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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23 views

Can standing waves be formed from a medium with one free end?

I know that standing waves are normally formed in a confined medium where both ends are fixed. However, I wonder if standing waves can be formed in a medium with one fixed end and one free end since ...
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48 views

Intuitive explanation of insulated boundary condition

Could someone please provide an intuitive explanation of why an insulated end of a rod must have a temperature gradient $\frac{dT}{dx}$ equal to zero? Based on the previous question I asked, it seems ...
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70 views

How to solve a Laplace equation with Neumann boundary condition for a potential flow?

I need to find the velocity distribution of a potential flow around an object of a given shape, and it’s dictated by Laplace’s equation: $\nabla^2\phi=0$, where $\phi$ is the velocity potential, ...
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54 views

Klein-Gordon equation on a compact, two dimensional domain

Consider the Klein-Gordon equation in two dimensions on any compact subset of $\mathbb{R}^2$ (that is, a Jordan domain). The equation is hyperbolic, and since the domain is compact it is not evident ...
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49 views

Boundary conditions for the $bc$ system

In this question, I will be referring to chapter 2 of Polchinski String Theory vol. 1. In equation (2.7.29), he states that the boundary conditions for the $bc$ system of the open string are \begin{...
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23 views

How to formulate a PDE for diffusion between two different materials?

Suppose I have two (connected) materials with different diffusion coefficients for which I am modelling diffusion. Consider the one dimensional case. I am not sure what conditions to impose at the ...
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34 views

Poisson equation of a point charge in spherical coordinates

I have to find the potential between two concentric grounded conducting spheres with a point charge betweem them. Gounded means that the boundary conditions are $\phi (r=a)=\phi (r=b)=0 $. I ...
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28 views

Boundary condition for $\Box\vec{E}(t,\vec{x})=0$ that preserves scale-invariance

In short, this is a question about the symmetry of a differential equation preserved by its boundary condition. In free space, the vector wave equation satisfied by the electric and the magnetic field ...
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72 views

I am stuck on this electromagnetism concept

Does $\mathbf{D}$ depend on the medium or not? If not, then why does the tangential component of $\mathbf{D}$ change across the boundary of two dielectrics with different relative permittivity when we ...
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33 views

Boundary condition of magnetic field intensity when one of the mediums is a perfect conductor

i'm studying boundary conditions of maxwell's equations at the interface of two mediums, for the case of the magnetic field intensity $\boldsymbol{H}$ one can obtain using the ampere-maxwell law that \...
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31 views

How does boundary condition and phase change on refraction related?

Refraction at an interface never causes a phase change—but a reflection can, depending on the indexes of refraction on the two sides of the interface I have seen many answers why the phase doesn't ...
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19 views

Quantum Mechanic: Sperically symmetric finite well [duplicate]

Consider the following 3D potential: $$V(r)=\begin{cases}-V_0 \ \ \ \ \ \ & r \leq a\\ 0 &r > a\end{cases}$$ we want to find the eigenfunctions for $\ell=0$, in particular we are interested ...
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Boundary layer effects of Phylic surfaces in viscous flow

In fluid flow over Phobic surfaces, the boundary layer thickness is decreased by the Slip Length, meaning the fluid is not stationary at the surface and lowering the energy transfer to the surface (...
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Reflection of sound waves from open end [duplicate]

I know that sound waves can be described as pressure waves. Now, let's consider the case of reflection of a sound wave from an open end, as in an organ pipe. My textbook says that, in case of ...
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2answers
39 views

Boundary conditions for the wave equation in spherical coordinates

Consider the wave equation (In this context, I'm talking about an acoustic wave), $$\frac{\partial^2 p'}{\partial t ^2} - c^{2} \nabla^2 p'=S.$$ Let us assume this is in free-space, i.e, there is no ...
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Cause of 180 degree phase shift of reflected light [duplicate]

When light originating from a medium of lower refractive index reflects off a surface of higher refractive index, why does the phase shift by 180 degrees? Stokes relations show that the reflection ...
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13 views

Equation and boundary conditions for the temperature problem in a cylinder with thermal power on its axis

I'm having trouble trying to establish the equation for the following problem: On the axis of a long cylinder, whose radius is r = r0, there is a conductor with a current that releases the thermal ...
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24 views

What is the potential outside a conducting cube with specific plates held at potential $V$?

I have a conducting cube of dimensions $[0,a]\times[0,a]\times[0,a]$. The plates at $z=0$ and $z=a$ are held at potential $\phi(x,y,z=a)=\phi(x,y,z=0) = V$. For the other plates $\phi$ is held at $0$. ...
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36 views

Set up particular BCs for a block in an elasticity problem

Let's consider an incompressible block of Neo-Hookean material. Let the initial reference geometry be described by $B=[0,b] \times [0,b] \times [0,b]$. The professor gave me the following task: Set ...
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Directions of Interfacial Tensions

source: Wikipedia/contact angle Here there are 3 interfacial tensions acting on the water drop which is known to wet a surface. But I am unable to understand the reasoning for the direction of these ...
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76 views

Time generalized Biot-Savart and Coulomb laws: Jefimenko's Equations, boundary-conditions, reflection, transmission

I am attempting to use the time generalised Biot-Savart and Coulomb laws (sometimes called the Jefimenko's equatios) to calculate the electric and magnetic field of a wire loop in a homogeneous sphere ...
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32 views

Brown-Henneaux central charge in Chern-Simons formulation

In the article of Pérez, Tempo and Troncoso, in the Chern-Simons formulation for three-dimensional general relativity, they compute the canonical generator of charges \begin{align} Q\left[\varepsilon\...
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53 views

Can superposition principle work with boundary conditions in electromagnetism?

Let's suppose I have two finite lines of charge, each of them at some voltage $V$. The first one would be at $x=0$ and the second one at $x=a$, and each of them has the same length $l$. If I want to ...
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6 views

Induced voltage on a third conductor due to the field of a capacitor

I just started solving PDE solvers for electrostatic problems using MATLAB, which solves for scalar unknown fields. I know how to set up Dirichlet and Neumann boundary conditions, and have produced ...
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1answer
23 views

Path integral with initial definite momentum

Generally, in path integral formalism a propagator between two states with definite position is computed, something like, $$K(x_1,t_1;x_0,t_0)=\int_{x_0(t_0)}^{x_1(t_1)}\mathcal{D}x(t)\exp\left(\frac{...
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167 views

Density of states and boundary conditions: how the density of states is physical if it depends on box size

This question is closely related to this one: Why is the density of states required conceptually? Should it be seen as a mathematical trick related to Fourier series? But it was suggested that I ask ...
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29 views

Periodic boundary condition and hadronic correlator

Recently I have been learning about lattice QCD in a self-taught way. I have a question about the 18th page of the following link: https://www.jlab.org/hugs/Slides/Sufian_HUGS2018.pdf It seems to me ...
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2answers
75 views

The metric exterior of a massive object

The only condition apart from perfect spherical symmetry that is required for the retrieval of the Schwarzschild-metric $g_{ik}$ is actually ($R_{ik}$ being the Ricci tensor, the contraction of the ...
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24 views

How is the Hamiltonian of propagating modes transformed with a boundary condition $\phi(0,t)=0$?

I am confused by the following. Imagine that I have wave propagating in a waveguide. I can have left and right moving waves. The total Hamiltonian reads: $$H=\sum_{\omega>0} \hbar \omega (a^{\...
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79 views

Solving Laplace equation on a triangle with mixed boundary conditions

From sources [Ref: 1 to 5], one learns that there is a class of boundary conditions (see Fig 1) on a triangle (in this case the 30-60-90 triangle) that allow for closed form solution for eigensystems ...
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34 views

Why in the case of free particle we are not taking sine and cosine in the solution of the wave function whereas in the case of 1D box we are using? [closed]

why in the case of free particle we are not taking sine and cosine as the solutions of the wave function whereas in the case of 1D box we are using it?
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27 views

Equations of motion/ Boundary conditions in presence of defect

Cosider the following Lagrangian $$\mathcal{L} = \Theta(-x) \left( \frac{1}{2} (\partial\phi_1)^2 - V_1(\phi_1) \right) + \Theta(x) \left( \frac{1}{2} (\partial \phi_2)^2 - V_2(\phi_2) \right) + \...
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47 views

How to prove that there is only trajectory given fixed boundary conditions, if you know the Lagrangian of the system?

In my particular problem, the Lagrangian of the system is: $$ L = \frac{m(\dot r^2 + r^2\dot \varphi^2)}{2} + \frac{m\omega^2 (r\sin \varphi)^2}{2} $$ From there, we can derive the equations ...
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53 views

Boundary conditions, physical wave-functions and domain of Hamiltonian

Context : In quantum mechanics, time evolution is described by a one-parameter unitary group, acting on the Hilbert space of states. Under Stone's theorem (with the right hypotheses), this group has a ...
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1answer
84 views

Infinite square well plane wave solutions?

In the Sommerfeld free electron model, we assume the electrons are free, independent and confined to a volume $V=l^3$. We then solve the Schrodinger equation for the infinite cubic well and apply the ...
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46 views

Infinite square well separation of variables issue

When solving the time independent Schrodinger equation for the infinite square well in 2 or 3 dimensions (I'll use 2 dimensions for brevity), we use separation of variables, first assuming that our ...
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2answers
145 views

Understanding the Israel Junction Conditions

The well known and frequently used Israel Junction conditions are the equivalent of Einstein's field equation on a membrane in the brane-world picture. All the sources have this notation: $K_{ab}^{(i)}...
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121 views

Does Bose-Einstein condensation depend on boundary conditions?

In a box with sides of length $L$, the energy eigenvalues depend on the boundary conditions. For periodic boundary conditions, they are $$E_{n_x,n_y,n_z}=\frac{\hbar^2}{2m}\left(\frac{2\pi}{L}\right)^...
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46 views

Neutron falling off a cliff edge

Consider a particle, say a neutron, of mass $m$ moving from some $x < 0$ with some speed $ v_0 $ in the positive $x$ direction along the ground, see Figure 1. At $ x = 0 $ there is a drop of ...
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24 views

1D Convective Heat Transfer

I am considering 1D convective heat transfer in a vertical heat pipe with a constant wall temperature Tw, with z varying from 0 to L. Applying the Boussinesq assumption, we arrive at the following ...
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3answers
365 views

What happens to light as it enters a denser medium?

I am a first year undergrad student doing optometry (never done any physics before in my life :( ). I got a question asking what happens when light enters a denser medium. I was told that the ...
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1answer
30 views

Properly implementing boundary conditions on a periodic 2d system

I'm trying to read the following paper: https://arxiv.org/abs/0903.4271. In this paper, they showed how to write a Schrodinger equation for a quantum particle constrained on a corrugated 2D surface. ...
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1answer
127 views

Poisson equation on the complex plane with mixed boundary conditions

I would like to compute the Green function for the Laplacian on the complex plane, with mixed Neumann and Dirichlet boundary conditions (see equation 4.150 of Blumenhagen, Lust, Theisen "Basic ...
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15 views

Boundary Condition at the Interface of 3 Mediums in Magnetostatic?

we know that the boundary condition of two adjacent media in magnetostatics (assuming no surface current) is: $$\mathbf n \times ( \mathbf H_2 - \mathbf H_1 ) = 0$$ the interface of two adjacent ...
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1answer
55 views

Endpoints in Fermat's Principle

Are the endpoints of the light ray path in Fermat's principle must be fixed? To clarify my question: Using Wikipedia definition for Fermat's Principle: Fermat's principle states that the path taken ...
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1answer
73 views

Propagator in string theory and boundary conditions

I would like to understand how to compute propagators for open and closed strings. My references are Tong's lecture notes, https://arxiv.org/abs/0908.0333 and Blumenhagen, Lust, Theisen book "...
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1answer
34 views

What is the governing differential equation on the interface of two areas with different permeabilities in magnetostatics?

The boundary condition requires that parallel component of magnetic field intensity to be continuous on the interface of two adjacent media with different permeabilities. for example on a 2d plane in ...
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18 views

Is it possible to transform continuity boundary condition to boundary value problem in magnetostatic?

The goal is to find the flux density distribution at all points on a 2d plane, (a magnetostatic problem). the problem consists of simple geometry areas(the interfaces are lines across only one ...
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1answer
50 views

Reflection and transmission wave on three joined strings

Suppose we have a system of three joined strings, of different linear mass densities and subjected to a constant tension force, such that the velocity of propagation is $v_1$ in the string 1 (at $x &...
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1k views

When vibrating a string, how exactly is the vibration inverted to create an identical wave traveling in the opposite direction?

I am currently trying to understand standing waves. The key to the phenomenon is that: $$A\sin(kx+\omega t) + A\sin(kx-\omega t)=2A\sin(kx)\cos(\omega t)$$ The shape of $2A\sin(kx)\cos(\omega t)$ is ...

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