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-4
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0answers
27 views

Show that boundary layers diffuse out from the plate with speed $\sqrt{\frac{\nu}{t}}$ [on hold]

I was wondering if someone could help me with this problem. I think it's quite simple, but I'm not sure how to do it! An infinite horizontal plate moves with speed $U$ in its on plane relative to ...
9
votes
4answers
489 views

When/why does the principle of least action plus boundary conditions not uniquely specify a path?

A few months ago I was telling high school students about Fermat's principle. You can use it to show that light reflects off a surface at equal angles. To set it up, you put in boundary conditions, ...
0
votes
1answer
33 views

Reflection of EM waves

In reflection of e m waves at the boundary, to show the reflected magnetic fields we put negative sign in the unit vector, example, if the B is along z direction we put (-k) in he reflected wave, ...
0
votes
1answer
33 views

Newtonian heating

Suppose a fluid over a heated surface which is being stretched and the flow starts. Now the boundary condition at the surface is assumed as $q_w$ is proportional to the surface temperature. It is ...
0
votes
0answers
41 views

Boundary conditions for Maxwell's equations at the interface between two media

Consider the following simple Maxwell's equations: $$ \nabla\cdot\mathrm{D}=\rho $$ $$ \nabla\times\mathrm{E}+i\omega\mathrm{B}=0 $$ $$ \nabla\cdot\mathrm{B}=0 $$ $$ ...
0
votes
0answers
35 views

Boundary conditions for laplace's equation rectangular conductor

Find the potential inside a rectangular conductor that is subject to the boundary conditions $$ (i)\ \phi (x = a,y,z) = \cos(\beta y)\cos(\gamma z) \\ (ii)\ \phi (x, y = b,z) = V_0\ (constant)\\ ...
0
votes
2answers
128 views

Why is $ \psi = A \cos(kx) $ not an acceptable wave function for a particle in a box?

Why is $ \psi = A \cos(kx) $ not an acceptable wave function for a particle in a box with rigid walls at $x=0$ and $x=L$ where $$ k = \frac {(2mE)^{1/2}} {\hbar} \, ?$$ I had plugged the wave ...
0
votes
0answers
23 views

What is the meaning of the point of inflection in this?

Consider the following flow velocity profiles near a boundary, and the boundary profile point of inflection,PI Regarding the effect of pressure gradient on boundary-layer profiles does the point of ...
5
votes
2answers
513 views

Why don't all free particles lose their kinetic energy?

I'm currently studying Action. I've been reading about how a particle has particular probabilities of ending at an infinite number of events. Say I have a free particle that isn't experiencing any ...
1
vote
1answer
47 views

Fixing time in Feynman phase space path integral

The phase space version of Feynman's path integral expression for the free particle propagator involves a (formal) sum over paths in phase space with fixed $q$ endpoints and (as far as I'm aware) ...
0
votes
0answers
36 views

1D drift-diffusion equation with single absorbing boundary

If we have just the simple diffusion equation (in 1D): $$ \frac{\partial P(x,t)}{\partial t} = D \frac{\partial^2 P(x,t)}{\partial x^2} $$ with an absorbing boundary at x=0 and initial condition ...
1
vote
0answers
25 views

Does a whirlpool(vortex in water) continue in air(vortex in air),and when does a vortex stop?

First part: The question is both about the continuity of the water vortex(whirlpool) to vortex in air in time and in space. About continuity in time,does the vortex of the water slowly produce a ...
1
vote
0answers
39 views

Normal of a null surface and null junction conditions in general relativity

I am trying to use the null junction formalism in general relativity (as explained in eg http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.43.3763&rep=rep1&type=pdf, "Junctions and thin ...
0
votes
1answer
55 views

Dirichlet boundary conditions in space-time?

In the context of string theory, and world sheets the Dirichlet boundary conditions can be written as: $$\frac{\partial X^\mu(\tau,\sigma_1)}{\partial \tau}=0$$ where $\sigma_1$ is the value of the ...
0
votes
0answers
31 views

Developing an analytic equation set for a complicated solenoid

So, I'm a bit stumped by this. I've been asked to develop an analytic function for the magnetic field of a complicated axisymmetric solenoid with free parameters to be determined by making a fit ...
0
votes
0answers
15 views

Pefectly electrically conducting Neumann boundary conditions

I have a rather subtle question regarding necessary boundary conditions. To solve Maxwell's source-free equations as an initial boundary value problem in a volume $\Omega$ bounded by a perfectly ...
1
vote
1answer
61 views

The definition of the vacuum state of quantum field by path-integral

In the review Entanglement entropy of black holes by Sergey Solodukhin (arXiv:1104.3712, equation 13), I see a definition of vacuum state of quantum field by path integral over half of the total ...
0
votes
0answers
21 views

Initial conditions for second order ODE with complex stiffness

I tried this on Math Stack Exchange. I'm trying to find initial conditions to ensure systems of the form stay bounded $$\ddot{x}_i+\sum_{j=1}^N k_{ij} x_j = 0, \quad k_{ij} \in \mathbb{C}.$$ For ...
2
votes
0answers
45 views

Variation of Gibbons-Hawking-York term. General boundary condition and total derivatives

It is actually a comment and question to the answer of Robert McNees in the following post: Explicit Variation of Gibbons-Hawking-York Boundary Term In deriving the variation of the extrinsic ...
2
votes
1answer
129 views

How do you know when you need to use distributions to represent charge densities? [closed]

I tried to solve a problem using Gauss' law in the following way. Let's assume we have a spherical shell of radius $R$ with a charge $Q$ being homogenously distributed on its surface. I am trying to ...
0
votes
1answer
31 views

Border conditions on the separation surface (Electromagnetism & Optics)

My teacher taught me that we can consider the following equation: $$E_{1t}=E_{2t}$$ to the descontinuity of the electric field tangent component on the separation surface of two means with ...
1
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0answers
82 views

A classically charged point particle interacting with electromagnetism and gravity

Consider a classically charged point particle interacting with electromagnetism and gravity. The relevant dynamical variables are $\chi^\mu (\tau)$ of the particle, the electromangetic potential ...
0
votes
1answer
39 views

Boundary condition for E field

My book says that the boundary condition for the E field is: $$\hat{n} \times (\textbf{E}_1 - \textbf{E}_2) = 0 $$ and then concludes that the above condition can be summarized by the statement, "The ...
0
votes
0answers
52 views

How to solve Laplace equation in a domain with one boundary along a curve?

Is there a way to solve the 2D Laplace equation $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} =0$ on $0 <x<\infty$ and $0 < y < \infty$, such that the domain is ...
2
votes
2answers
128 views

Detailed conditions for symmetries of Lagrangian

Edit: To clarify the question, I am asking why we are justified in calling a continuous symmetry a symmetry of a system when it changes the Lagrangian by a total derivative of a function of $t, q(t)$ ...
1
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0answers
31 views

Magnetic dipole near black hole

Usually it is said that black holes cannot have electric or magnetic dipole, only electric charge and angular momentum are allowed quantities besides mass So, it would seem that black holes behave as ...
1
vote
0answers
30 views

Boundary conditions for vector wave equations

Assume the time-harmonic case of Maxwell's equations, one can obtain the following vector wave equations: $$ ...
0
votes
0answers
28 views

Why are closed strings with different perioidicities equivalent?

I was typing up some lecture notes the other day when I saw something unclear. While talking about bosonic open and closed strings and the Polyakov action, the notes say we don't need to distinguish ...
3
votes
2answers
144 views

Minimizing the Lagrangian action of an impossible problem

I'm working my way through Structure and Interpretation of Classical Mechanics (SICM), and am stuck on an exercise in Section 1.4: Exercise 1.6. Minimizing action: Suppose we try to obtain a ...
2
votes
0answers
63 views

Current Density Boundary Conditions and its Implications

According to Ohm's Law, one can say $ \overline{J} =\sigma \overline{E} $ if the field is in a conductor, and $ \overline{J} =0 $ if it's in empty space. Now if we take the surface of a conductor and ...
1
vote
1answer
49 views

Can Ampere's Circuital law be used on an infinite number of alternating Helmholtz coils?

I have the following surface current density $$ \bar{\sigma}_s = \hat{\phi} \sin(kz) |\bar{\sigma}_s| $$ to approximate an infinite number of alternating Helmholtz coils stacked along the z-axis with ...
1
vote
1answer
74 views

Boundary conditions of the radial Schrodinger equation

Consider the radial differential equation $$\bigg( - \frac{d^2}{dr^2} + \frac{(\ell+\frac{d-3}{2})(\ell+\frac{d-1}{2})}{r^2} + V(r) + m^2 \bigg) \phi_\ell (r) = \lambda\ \phi_\ell (r),$$ which I've ...
0
votes
0answers
21 views

What are maximally dissipative boundary conditions?

I ran into this term when reading about the initial boundary value problem in general relativity. They seem to be relevant when you need to impose boundary conditions on a timelike boundary, for ...
2
votes
2answers
247 views

Why do electric field lines start and end at 90 degree at the surface of a conductor? [duplicate]

There is one property of electric lines of forces which states that: Electric field lines start and end at 90 degree at the surface of the conductor. But why is that so? Is there any proof for ...
2
votes
1answer
269 views

Newton's law of cooling for the heat equation boundary condition

Newton's law of cooling says the temperature of an object satisfies $$ \frac{dT}{dt} = -k(T(t) - T_0),\tag{1} $$ where $T_0$ is the surrounding temperature. See these HTML notes for example. Now if ...
12
votes
2answers
290 views

Is there a physical interpretation of Neumann boundary conditions for the free Schrodinger equation on a domain?

Let $\Omega$ be a domain in $\mathbb{R}^n$. Consider the time-independent free Schrodinger equation $\Delta \psi = E\psi$.[*] Solutions subject to Dirichlet boundary conditions can be physically ...
5
votes
2answers
93 views

Why do we not require higher derivatives to match at boundary when solving the Schrödinger equation in a given potential?

When solving the time independent Schrödinger equation for a given potential in 1D, the main part of the solving involves matching boundary conditions. Usually, we require the value and the first ...
1
vote
1answer
39 views

How do i mathematically represent reflection in a (diffusion) Problem?

I am trying to formulate boundary conditions and it occurred to me that I never had to implement a reflective boundary before. The example is a one dimensional diffusion, where at $x=0$ the ...
0
votes
1answer
53 views

Fourier series for a wave on an infinite string?

From "Vibrations and Waves" by A.P. French I know that any wave on a string length $L$ can be represented by: $$y(x,t)=\Sigma^\infty_0 A_n \sin(\frac{n\pi x}{L})\cos(\omega_nt-\delta_n)$$ But can we ...
0
votes
2answers
66 views

When does $\mathbf n\times(\nabla V_2-\nabla V_1)=0$ imply $V_1=V_2$

I was reading a paper on electrohydrodynamics which has the following sentence (in my own words): At the interface/boundary, the requirement of continuity of the tangential component of the ...
1
vote
0answers
17 views

Phase change by reflection [duplicate]

Let's consider a light ray falling on a cuboid made of glass at the angle $\alpha$. Then there will be a reflected ray $A$. The ray will also refract. Let the refracted ray be $B$. Ray $B$ will be ...
2
votes
1answer
36 views

Could you give boundary conditions to the gravitational potential given the density distribution?

We´re doing a project that's all about solving differential equations with separation of variables. We´re trying to find the gravitational potential given the density distribution (that has azimuthal ...
0
votes
0answers
50 views

Electromagnetic boundary conditions for modelling symmetrical geometry

I stumbled upon this article: http://www.comsol.com/blogs/exploiting-symmetry-simplify-magnetic-field-modeling/ Since the article does not contain any mathematical formulations, I was wondering how ...
1
vote
0answers
31 views

How to specify boundary conditions as function of curvature in dynamic elastic beam pde?

In this article (already mentioned in this question) the dynamics of a planar elastic beam with "cantilever constrains" (one clamped end and one free end) is modeled. Using the Euler-Bernoulli Beam ...
1
vote
0answers
32 views

Neumann Green's function inside semi-infinite conductor [closed]

Consider a semi-infinite conductor with uniform conductivity $s$ occupying the space $z>0$. What is the Green's function with Dirichlet and Neumann boundary conditions inside the region $z>0$? ...
2
votes
1answer
40 views

How come a current sheet of $J_s = J_0 \hat{x}$ produces plane wave solution?

Given in the picture. There is a current sheet $J_s = J_0 \hat{x}$. Supposedly Jo is not oscillating. So, how does this thing create a plane waves propagating away from the current sheet? Shouldn't ...
0
votes
0answers
15 views

Solution for elastic wave on plane of ideal contact between two half spaces of elastic, homogeneous, isotropic, linear solids

Please give the solution for elastic wave of arbitrary polarization incident at arbitrary angle on plane of ideal contact (meaning no slip, homothermal, nondissipative, and no transfer of material ...
0
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0answers
16 views

Zero stress boundary conditions for the acoustic wave function

When is it appropriate to use zero normal stress boundary conditions when solving the acoustic wave equation. That is when the pressure is equal to zero.
3
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2answers
106 views

Idea behind Compactified Boson

On p. 167 of his Conformal Field Theory, Di Francesco introduces "Compactified Boson". He says: The invariance of the free-boson Lagrangian [...] with respect to translations $\varphi(x) ...
1
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0answers
38 views

Faraday's law in free space explaining away the constant vector?

Let's say that I have a plane electromagnetic wave travailing in free space, and I know the electric field part to be $\vec E$. If I am using Faraday's law to get the magnetic field part I will get ...