Questions tagged [boundary-conditions]

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Continuum limit of Euler-Lagrange equation for Lagrangian density of 1D harmonic lattice

I'm trying to follow a derivation of the Euler-Lagrange equation at the continuum limit, and find some details hard to understand. The 1D lattice has a mono-atomic basis with atomic spacing $\mathfrak{...
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1answer
40 views

Why equal sign wasn't used for boundaries for radius ($r < b$) between internal and external conductor?

Why boundaries for radius between internal and external conductor are set to $a \leq r < b$ instead of $a \leq r \leq b$? Example: An air coaxial line made of copper ($μ \sim μ_0$) is given. A ...
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1answer
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Finding out bound states for a given potential well [closed]

Most importantly: Why is equating the two wave functions the correct approach? What's the physical interpretation of that this approach ? i.e why does it make sense? What is the condition under which ...
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2answers
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I don't understand the discontinuity in electric field across a surface

In Griffith, it was given that when we cross a surface charge density, a discontinuity in the electric field occurs. The proof was given from Gauss law. $$E_{\rm above}^\perp-E_{\rm below}^\perp=\frac{...
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1answer
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Why is $Q=p$, $P=-q$ a canonical transformation from the perspective of 2 variational principles satisfying boundary conditions? [duplicate]

This is to ask a more general question: Landau-Lifshitz say that for the variational principles $$\delta\int_{t_1}^{t_2}p\mathrm{d}q-H\mathrm{d}t =0$$$$ \delta\int_{t_1}^{t_2}P\mathrm{d}Q-H'\mathrm{d}...
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0answers
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Derivation of MTZ Black Hole

I am trying to derive from scratch the MTZ Black Hole: https://arxiv.org/abs/hep-th/0406111 I have obtained equations (2.3) and (2.4) in terms of the metric functions and the scalar field. The metric ...
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2answers
112 views

Field variations at boundary - Glass of water problem

My question is very simple: if I have a vector field $\boldsymbol{\phi}(t,\boldsymbol{x})$ defined inside an $n$-dimensional manifold $\mathcal{M}_n$ to which $\boldsymbol{x}$ belongs, why should it ...
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1answer
26 views

Conceptual question on heat transfer at steady state after a heated body is immersed in a fluid medium at low temperature

Recently during a discussion with a colleague we got into an argument. The discussion involved imagining a heated solid body at some temperature $T$ which is immersed in a large fluid medium ...
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1answer
23 views

Finding the potential of a waveguide with given boundary condition

During the review of some EM exercises I stumbled over a very interesting problem I just can't find the solution for. Suppose we are looking at a waveguide with side length $\pi$. The boundary ...
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0answers
52 views

How to obtain a solution for the following IBVP?

I am trying to solve the following advection-diffusion equation for transient flow conditions for radial flow. The governing equation is as follows. $$\frac{\partial T}{\partial t} = \frac{\partial^2 ...
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1answer
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Laplace and First uniqueness theorem proof: Why potential difference?

On Griffith's "Introduction to Electrodynamics" page 120 the author states that when proving the First Uniqueness theorem: The solution to Laplace's equation in some volume V is uniquely ...
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Coupled solid-fluid heat transfer over a rectangular plate heated from the bottom (Boundary value problem)

I have the two-dimensional temperature Laplacian $(\nabla^2 T(x,y)=0)$ coupled with another fluid equation (which is one-dimensional). The Laplacian is defined over $x\in[0,L], y\in[0,l]$. On ...
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surface charge density $\hat{\rho}$ and surface current density $\hat{\mathbf{j}}$

I am currently studying the textbook Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th edition, by Max Born and Emil Wolf. Page 5, chapter 1.1.3 ...
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2answers
90 views

What does the notation $|\text{grad} \ F|$ mean?

I am currently studying the textbook Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th edition, by Max Born and Emil Wolf. Page 5, chapter 1.1.3 ...
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1answer
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Does light return to its starting point in a closed universe?

I was reading about the possibility that our universe could be a closed sphere. from Sean Carroll “in a closed universe, one that wraps around on itself to form a compact geometry, like a three ...
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1answer
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Why would the two areas not shrinking together cause the total charge to become infinite?

I am currently studying Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th edition, by Max Born and Emil Wolf. Chapter 1.1.3 Boundary conditions at ...
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3answers
125 views

Euler-Lagrange Equation: From boundary value to initial value problem

In the principle of stationary action, the initial and final points in configuration space are held fixed. This is a boundary value problem. However, this principle leads to the Euler-Lagrange ...
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2answers
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Any boundary conditions missing from this problem? [closed]

Recently I was solving some boundary value problems in Electrostatics. I stumbled upon a problem with an infinitely long cylinder (axis along the $z$-direction and radius $a$) with a plate inside it (...
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2answers
37 views

Magnetic field and permeability

Why does only the parallel component of the magnetic field changes when magnetic field lines travel from one medium to another like from vacuum to iron?
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1answer
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Momentum modes in antiperiodic boundary condition

Fermions with antiperiodic boundary conditions have momentum in the form $$\frac{n \pi}{L}$$ where $n=-(L-1)...-1,1,...(L-1)$, i.e. odd integers. I am dealing with a Hamiltonian in momentum space with ...
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1answer
37 views

Doubt on the form of energy-momentum tensor in the context of thin shells and traversable wormholes

In reference $[1]$ the author constructed a junction condition to an external Schwarzschild with cosmological constant and a traversable morris-thorne wormhole. The form of the energy-momentum tensor ...
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1answer
48 views

Modified Hamilton's Principle overconstraining a system by imposing too many boundary conditions

In Hamiltonian Mechanics, a version of Hamilton's principle is shown to evolve a system according to the same equations of motion as the Lagrangian, and therefore Newtonian formalism. In particular, ...
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0answers
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Why do waves reflect in phase off of soft boundaries?

Based on Newton's third law of motion I understand how waves reflect pi out of phase off of hard boundaries (see gif 1), but what would be the reasoning as to why waves reflect in phase off of free ...
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45 views

Timoshenko Beam with sliding boundary conditions at angle

I would like to calculate the effective stiffness of a structure which can be represented as follows: (Image source: Link to paper with image.) The yellow part is flexible and the blue is a rigid-...
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2answers
70 views

Quantum free particle in spherical coordinate

I am trying to understand free particle in both cartesian and spherical coordinate. So a free particle going in, say $x$ direction with some energy $E$. We know the wavefunction of such particle is: $$...
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1answer
28 views

How can one eliminate/minimize water wave reflections in a ripple tank?

Is there any way to prevent or minimize reflections of water waves in something like a ripple tank? I'm thinking maybe something analogous to acoustic sound panels (I believe they use foam properties ...
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0answers
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One-point function in CFT on an infinite strip through scaling analysis

In Philippe Di Francesco's book on Conformal Field Theory in section 11.2.3 on the Infinite Strip, the one point function of a primary operator (with scaling dimension $\Delta$) is calculated by ...
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Do non-vanishing boundaries conditions necessarily force a discrete/quantized set of solutions for the Schrödinger Equation? [duplicate]

I was reviewing past exams and I found a question where I could give no satisfactory answer. The core of the question is the following: If there are boundary conditions, does this necessarily imply, ...
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34 views

Inhomogeneous Classical Gravitational wave equation discretization and boundary conditions?

On a previous posting I asked this community for information on forms of gravitation (classical or relativistic) that take into account the wave like nature and finite speed of a propagation for ...
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0answers
32 views

According to Hartle-Hawking state, could we build a sum over all possible metrics (including non-compact ones)?

Physicists Stephen W Hawking and James B Hartle 1 proposed that the universe, in its origins, had no boundary conditions both in space and time. To do that, they proposed a sum over all compact ...
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3answers
610 views

Is “Particle in a box” actually a misnomer?

In the usual statement of the Particle in a Box problem, we assume two infinite potential barriers, to hold its wavefunction constrained, so it goes to zero on both ends: But instead of invoking some ...
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Interface conditions with two interfaces following each other

Consider three media: A vacuum ($\epsilon_0, \mu_0)$; An isotropic medium ($\epsilon_2, \mu_2)$ to contain the liquid in medium 3; A liquid ($\epsilon_1, \mu_1$). On the first interface (medium 3 to ...
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How is the translational invariant thermodynamic limit different from periodic boundary conditions?

In a lot of papers physicists simulate spin systems in the thermodynamic limit (infinite chain) with translational invariance using tensor networks etc, in essence very complex methods. For example ...
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Potential at infinity and zero of an infinitley long Cylinder with dirichlet boundary conditions

I have an ifinitley long cylinder in z direction with the boundary condition $\phi(\rho=R,z,\varphi)=\phi_0+\phi_1 \cdot \cos(v\varphi)$ or $\phi(\rho=R,z,\varphi)=cos(kz)(\phi_0+\phi_1 \cdot \cos(v\...
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1answer
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Boundary condition not working on infinite place (uniformly charged) with dielectric on one side and vacuum on the other

To describe this situation mathematically, we have an infinite place at $z = 0$. For $z > 0$ a vacuum is present while for $z < 0$ a linear dielectric with dielectric constant $\epsilon_{r}$ is ...
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3answers
93 views

Why do we take the wave function to be zero at the edges of the box when solving the Schrödinger Equation for particle in a box?

The evanescent wave would be penetrating the box so why don't we account for that even if it decays, there might be some part protruding the box with walls of infinite potential.
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1answer
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General relativity, quantum gravity, and boundaries?

Does it make sense to have a spacetime in general relativity that has a finite extent or boundary. I'm not talk about a hyper-torus or hypersphere shaped spacetime but rather has there been studies of ...
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1answer
38 views

Gauge transformation and phase matching in cylinder coordinate for the Aharonov-Bohm Effect

I'm trying to get through Sakurai The Aharonov-Bohm Effect where on page 141. According t0 Eq 2.7.53, if the original equation was transformed by a gauge $\tilde A=A+\nabla \Lambda$, then the ...
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0answers
87 views

pn junction: band bending vs. fermi level bending [closed]

① I have two semiconductors of the exact same material. Hence, the conduction band and the valence band are exactly on the same level. The same holds for the vacuum level. Now the left one is p-doped, ...
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1answer
30 views

How do we get Maupertuis Principle from Hamilton's Principle?

Maupertuis principle says that if we know the initial and final coordinates but not time, the total energy and the fact that energy is conserved, we can choose the "right" path from all mathematically ...
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1answer
38 views

What does the Neumann boundary condition imply for the electric flux lines?

If I prescribe a Neumann boundary condition in an electrostatic problem, that means that I am prescribing a surface charge density, and in the way I understand it that means that the electric flux ...
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4answers
45 views

Is there any meaning for the path found by Hamilton's Principle to an impossible state?

Hamilton's principle is written as a statement about the path taken between two states of the system which occur. Is there any meaning found in solving the variational problem for points which can ...
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0answers
20 views

Derivation of boundary conditions of electric and magnetic fields

In the proof of boundary conditions given in some books on electrodynamics, the approach was to a take a pillbox and use the Maxwell's law on this pillbox-like surface and apply some limits on the ...
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23 views

Open boundary condition or no condition at all?

I want to write a model for a 1-D wave $\rho(x,t)$, with boundary conditions at $x=0$ and initial conditions at $t=0$. I want this wave to be free to propagate after any distance $L$. Should I just ...
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1answer
55 views

Intuitive explanation for why reflected waves change phase by $π$? [duplicate]

I have seen the equations that show the coefficient of reflection etc. But I'm searching for an intuitive rather than solely mathematical explanation for why waves change phase by π when reflected (...
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0answers
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Delta functions for surface of non trivial shapes

Say I'm trying to solve Laplace's equation on an object that isn't a sphere or a trivially characterized surface. I will want to solve: $$\nabla^2G\left(\mathbf{r},\mathbf{r}'\right)=-4\pi\delta\left(...
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1answer
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Why is an experimental basis used to solve the scattering state problem of the finite well when they do not represent a physical particle?

The first time I was doing the finite square well problem in QM, there was a dissection of the problem into bound and scattering states. For the bound states, the appropriate boundary conditions were ...
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0answers
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Infinite sheet in static magnetic field [duplicate]

What happens if I add a infinite sheet of a perfect conductor in a constant applied magnetic field perpendicular to the surface? e.g. I have a sheet in the x-y plane and I apply $B_0 \hat{\mathbf{z}}$....
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1answer
113 views

Why are there no even harmonics in a closed pipe?

I have seen a diagram on sites such as hyperphysics.com that show that there is a missing bit every time so that it makes every harmonic odd. I was hoping I could get a more intuitive explanation. We ...
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Is a 3-terminal magnetic tunnel junction possible?

I am wondering, whether a 3 terminals magnetic tunnel junction can be implemented, as shown in the figure below. At the bottom I have a normal metal (NM), on the top - a magnetic junction (FM = ...

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