Boundary conditions Definition and 418 Threads

Homework Statement Consider a plane monochromatic wave incident on a flat conducting surface. The incidence angle is ##θ##. The wave is polarized perpendicular to the plane of incidence. Find the radiation pressure (time-averaged force per unit area) exerted on the surface. Homework Equations...

(Mentor note: moved here from noon homework thread hence no template) I was studying vibration of a one-dimensional monatomic chain and the textbook used periodic boundary condition (PBC). I wanted to justify the use of PBC, so I came up with this: atoms deep inside the crystal sees an...

I've been trying to come up with wave equations to describe the motion on vibrating rectangular (more specifically, square) membranes. However, most paper I find assume fixed edges. What are the boundary conditions I need to apply to the 2D wave equations in order to have an free boundary in a...

I'm trying to read this paper. Right now my problem is with equations 3.16 and 3.17. I understand that in equation 3.16 we're putting some boundary conditions on the fields, but I have two problems with these boundary conditions: 1) The fields depend on both ## t_E ## and ## x##, i.e. ##...

I have a couple homework questions, and I'm getting caught up in boundary applications. For the first one, I have y'' - 4y' + 3y = f(x) and I need to find the Green's function. I also have the boundary conditions y(x)=y'(0)=0. Is this possible? Wouldn't y(x)=0 be of the form of a solution...

This question is inspired by Gilbert Strang's Course on Computational Science and Engineering, MIT 18.085. Consider the three matrices Fixed-Fixed $$K=\begin{bmatrix} 2 &-1 & 0 &0 \\ -1&2 & -1 &0 \\ 0 & -1 &2 & -1 \\ 0 & 0 & -1 & 2 \\ \end{bmatrix} $$ Free-Fixed $$T=\begin{bmatrix} 1 &-1 & 0 &0...

Hi everyone, This is my first time posting here I am looking to get some help with Abaqus, I wish to compare two models and find the residual stress which causes a original model to deform to the other. The deflection between the two models can be calculated by other software. I plan to do...

Hi all. I don't have as much experience with thermal analyses in ANSYS, and I can't quite figure out a problem that I'm working on. I'm trying to find the temperature distribution in a box that has a circle in the center. I want to define constant temperature b.c.'s at the walls and the circle...

Hi all! I have to calculate the natural frequency of the system. Any idea of boundary conditions of this case? There is beam supported by two springs on the left side.

If we have an infinite square well, I can follow the usual solution in Griffiths but I now want to impose periodic boundary conditions. I have \psi(x) = A\sin(kx) + B\cos(kx) with boundary conditions \psi(x) = \psi(x+L) In the fixed boundary case, we had \psi(0) = 0 which meant B=0 and...

In calculus of variations, extremizing functionals is usually done with Dirichlet boundary conditions. But how will the calculations go on if Neumann boundary conditions are given? Can someone give a reference where this is discussed thoroughly? I searched but found nothing! Thanks

Homework Statement Homework Equations 3. The Attempt at a Solution [/B] I know dV=1/C∫idt and that we integrate the voltage from V to V0. What I don't get are the boundary conditions for t - How do we get what we get in the parenthesis? My closest assumption is that the t/T values refer to the...

Suppose we are solving a diffusion equation. ##\frac{\partial}{\partial t} T = k\frac{\partial^2}{\partial x^2} T## On the domain ##0 < x < L## Subject to the conditions ##T(x,0) = f(x) ## and ##T = 0 ## at the end points. My question is: Suppose we solve this with some integration scheme...

Homework Statement Find the Green's function $G(t,\tau)$ that satisfies $$\frac{\text{d}^2G(t,\tau)}{\text{d}t^2}+\alpha\frac{\text{d}G(t,\tau)}{\text{d}t}=\delta(t-\tau)$$ under the boundary conditions $$G(0,\tau)=0~~~\text{ and }~~~\frac{\text{d}G(t,\tau)}{\text{d}t}=0\big|_{t=0}$$ Then...

Homework Statement I'm having issues with a Laplace problem. actually, I have two different boundary problems which I don't know how to solve analytically. I couldn't find anything on this situations and if anybody could point me in the right direction it would be fantastic. It's just Laplace's...

The equation is Uxx + Uyy = 0 And domain of solution is 0 < x < a, 0 < y < b Boundary conditions: Ux(0,y) = Ux(a,y) = 0 U(x,0) = 1 U(x,b) = 2 What I've done is that I did separation of variables: U(x,y)=X(x)Y(y) Plugging into the equation gives: X''Y + XY'' = 0 Rearranging: X''/X = -Y''/Y = k...

This isn't homework but could be labeled "textbook style" so I'm posting it here. Homework Statement I'm trying to solve \frac{\partial^2 u} {\partial x^2} +\frac{\partial^2 u} {\partial y^2}=0 on the domain x \in [-\infty,\infty], y\in[0,1] with the following mixed boundary conditions...

Laplacian for polars: $$\frac{1}{r}\frac{\partial}{\partial r}\left( r\frac{\partial \phi}{\partial r}\right) + \frac{1}{r^{2}}\frac{\partial^{2} \phi}{\partial \theta^{2}} = 0$$ This is in relation to a problem relating to a potential determined by the presence of a wedge shaped metallic...

I'm modelling a system with a nanosized semiconductor in 1d, inside which I want to find the electrostatic potential. Having found this I am unsure what boundary conditions to put on this, when it is connected to a metal on one side and to vacuum on the other. So far I have put that it is...

I've been stuck over this integral for around an hour while studying the derivation of the second coefficient of the virial equation: ∫∫dx1d x2 γ(x1,x2) where γ(x1,x2) is 1 when x1 - x2 < constant. = V∫ dx2 γ(x1,x2) where V is the integral of dx1. Given: periodic boundary condition: x1 + V = x1...

Hello, forum! I'm just starting a new course on heat transfer and we're using Incropera's book. Last time I studied heat transfer was in my transport phenomena course, using BSL, so it was kind of a culture shock using the new book, because the methods used are kind of different in some cases...

Hi there, I'm solving the equation for the transverse vibrations of a Euler-Bernoulli beam fixed at both ends and subject to axial loading. It's a similar problem to that described by Rao on page 355 of his book "Vibration of Continuous Systems" (Google books link), except the example he uses...

Hello. I'm trying to wrap my head around how Lagrangians work in classical field theory. I have a book that is talking about the gauge invariance of the Lagrangian: \mathscr{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-J^\mu A_\mu. It shows that we can replace A^\mu with A^\mu+\partial^\mu\chi for...

The Euler-Lagrange equations give a necessary condition for the action be extremal given some lagrangian which depends on some function to be varied over. The basic form assumes fixed endpoints for the function to be varied over, but we can extend to cases in which one or both endpoints are free...

I am trying to solve the differential equation ##\frac{d^{2}y}{dr^2}+(\frac{1}{r}+1)y=0## with the boundary conditions ##y(r) \rightarrow r \frac{dy}{dr}(0)## as ##r \rightarrow 0## and ##y(r) \rightarrow \sin(kr+\delta)## as ##r \rightarrow \infty##. I know that the shooting method is the...

Hi! I can't understand how to implement boundary conditions in a 2D axisymmetrical model. How should be the value of pressure, x-velocity and y-velocity at the axis of symmetry? Thank you!

I have come across the paper attached in which a 1D fluid piston is modeled. I have question on the boundary conditions (BCs) of the system. Essentially, the problem consists of a fluid chamber in contact with a spring (a mass -spring system). ALE is used to move the mesh. I am not certain...

I want to model the diffusion-controlled combustion of a small carbon particle. The system I want to model is similar to this one However, I'm not going to use the stagnant gas film model as shown in the figure, since I lack data for the film thickness, and I want to evaluate the problem...

Hello, Can anyone help me find the boundary conditions of the below given beam please. Its a clamped-free beam but the overhanging sectiona and the mass makes it confusing. Actually I am puzzled about finding the initial conditions.

Good evening, PF! I am supposed to model the following system. I will be using the same notation as in BSL. Fluid A enters the reaction zone at z = 0 at a concentration CA0. A reacts to form B in the first order reaction A → B at a rate of R_A = k_1''' C_A. We assume the whole mixture to be...

Hello, PF! Recently, while reading chapter 10 (microscopic energy balances) of the second edition of BSL, I found a minor discrepancy which is confusing me, especially when considering the mathematical analogies of heat and mass transfer. In section 10.1, the authors introduce Newton's law of...

How to find value of integration constant?I know with the help of boundary conditions,but How boundary conditions help in finding integration constant?

Homework Statement How do the normal and tangential components of the Poynting vector in matter, S = E x H , behave at an interface between two simple media where no free current flows, but either free charge or polarization charge is present at the interface? Homework Equations E1(para) -...

Hi, just want to confirm that with the eigenfunction boundary condition $ p(x) v^*(x)u'(x)|_{x=a} = 0 $, the order of (solutions) v, u doesn't matter? I ask because a problem like this had one solution = a constant, so making that the u solution makes $ p(x) v^*(x)u'(x) = 0 $ no matter the...

I am reading Jackson Electrodynamics (section 1.10 in 3rd edition) and he is discussing the Poisson eqn $$\nabla^2 \Phi = -\rho / \epsilon_0$$ defined on some finite volume V, the solution using Greens theorem is $$\Phi (x) = \frac{1}{4 \pi \epsilon_0} \int_V G(x,x') \rho(x')d^3x' +\frac{1}{4...

I've forgotten a lot of field theory so I've been rereading it in a couple of electric field theory textbooks. What seems like a simple problem falls between the cracks. I hope some readers can help - it will be appreciated. My application seems simple (solution will require numerical FEA but...

I am working with a fixed fixed bar with a distributed axial load to the right as w(x)=CX/L. I am having a hard time determining the force boundary conditions. I know that U(0)=0 and U(L)=0. However, I need to come up with something in regards to U'(Value). Any help would be appreciated.

Hi, I have two coupled differential equations d^2 phi(z)/dz^2=lambda*phi(z)*(phi(z)^2+psi(z)^2-sigma^2) d^2 psi(z)/dz^2=lambda*psi(z)*(phi(z)^2+psi(z)^2-sigma^2+epsilon/lambda) where lambda, epsilon and sigma are arbitrary constants. The equation subject to the bellow boundary conditions...

If you seal a loudspeaker at the end of a tube and close the other end of the tube you will get standing waves; but what are the boundary conditions at the speaker for the sound pressure wave? Pressure =0 or Pressure = MAX? I find no mention of this in the literature. To find out I performed a...

Consider the heat equation dT/dt - aΔT + v⋅∇T = S where S is a source term dependent of the radiation intensity I and the temperature T. The fluid velocity v is prescribed. We also consider the radiative transfer equation describing the radiative intensity I(x,ω,t) where ω is the ray direction...

How would I go about finding temperature distribution in a thin square plate during the the first few milliseconds (or actually a fraction of a millisecond) after t=0s. Initial temperature distribution throughout the plate is known, there is heat flux to one side = Qinj, while heat flux from all...

I am trying to set up the mass matrix for a 1D system which I want to solve using finite elements. So the mass matrix is defined as M = \int{NN^T}dL, where N is the finite element linear basis functions. I use hat functions. Say I have 10 elements, corresponding to 11 nodes running from -5...

I am not sure if this should be posted here. If not I hope you accept my apologizes and the admin move on the post as soon as possible. I am studying manifold with boundaries and boundary conditions in a quantum field theory approach. Could you recommend me books or papers about that? I lack...

Hi! When we model bloch-waves in a solid we assume that there exist some kind of periodic boundary conditions such that the wave function is periodic. In 1D, ##\psi(x)## repeats itself for every ##L##, ##\psi(x) = \psi(x+L)##, such as here: OK, fine, we get pretty wave solutions if we assume...

we have that Ht1 (x,y,z) - Ht2 (x,y,z) = Js and for the special case Ht1 (x,y,z) - Ht2 (x,y,z) = 0 where there is no surface current. At a boundary with Js =0, which for simplicity let's asume is at at x = a, then knowing that Ht1 and Ht2 are the magnetic fields to the left and right of the...

Homework Statement Two elastic bars are joined. A step wave is coming in from left. Derive the shape and magnitude of the reflected wave if the right bar is approximated by a rigid body (point- mass) that is free to move in the axial direction. The Attempt at a Solution I have problem with...

Consider the radial differential equation ##\bigg( - \frac{d^2}{dr^2} + \frac{(l+\frac{d-3}{2})(l+\frac{d-1}{2})}{r^2} + V(r) + m^2 \bigg) \phi_l (r) = \lambda\ \phi_l (r)##, which I've obtained by solving the Schrodinger equation in ##d## dimensions using the method of separation of...

Suppose that we want to solve the eigenvalue equation with Dirichlet boundary conditions ## \bigg(-\frac{d^2}{dx^2}+V(x)\bigg) \phi_n = \lambda_n \phi_n,\ \ \ \ \ \ \ \ \ \ \ \ \ \phi_n(0)=0,\ \phi_n(1)=0, ## where ##0 < \lambda_1 < \lambda_2 < ...## are discrete, non-degenerate eigenvalues...

I have a question I'm a little embarrassed to be asking: what is meant in condensed matter when someone describes a system with "open boundary conditions," say in one-dimension for simplicity? I am comfortable with the statement of fixed (Dirichlet) or free (von Neumann) boundary conditions, as...

Hi all! I am trying to solve a system of partial differential equations in Matlab, with both derivatives in time and space domains. I am using the pdepe function for that. The system is, to be simple, a sort of solar thermal panel, made of three layers: an absorber plate, a fluid layer of...