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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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:
$$ \frac{\partial^2 y}{\partial x^2} - \frac1{c^2}\frac{\partial^2y}{\partial t^2} - \gamma\frac{\...
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1
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No-slip condition tangential and normal component decomposition
No-slip condition on a corrugated surface (modelled by a sinusoidal function $b(x)$))
$\vec{ u} (x,b(x)) =u \vec{i}+ w \vec{k} = 0 \vec{i} + 0 \vec{k}$
expressing in terms of the stream function :
$$\...
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Are analytic boundary conditions able to be found in the majority of numerical physics problems?
A typical PDE usually is specified with initial and boundary conditions to be able to yield an exact solution, or at least be approximated:
$$
\frac{\partial u(x, t)}{\partial t} + F\left(u(x, t), \...
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Why can static solutions be found only in problems with Dirichlet boundary conditions in non-relativistic point particle mechanics?
Qmechanic in this answer says that only Dirichlet b.c.s are consistent with static solutions of point particle mechanics. Why is this so?
E.g. The standard classical harmonic oscillator problem with b....
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Least action principle and uniform motion
I'm trying to apply the principle of least action to the case of a uniform motion under no potential. Assume the object starts with initial velocity $v_0$, moving from point $A$ to point $B$. We know ...
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Does a rigid body rotates in an irrotational flow?
Consider a given flow field such that vorticity $\mathbf{\omega} = \text{Curl} ~\mathbf{u} = 0$. In this case, we can consider an arbitrary shape fluid element and look at how it evolves in short ...
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Infinite wire with alternating current and conducting half-plane
I'm kinda confused about the following. Suppose I have a harmonic current-density of
$$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}$$
where $\psi(t)$ are a real Grassmann fields.
I get two answers ...
user84158
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Extending Geodesic Across Boundary in Discontinuous Spacetime
Imagine a metric space to be discontinuous: a Schwarzschild outer metric that changes to a flat FLRW metric once inside the boundary of a $2$-sphere of fixed $r$. If an object (or ray) just hitting ...
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Aharonov-Bohm effect and periodic boundary conditions for particle on a ring
For a particle on a ring, we have the periodic boundary conditions $\psi(\phi+2\pi)=\psi(\phi)$. If we also have a magnetic field penetrating perpendicularly the ring, then when the particle goes ...
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How to add walls or other solid obejcts to Eulers equations? [closed]
I've recently started learning about the physics of fluids and I've found that euler's equations exist and that I can use them to compute the flow of fluids with zero viscosity. But I have a problem ...
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Polchinski's doubling trick for extending open string theory to the whole complex plane
Open string theory can be described on the upper-half complex plane. To simplify the description of open string theory, Polchinski asserts (eq. 2.6.28 in his Vol. I String Theory book) that it is ...
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On obtaining the standard boundary condition for the radial equation
The solution of spherically-symmetric one-particle problems is often facilitated by looking for solutions of the form
$$\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:
$$ \{\nabla^2 + ...
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Killing vectors for a boundary condition?
So here's something I was wondering about. Let's say I have to have boundary conditions in flrw metric which does not leave the universe isotropic and homogeneous. This would go against observations. ...
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Shape on string for function in range $(-L,+L)$ [closed]
What combination of wave equation solution should I assume for string fixed at $x=-L$ and $x=+L$. I was trying with for example:
$$\psi_m(x,t)=\cos(k_mx+\alpha_m)[A_m\cos(\omega_mt)+B_m\sin(\omega_mt)]...
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How many boundary conditions do we need to solve Laplace's equation?
I'm struggling to understand which boundary conditions are necessary to solve Laplace's equation. In order to better understand them, I have thought of a scenario that I would like to ask a couple of ...
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Finding the potential of a point charge and a conductive sphere solving Laplace's equation
How can we solve this, specifically using the method of separation of variables, instead of the method of the images? Specifically, outside the sphere
We have a conductive sphere of radius R at the ...
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Temporal Gauge with periodic boundary conditions
In Yang-Mills theory with periodic boundary conditions in time, is the temporale gauge, i.e. $A_0 = 0$, well defined?
Periodic boundary conditions would be
$$A_\mu(T_2,x) = A_\mu(T_1,x).$$
Naively I ...
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The WKB approximation and boundary conditions
To estimate a quantization rule using the WKB approximation, one is usually working in the 'classically allowed region'. You apply the WKB approx in the middle of the region and use the Airy ...
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Legendre series solutions for the Laplace equation: can Neumann boundary conditions be applied?
In Jackson E&M, it is shown in equation (3.33) that the Laplace equation with azimuthal symmetry can be expanded in Legendre series in spherical coordinates,
$$ \Phi(r,\theta)=\sum_{l=0}^\infty\ [...
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QED photon path (direction of photon emission)
In QED we look at all possible path a photon could go from S to P, and I understand the most significant contributions to the final arrow are the few near straight paths connecting S and P while other ...
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Logical consistence in Neumann BC in the Nambu-Goto action
Background: In Zwiebach's A First Course in String Theory, 2nd ed. equation $(6.50)$ define a kind of momentum along the $\sigma$ direction on the worldsheet (Here $\dot{X} = \partial_\tau X, \, X' = \...
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Simulating Dirichlet Boundary Conditions for Lattice Boltzmann Method
So, I'm going through A.A. Mohammad's book "Lattice Boltzmann Method : Fundamentals and Engineering Applications with Computer Codes." And I'm currently coding the D2Q9 lattice with respect ...
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Solving heat equation with fixed temperature at $x=0$ [closed]
Suppose, there is an insulated 1d rod of length $L$ with initial temperature of $T_{0}$. Suddenly, the beginning of the rod is heated such that its temperature is fixed at $T_{f}$, while the other end ...
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Mean first passage time (MFPT)
I want to know the mean first passage time (MFPT) on a unit interval for two boundary conditions (please see attached figures a and b for your reference). This is in the context of the hydrodynamic ...
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Confusion in calculation of density of states
So, when we calculate density of states of the electrons/ phonons, or while during a similar calculation of no. of modes per frequency in the Blackbody radiation, we assume Standing wave solutions:
$...
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Physical and mathematical relation between $(\tau, \sigma)$ parameters and coordinates $X^\mu$ in String Theory
When we define the parameter space for a string Worldsheet $\Sigma$ to be diffeomorphic to, say, $\mathbb{R} \times [0,1]$ or $\mathbb{R}\times S^1$, and use standard coordinates $(\tau, \sigma)$, $\...
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Junction Conditions: In what cases is matching the extrinsic curvature at a boundary tantamount to matching metric derivatives at the boundary?
My understanding of the Israel junction conditions are as they are laid out in Poisson's "A Relativist's Toolkit", namely that if one wishes to join 2 different spacetimes across some ...
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What is the opposite of "periodic boundary conditions"?
I am writing a paper on solid-state physics and use periodic boundary conditions (PBC) for a calculation. To motivate why I use PBC, I write a paragraph about what would happen if I did not use PBC. I ...
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References about Boundary Conditions ADM Relativity (Thiemann)
I'm reading chapter $1.5.1$ (Section about Boundary Conditions) at Thiemann's Loop Quantum Gravity book and its taking a while to understand whats going on.
Along page 61 there's statements like $\...
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How to solve dual boundary condition contradiction at the corner of an axially loaded cylinder?
Subject: Linear elasticity
Consider an isotropic elastic cylinder rigidly bound to the surface on one end. The other end is under uniformly distributed static load (pressure for example).
I am trying ...
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Surface charge density on an open conductor
In Griffiths' "Introduction to electrodynamics" (3rd edition, page 102) he states that:
since the electric field inside a conductor is zero, boudary condition 2.33 ($\mathbf{E_{above}-E_{...
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Helmholtz theorem: “There is no function that has zero divergence and zero curl everywhere and goes to zero at infinity”. How do I know this?
I am trying to proof Helmholtz’s theorem and I am currently using David J. Griffith’s book on electrodynamics to do this. Within the book’s Appendix B, (I have not finished the book, and am on the ...
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Flow down an incline - Understanding boundary conditions
After working with some problems regarding flow, I came up to a similiar problem as the one presented here:
In solving the problem, we assume a laminar flow in steady state.
When using Navier-Stokes ...
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