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Questions tagged [boundary-conditions]

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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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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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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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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
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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
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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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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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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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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
64 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
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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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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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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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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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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
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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
64 views

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 ...
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1answer
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How do I solve for the voltage on a geometry that has varying thickness of a conductive metal?

I currently have a 3D geometry which is made from an isotropic material. In my case this material is simply a highly conductive metal. We can think of this geometry almost as a thin film with slightly ...
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1answer
76 views

Sound wave equation: Neumann boundary conditions

In this paper it's described the solution of the damped wave equation in cylindrical coordinates $$ \nabla^2\left(c^2\rho_1+\nu\frac{\partial\rho_1}{\partial t}\right)-\frac{\partial^2\rho_1}{\...
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2answers
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How to mathematically express permittivity as a function of frequency

I am a amateur physics student. I am modelling the wave propagation of an EM wave at different frequencies through water placed between two acrylic rectangular materials using COMSOL. But since the ...
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1answer
133 views

Brownian dynamics simulations in confined geometries [closed]

I am currently trying to implement a 2D Brownian dynamics simulation in confined geometries (corrugated channels, of the form $A\cos(2 \pi x) \ + B\ $ in this case). The concept is to compute the ...
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2answers
157 views

Combined Poiseuille-Couette flow

I stumbled upon this exercise in James Fay "Fluid Mechanics" book, which I'm using to learn fluid dynamics by my own, and I am struggling a bit with it, any help will be appreciated: The figure ...
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What are the “basic things” we need in physics to define any kind of energy, including “mass”?

After having commented an answer there: Relative potential energy vs Absolute potential energy, I realised that the energy concept may be much more subtle than what we usually believe, even if we ...
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What are the boundary conditions for the Hydrogen Atom which cause the polar power series to need to terminate?

I am trying to solve the Hydrogen Atom, and I am stuck in the polar equation. To simplify, I have taken the special case in which $m=0$ to get the Legendre Equation: $$(1-x^2)P''(x)-2xP'(x)+AP(x)$$ $$(...
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1answer
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Why does a sound wave on encountering a low pressure region gets reflected with a “phase change of π”?

I am particularly concerned with the reason for phase change of π. This is observed in organ pipes at their open ends. The waves on encountering the open atmosphere(low pressure region) reflects back ...
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2answers
364 views

If $A^\mu$ is not determined uniquely by Maxwell's equations, what happens if we solve for it numerically?

Given a solution $A^{\mu}(x)$ to Maxwell's equations \begin{equation} \Box A^{\mu}(x)-\partial^{\mu}\partial_{\nu}A^{\nu}=0\tag{1} \end{equation} which also satisfies some specified initial conditions ...
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1answer
36 views

Must the electromagnetic 2-form be harmonic in vacuum?

The Maxwell equations in vacuum are $dF=0$ and $d*F=0$. Is this not the same as saying $F$ is both closed and co-closed, and hence harmonic? But Hodge's theorem says the space of harmonic $p$-forms on ...
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1answer
82 views

Phase shift during wave reflection [duplicate]

I know that Electromagnetic waves undergoes a phase shift when reflected from a denser optical medium. Does this conclusion also hold for other mechanical waves like sound wave reflecting from water ...
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1answer
78 views

Wavefunction of a shifted radial harmonic oscillator [duplicate]

Suppose we know the solution of Schrodingers equation for a radial potential $V(r)$. Then the energy eigenstates are $\psi(r,\theta,\phi) = \frac 1ru(r)Y_\ell^m(\theta,\phi)$ where the radial ...
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1answer
28 views

How does the image of $\sqrt{2/L} \ \sin\left({k_n x}\right)$ satisfy the boundary conditions for the infinite square well?

I understand mathematically how $\sqrt{2/L} \ \sin\left({k_n x}\right)$ satisfies the boundary conditions for the infinite square well in terms of the fact that $\psi(0) = \psi(a) = 0$, and excuse the ...
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Is the geodesic equation valid for a motion of an object with an arbitrary initial condition?

As far as I know, the geodesic equation of motion can be directly derived from the equivalent principle. For instance, as shown by Steven Weinberg, the geodesic equation can be obtained by ...
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1answer
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Why are there only odd numbered harmonics in one closed end resonant tube? [closed]

Why do we only have odd numbered harmonics at one-end closed tubes, however, if we do a frequency spectrum we have some periodic spikes between the odd harmonic spikes, just like the picture below ...
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0answers
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Impurity diffusion from solid to air

I have to put up a simple 1D model to simulate the diffusion of impurities in a bulk material, to a top thin layer made of another material, and compare it to experimental data. I first did it with ...
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1answer
50 views

Derivative of wavefunction of a quantum system

If $\psi(x)$ represents the wavefunction of a 1D quantum system, it satisfies the Schrodinger equation, has a unit norm, and $\lim_{x\rightarrow \infty }\psi(x)=0.$ Then is it true that $\lim_{x\...
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Boundary conditions for backscattering?

In solving the well-known problem of normal incidence reflection, it is typical to say that there should be no backwards-moving wave on the far side of the reflective surface, or in other words that ...
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Switching between advanced and retarded solutions mid-integral?

In wave mechanics, an advanced solution can be thought of as a wave that propagates until it is "caught" and stopped by the forcing function, and a retarded solution can be thought of as a wave that ...
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306 views

Using a time-like boundary as a computer?

Question and Summary Using classical calculations and the Robin boundary condition I show that one calculate the anti-derivative of a function within time $2X$ (I can compute an integral below) $$\...
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1answer
90 views

Clarification of Path Integral formulation

I am reading from Schwarz book on QFT the Path Integral chapter and I am confused about something. I attached a SS of that part. So we have $$<\Phi_{j+1}|e^{-i\delta H(t_j)}|\Phi_{j}>=N \exp(i\...
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1answer
113 views

Boundary conditions for calculus of variations in phase space and under canonical transformations

This might be a stupid question, but I just don't get it. In Hamiltonian mechanics when examining conditions for a $(\boldsymbol{q},\boldsymbol{p})\rightarrow(\boldsymbol{Q},\boldsymbol{P})$ ...
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0answers
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Property of surface Green function in electrostatic field

Let's consider a 2D-square with 4 equal subsquares containing different dielectrics. Inside the square domain, the unknown electric potential function $\Phi$ satisfies the Laplace equation: $$\nabla^...
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2answers
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Boundary condition: displacement

I have a controversial case. I have a rod, which is fixed from one end (constraint). From another end, I apply a compressive force, by pressing the rod down. So in a way I have a constraint, but at ...
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1answer
135 views

Noether's Theorem, Boundary-Bulk, and Open Thermodynamic Systems

Before going any further, I should emphasize that I know we cannot use the action principle for locally dissipative systems or even Noether's theorem for that matter. There are plenty of stackexchange ...
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1answer
206 views

Difference between bound and free charge/current in a perfect conductor

For the case of charge, it seems clear that in a perfect conductor the free charge refers to the excess charge that has been dumped into the conductor, while the bound charge refers to the charge that ...
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1answer
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If the path integral formulation includes future events, why doesn't that imply retrocausality?

I know that such events would cancel out in the math, but if an extreme event were to happen in the future (say a black hole forming or something on that par), would a particle in the present react to ...
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Equation of reflected wave (fixed end/free end)

I have an equation of a wave as $y = 2 \sin\left( \dfrac{\pi}{6}x - \dfrac{\pi}{4}t \right)$. I want to find the equation of the wave which is formed when it gets reflected from (i) a fixed end or (ii)...
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Confusions about Israel's paper on junctions / singular shells in GR

After reading Israel's well-known paper superficially (see also the erratum) I thought that his formulas (38)-(40) were necessary and sufficient conditions to join two manifolds along a time-like ...
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How can the solutions to equations of motion be unique if it seems the same state can be arrived at through different histories?

Let's assume we have a container, a jar, a can or whatever, which has a hole at its end. If there were water inside, via a differential equation we could calculate the time by which the container is ...
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On the boundary conditions for closed and open strings

In the process of finding the equations of motion for a string from the Polyakov action (say in the conformal gauge), we have to implement some boundary conditions on the target spacetime coordinates $...
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What happens when cosmic background microwaves enter a bound system?

Gamma radiation after spending eons in expanding space have now “stretched “ into microwaves. What happens when these waves enter a bound system like a galaxy or galaxy cluster. Expanding space ...
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1answer
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Does aerodynamic heatings at the wall depend on the material of the wall itself?

I am reading "Fundamentals of Aerodynamics" 5th edition, J.D.Anderson. He said: "The slope of the temperature profile at the wall is very important; it dictates the aerodynamic heating to or from ...