Questions tagged [boundary-conditions]
The boundary-conditions tag has no usage guidance.
653
questions
0
votes
2answers
229 views
Phase change on reflection only 0 and $\pi$ allowed
We know that when a wave on a string is reflected from a hard boundary, the phase change is $\pi$, and from a soft boundary, the change is 0. My question is: this two conditions (hard and soft ...
1
vote
0answers
32 views
Lagrangian with a boundary contribution, external work and interaction
I am considering a linear second order partial differential equation of the form
\begin{align}
F(q,p)&=-a\,q-\nabla\cdot p=0\\
p(q,\nabla q)&=b\cdot\nabla q
\end{align}
with $a$ scalar and $b$ ...
0
votes
1answer
524 views
How does one show specific thickness and wavelength determine full transmission of electromagnetic waves?
How does one show that thickness and wavelength determine the full transmission between two different dielectric media if the boundary condition equations between two dielectric media are independent ...
0
votes
0answers
403 views
Is the principle of least action fully equivalent to the Euler-Lagrange equations?
I am citing from Landau and Lifschitz, this statement that will seem to you well-known, trivial, etc:
"Between these positions, (i.e. $q_1$ and $q_2$) the system moves then in such a way that the ...
0
votes
1answer
91 views
Wave guide boundary conditions
Why only the normal component of Electric field and the parallel component of Magnetic field exist at the surface of a wave guide or any conductor?
0
votes
1answer
99 views
Concentration distribution in a phase separated mixture. Can't get the correct ODEs and boundary conditions
I wish to compute the equilibrium concentration distribution of a binary mixture that has phase separated. I start with writing the free energy as a functional depending of the concentration. I use ...
1
vote
2answers
607 views
Solution of one dimensional wave equation by variable separation method
When solving the One dimensional wave equation by variable separable method, we equate left-hand side and right-hand side to a constant which is negative in nature. Why has the constant be only ...
0
votes
0answers
509 views
Boundary conditions of stream function
I have to do an problem about solving numerically the flow that goes under an airfoil. The airfoil has a flap deployed downwards and I need to solve the mesh that it's under the airfoil.
I have drawn ...
2
votes
0answers
67 views
Heat Transfer in Cylindrical Coordinates
Lets say one has an infinitely long cylinder with some boundary heat terms on $r=r_0$ of the form $T(r=r_0, \phi,z)=T_0(\phi,z)$. What is the general solution for this type of equation? The general ...
2
votes
0answers
95 views
What is “above” and what is “below” the surface of a sphere?
When studying Electromagnetism using D.J. Griffith's Introduction to Electrodynamics, the boundary conditions for the electric potential across a surface charge density are expressed using the normal ...
6
votes
4answers
3k views
Question on open organ pipe
Although open organ pipe is open on both ends, how standing waves are produced in a open organ pipe. Can someone explain with more clarity?
2
votes
2answers
205 views
Are solutions coordinate invariant?
In the case of electromagnetism, we can solve the sorceless wave equation in Cartesian coordinates ($x$,$y$,$z$) getting plane waves as solutions: $$ u(x) = A(x-ct) + B(x+ct) $$ and actually I am not ...
1
vote
2answers
136 views
What are the end points in the action integral of field theory?
In the mechanics of particles when we apply the principle of the least action the two end points are two spatial coordinates. Therefore, if we consider the variation of the action with respect to the ...
10
votes
5answers
39k views
Does light change phase on refraction?
I have seen a lot about when light undergoes a phase change when it is reflected. But does it undergo a phase change when refracted and if so why and if not why not?
0
votes
2answers
188 views
Conducting cylinder by dielectric interface
To help me with a project I'm working on, I attempted to solve what I thought was an easy problem - There is an infinite, conducting cylinder of radius R at some potential V, located distance b from a ...
0
votes
1answer
950 views
What are the boundary conditions for EM waves normally incident on the interface between two dielectric media?
An EM wave, amplitude $E_0$, frequency $\omega_0$, is incident upon a material with relative permittivity (dielectric function)
$$\varepsilon \left( z \right) = \left\{ \begin{gathered}{\varepsilon ...
4
votes
4answers
2k views
Rigorously prove that electric field is zero in a perfect conductor
I have ran into a problem while trying to prove that the electric field is zero in a perfect conductor
My argument went something like this:
We know that: $$\vec J = \sigma \vec E$$
In a ...
0
votes
2answers
164 views
Why does $\hat n \times (\vec E_1 - \vec E_2) =0 $ imply that the tangential electric field components are equal?
On page 8: http://local.eleceng.uct.ac.za/courses/EEE3055F/lecture_notes/2011_old/eee3055f_Ch4_2up.pdfele
I don't understand why $E_{t1} = E_{t2}$ is equivalent to $\hat n \times (\vec E_1 - \vec E_2)...
1
vote
1answer
364 views
Large rotation Euler-Bernoulli beam boundary condition
Is given in Wikipedia as
$$EI\frac{d^4u}{dx^4}-\frac{3}{2}EA\left(\frac{du}{dx}\right)^2\frac{d^2u}{dx^2} = q(x) ,$$ where $q(x)$ is the transverse load (assuming uniform cross-section and no axial ...
4
votes
1answer
983 views
Total derivative in action of the field theory
Consider a classical field theory.
When applying the least action I see that a term is considered total derivative.
We say that
$$\int \partial_\mu \left(\frac {\partial L}{\partial\left(\partial_\...
3
votes
2answers
563 views
Usage of Poisson's equation?
I revisited electrostatics and I am now wondering what the big fuzz about Poisson's equation
$$\nabla^2 \phi = -\frac{\rho}{\varepsilon_0}$$
is. Wiki says
One of the cornerstones of electrostatics ...
3
votes
2answers
604 views
Challenging Magnetostatics Problem - the “blind spot” of a magnetic dipole
I'm reviewing for an electromag exam and I stumbled upon a problem that's really hard to figure out. Here it is:
A small magnetic dipole with moment $\vec m = m_o \hat z$ is in a region with uniform ...
2
votes
2answers
1k views
What is the phase shift incurred by a sound wave as a result of reflection?
While studying waves I read the fact that a sound wave gets shifted by $\pi$ as a result of reflection against a surface. But I am unable to prove that fact.
Assuming the interface to be a node I can ...
0
votes
1answer
113 views
Boundary value problem
Consider the boundary value problem
\begin{align} \frac{du}{dt}&= \frac{d^2u}{dx^2} , \\
u(0,t)&=0 \\
u(L,t)&=0 \\
u(x,0)&=f(x) \end{align}
I know how to solve it using ...
3
votes
3answers
2k views
Quantum mechanics in electric field
Consider a charged particle with charge $q$ trapped in a box of length $L$ with finite constant potential $ V_0 $ on both ends. A constant (static) electric field of magnitude $F$ is applied from $- \...
3
votes
1answer
194 views
Inconsistency in the delta potential
I encountered an inconsistency in the one-dimensional delta potential. Suppose we have a one-dimensional infinitely deep square well from $-L$ to $+L$. We know the eigenstates are sine and cosine ...
1
vote
0answers
71 views
Can we take transport equation of imaginary quantity?
In the RANS equation we approximate the nonlinear fluctuating terms to eddy viscosity times strain rate. Then by using turbulence models like Spalart-Allmaras etc, we take the transport equation of ...
4
votes
0answers
87 views
Boundary conditions for enthalpy waves inside a pipe
So I'm trying to solve a form of the wave equation for sound produced by a vortex distribution $\vec{\omega}$ convecting at velocity $\vec{v}$ .
$$\left(\frac{1}{c_0^2} \frac{\partial^2}{\partial t^...
4
votes
0answers
418 views
Green's function for a dielectric with a charge [closed]
Suppose there are two infinite planes, one in $z=a$ and the other in $z=b$, with $a<b$. Between the planes, there is a dielectric medium with constant $\epsilon_1$. The differential equation for ...
3
votes
1answer
725 views
How to choose the Correct Green's Function?
In order to solve the Green’s function of the Helmholtz operator $$(\nabla^2+k^2)G(\vec r-\vec r’)=\delta^{(3)} (\vec r-\vec r’)$$ one can obtain four different Green’s functions corresponding to four ...
2
votes
0answers
178 views
Electrodynamics boundary conditions with complex $\epsilon$ and $\mu$
I wonder if the usual derivation for boundary conditions at an interface given in EMT textbooks hold for complex permittivity and/or permeability? Do the fields carry phase information themselves(i.e. ...
5
votes
1answer
517 views
Hamilton's equations from the action with boundary conditions involving position and momentum
Generally, when you are given the action
$$ S=\int_{t_1}^{t_2}\mathrm dt (p\dot q - \mathcal H )$$
the boundary conditions are
$$q(t_1)=q_1\quad\text{and}\quad q(t_2)=q_2.$$
This is useful ...
2
votes
0answers
107 views
Boundary Conditions for axisymmetric stream functions in a pipe
I'm solving the equation
$$
\frac{\partial^2 \psi}{\partial r^2}+\frac{\partial^2 \psi}{\partial z^2}-\frac{1}{r}\frac{\partial \psi}{\partial r} =-\omega_\phi
$$
in a cylindrical pipe, where $\psi(r,...
1
vote
1answer
539 views
Particle Outside the Box
What prohibits, mathematically, that a particle cannot be found outside the box ? Here, I am referring to particle in a box problem (infinite potential on both ends & zero potential along the ...
6
votes
2answers
3k views
Conductors and Uniqueness Theorem
I'm working with Griffiths Electrodynamics, and he introduces a uniqueness theorem:
First Uniqueness Theorem: The potential $V$ in a volume $\Omega$ is uniquely determined if (a) the charge density ...
14
votes
1answer
999 views
How do I enforce the no-slip boundary condition in time dependent incompressible pipe flow?
This is a technical problem which must have been solved already. It won't be in beginners textbooks but there should be a solution somewhere. I welcome reading suggestions. Maybe someone with ...
7
votes
1answer
1k views
Am I missing a trick to solving a 3D potential well problem?
I was playing around with a 3-D potential $V$ such that $V_{(r)} = 0$ for $r<a$, and $V_{(r)} = V_0>0$ otherwise. By using the Schrödinger Equation, I showed that: $$\frac{-\hbar}{2m}\frac{1}{r^...
0
votes
1answer
322 views
Reflected waves and phase changes
If a wave passes from a lightweight string to a higher density string, we say that the reflected wave has a pi phase change.
Can we say that it has minus pi phase change? If yes, why would that not ...
2
votes
1answer
138 views
String boundary conditions
I'm reading Polchinski and am confused about equation (1.3.13),
$$\gamma_{\tau\sigma}\partial_\tau X^\mu-\gamma_{\tau\tau}\partial_\sigma X^\mu=0~~~~~\text{at}~~~~~\sigma=0,l.$$
It says that this ...
1
vote
0answers
135 views
Contradictory boundary conditions in electrostatics problem?
Consider the following problem: A conducting cube of side $a$ is grounded. Inside there's a horizontal (i.e., perependicular to the $z$ axis) sheet with uniform surface charge density $\sigma$. The ...
1
vote
1answer
763 views
Junction conditions in GR including electromagnetism
I have recently learned about the Israel junction conditions in GR (as explained in for example Gravitation by MTW). I then tried to generalize it when including Electromagnetism, i.e. matching two ...
1
vote
2answers
5k views
No-slip boundary condition for viscous fluids
When dealing with fluid mechanics of viscous fluids, both theoretically and numerically, I've always been told that the boundary condition applied at solid walls has to be a no-slip one. My teachers ...
3
votes
3answers
610 views
How does a photon “know” that it's left one charge and that it's going to another one?
How does it know the same charge it left will be the same charge it will return to? My understanding is photons are neutral and have no charge.
i.e.
Like charges repel, unlike attract. All charged ...
1
vote
1answer
160 views
Does charge distribute itself uniformly on a conductor?
An excerpt from a beginning E&M book
[...] In other words, the surface of a conductor is an equipotential surface under static conditions. [...] Summarizing the boundary conditions at the ...
2
votes
1answer
268 views
Magnetic Field in the presence of a conductor
I am studying for my quals and came across an old question that reads like the following:
There are two regions in space separated by an infinite conducting plane. Region 1 has a magnetic dipole ...
3
votes
1answer
566 views
Conserved charges given conserved current via Noether's theorem
Let $j^{\mu}_{a}$ be the conserved current associated with an infinitesimal symmetry transformation, cf. Noether's theorem. The conserved charge associated with $j^{\mu}_{a}$ is $$Q_a = \int d^{d-1}x ...
3
votes
1answer
96 views
Self-adjoint extensions with 'teletransporting' boundary conditions
When choosing a self-adjoint extension of a Hamiltonian, in general one can obtain domains in which
(i) the probabilities teleport* between points on the boundary
and
(ii) boundary conditions ...
2
votes
1answer
720 views
Dielectric sphere placed in another dielectric medium with uniform external field: is there a surface charge density?
Consider a dielectric sphere placed within a dielctric medium. There is a uniform electric field $E_0$ present throughout in the medium.
Would there be surface charge on the sphere?
2
votes
4answers
1k views
The nature of “hard wall” boundary condition for Schrodinger's equation
For a quantum particle in an one-dimensional infinite well of width $L$, the potential has the formal expression:
$$
V(x) = \begin{cases}
\infty, & x < 0 \\
0, & 0 \le x \le L \\
\infty, &...
1
vote
2answers
92 views
Snells Law: Does the $k$ vector change on the boundary between mediums?
I was using Waves - Berkley Physics Volume III, and in explaining Snell's Law the author claims that as a wave is on the boundary between glass and air (going from glass to air) that the number of ...
