - D\nabla p(\...
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Clamped spherical pressure vessel

I am struggling to find the right terms to search my questions, so even just pointing me to reference material would be appreciated. 1) Suppose I have a spherical cap (thin), clamped at the edges, ...
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1answer
31 views

Boundary condition for partial reflection

I want to solve a wave equation for the wave $\psi(x,t)$. One boundary is moving, therefore I impose the velocity $$v(x=0)=v_a\cos(\omega t)$$ the other boundary is fixed, but reflecting. If the ...
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1answer
31 views

Diffusion equation with walls (if possible with gravity), analytical solution

The solution of diffusion equation $$ \partial_t\rho=D\nabla^2\rho$$ with a point source $$ \rho(0,z)=\delta(z)$$ is in 1 dimension $$ \rho(t,z)=\frac{1}{\sqrt{4\pi Dt}}e^{-\frac{z^2}{4Dt}}$$ My ...
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1answer
71 views

Why do bound charges not appear in boundary conditions on a surface between two dielectrics?

I'm studying electromagnetism, more specifically dielectrics. However, the concepts of bound charge and free charge have been somewhat confusing for me yet. The Griffiths' book deduces one of the ...
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32 views

Dirichlet boundary conditions Polyakov action

The most general solution for the equations of motion for Dirichlet is given by: $$ X^{\mu}=a^{\mu}+\frac{1}{\pi}\left(b^{\mu}-a^{\mu}\right) \sigma+\sqrt{2 \alpha^{\prime}} \sum_{n \neq 0} \frac{\...
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134 views

Importance of an extra total derivative term in Liouville theory

In this paper on boundary Liouville theory, the authors have introduced an extra term, $-\partial_{\sigma}^2\phi$, (the last term in the equation below) in defining the stress tensor of the Liouville ...
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2answers
169 views

Why is $x^n \psi^{(m)}(x)$ zero at infinity? [duplicate]

Prove: Product of any polynomial of $x$ with $\psi(x)$ or any of its derivatives goes to zero in the limit $x\to\pm\infty$. This comes from a footnote written by the professor in his quantum ...
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2answers
35 views

In band gap theory, why can we use periodic boundary conditions

In band gap theory, why can we use periodic boundary conditions when finding the wave function of free electrons in a conductor? Why do you think it is smoothly connected at both ends of the conductor?...
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28 views

Understanding boundary conditions and forced system - Wave Equation

I'm trying to solve the wave equation for a infinite string which is attached to a mechanism that moves as $y(t)=A\cos(\omega t)$ at $x=0$. The doubt I have is: can I see this system as an ...
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3answers
98 views

Initial conditions for wave equation

One of the common initial conditions given for the wave equation, $$\frac{\partial^2 p}{\partial t^2} - \nabla^2 p = 0,$$ is $p(\overline{x},t=0) =0$ and $p^\prime (\overline{x},t=0) =0$. What is ...
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1answer
17 views

Interface boundary conditions of superconductor

Are the usual interface conditions for electromagnetic fields, i.e. $$\mathbf{n}_{12}\times(\mathbf{E}_1-\mathbf{E}_2) = 0,$$ $$\mathbf{n}_{12}\bullet(\mathbf{D}_1-\mathbf{D}_2) = \sigma_s,$$ $$\...
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1answer
24 views

Laplace equation outside sphere

When solving the Laplace equation on sphere coordinates you get: $$ u(r,\theta) = \sum_{n=0}^{\infty}\left( A_n\,r^n + \frac{B_n}{r^{n+1}} \right) P_n(\cos\theta) $$ And it is clear that if you have ...
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1answer
45 views

boundary conditions in QM and statistical physics

I don't understand something about boundary conditions in problem that I discuss it below. in QM we solve the particle in Potential well and we obtain that we should have $k=\frac{n*pi}L$ that $n\in{...
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37 views

A simple question about equation of motion in polchinski's String theory?

In page 14 to get the equation of motion, it takes the variation of the action $$ S_P[X,\gamma]=-\frac{1}{4\pi\alpha'}\int_Md\tau d\sigma(-\gamma)^{1/2}\gamma^{ab}\partial_a X^\mu\partial_b X_\mu $$ ...
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1answer
47 views

Doubts in an introduction to classical field theory

I started to study classical field theory using the book "Field Quantization" of Greiner and Reinhardt, and I have some doubts. First, the book write the Lagrangian $L(t)$ as a functional of a field $\...
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15 views

Idea behind boundary states in BCFT

In Blumenhagen's book on CFT, in the BCFT chapter he introduces the concept of a boundary state. TO do this, he first explains how there is a duality between the one-loop open string worldsheet and ...
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23 views

Fields and gauge transformations vanishing at infinity

I find that, in field theory, it is very often assumed that the fields (classical) vanish at infinity. The same assumption is also applied to gauge transformations, for example, when saying that the ...
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2answers
87 views

Is speed of light continuous when entering a medium?

I know that light has the speed $c$ in vacuum and speed $c/n$ in a medium with refractive index $n$. I wonder how this exactly happens - is there some kind of smooth transition? If so, on which scale?...
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Why does the Schrödinger equation work so well for the hydrogen atom despite the relativistic boundary at the nucleus?

I have been taught that the boundary conditions are just as important as the differential equation itself when solving real, physical problems. When the Schrödinger equation is applied to the ...
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0answers
50 views

Constraints vs Boundary Conditions

I have a very broad question about how the mathematical framework that classical theories of physics utilize to solve problems. The question is: What are the intrinsic differences between the ...
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1answer
40 views

Why shear stress is assumed constant in the inner layer

In the derivation of the log-law and the viscous sub-layer velocity profiles, it is customary to assume that the shear stress is constant and equal to the wall shear stress. Is there any physical or ...
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1answer
44 views

Localization Principle (SUSY)

Mirror Symmetry p.200/201 Last section p.200/first p.201 It says, that the localization principle would not work if one would not impose periodic boundary conditions for the fermion integration, ...
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27 views

Non-viscous incompressible Fluid between two coaxial cylinder

Consider a non-viscous incompressible fluid lies between two coaxial cylinders. The domain occupied by the fluid is defined as $0<z<\xi$, $A<r<B$. The coaxial cylinders slowly rotate ...
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77 views

What does a negative end correction mean?

I was asked this question in one of my tests: "In an experiment to measure speed of sound by a resonating column a tuning fork of frequency 500 Hz is used. The length of air in the column is varied ...
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0answers
22 views

Elasticity free boundary terms

Suppose I have a 2D elastic body, and $\mathbf{u}(x,y)$ is a displacement field of the body. I am trying to derive the equilibrium equations for linear elasticity; I define an elastic energy $$E[\...
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1answer
69 views

Symmetric potential well different solutions

I have solved $H|\psi\rangle=E_{n}|\psi\rangle$ with $V(x)=0$ from $-a<x<a$ and $\infty$ otherwise. If I propose a solution of the form $\psi(x)=A_{n}e^{ikx}+B_{n}e^{-ikx}$ I arrive to the ...
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2answers
49 views

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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33 views

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
66 views

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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19 views

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
38 views

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
91 views

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
55 views

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
147 views

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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28 views

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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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
35 views

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
93 views

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
42 views

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 ...
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1answer
82 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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18 views

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
44 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 ...