Separation Definition and 448 Threads

I am trying to follow through a derivation of the solution to the wave equation governing a string with fixed ends via separation of variables but I am stuck at the step which concludes ‘m’ must be a natural number 1,2,3,… etc. as opposed to an integer. After analysing each case: m > 0, m = 0...

A bit messy but the bottom is supposed to be the potential function

The mentioned coordination complex has been prepared where KCl is a significant byproduct. The salt and the complex appear to be readily soluble in the same solvents. Despite the organic ligand (PDTA), it is not soluble in polar organic solvents, at least to a degree where it can be used for...

IIRC, He_3 is usually separated from much less rare He_4 by cryogenic cooling of gas mix to 'liquid', at which point the mix divides to two phases, one with each isotope... IIRC, Hydrogen and Deuterium, as gas mix, may be progressively separated at near-ambient conditions by differential...

Hello. Let's say we have two masses, each moving in 90 percentage of light speed in opposite direction. Then what will be the speed of the one mass according to an observer in the other mass?

I vaguely (strong word there because I can no longer remember the source, but the idea sticks in my head for 30 years now) recall reading (somewhere long forgotten) that method of separation of variables is possible in only 11 coordinate systems. I list them below: 1.Cartesian coordinates...

Problem: Solution: When I looked at an example problem, they started writing the potential in terms of the Legendre polynomials. The example problem: This is what I did: $$V_0 \alpha P_2 (\cos(\theta)) \Rightarrow \frac{\alpha 3 \cos ^2 (\theta)}{2} - \frac{\alpha}{2} \Rightarrow \frac{\alpha...

using the equation ##u(x,y)=f(x)g(y)##, first, I substitute the value of ##u_{xx}## and ##u_{yy}## in the given PDE. after that solve the ODEs but I can't understand about the ##u_{t}##.In my solution, I put ##u_{t}=0## because u is only the function of x and y. Is it the right approach, to me...

Is it possible for all the events in a field to be light-like separated from each other? What would such a field be like?

Hello, I want to see if hydrocyclones can be used in a application, where we have small metall particles that we want to extract from a closed loop water system. I found a book "Chemical Engineering Design" written by Richardson, where there are some equations given that makes it possible to...

I have been reading about the Ranke-Hilsch vortex tube. Details of the explanation tend to differ somewhat among different sources, but it got me thinking about the following thought experiment. Air enters a tube of about 0.5 x 4 cm cross section. It passes through a section that is channelized...

A quasar with a bolometric flux of approximately 10−12 erg s−1 cm−2 is observed at a redshift of 1.5, i.e. its comoving radial distance is about 4.4 Gpc. Assume that the quasar in the previous question is observed to have a companion galaxy which is 5 arcseconds apart. What is the projected...

Hi, I asked this question elsewhere, but I didn't understand the answer. It seems to be easy to understand, but for some reason I'm really confuse. I'm not sure how to find the average position of an electron and the average separation of an electron and his proton in a hydrogen atom. To be...

Im wondering if plasma is possible to be separated into a positive nucleus and negative electrons and contained within a magnetic bottle ? If possible, what is the most efficient method of achieving it ?

I just want to know if there is a consensus on experimental evidence of spin-charge separation in 2d?

Hi all, At my workplace we have a waste stream that includes both natural and synthetic corks. The synthetic corks we can further process, so the natural corks have to be selected out. We have two Polish workers who are able to do this just by sight, 8 hours a day. However, we tried to put...

Suppose I have 2 variables q and t (time), where q is some reparameterization of x (position) : ##x \to q = x f(t)##. Suppose I have a partial differential equation : $$\frac{\partial u(q,t)}{\partial t} = k \frac{\partial u(q,t)}{\partial q}$$ where k = constant Then I do a separation of...

The relative density data: $${\rho_{H_{2},H_{2}O}}=\frac{\rho_{H_{2}}}{\rho_{H_{2}O}}=8.988\times10^{-5}$$ With avogadro number, thus I can obtain number of molecules per 1 ##m^3## of Hydrogen gas, that is: $$N = \frac{{{\rho_{H_{2},H_{2}O}}}\times{\rho_{H_{2}O}}}{M_{r}}\times{N_{A}} $$ thus, I...

Can anyone tell me what the commonly used methods of isotope separation are for Potassium 41? I know there are many different methods used for isotope separation, but I'm wondering which method is most practical (cheapest and purity) in the case of potassium, specifically K41. Also roughly what...

I am reading on this part; and i realize that i get confused with the 'lettering' used... i will use my own approach because in that way i am able to work on the pde's at ease and most importantly i understand the concept on separation of variables and therefore would not want to keep on second...

If the maximum separation between two points on an infinite line is finite, then what is its value? So the maximum separation is infinite. Does this mean two points on an infinite line can be separated by an infinite distance? Why, why not?

I used the concept of electrostatic induction, which would cause the charges in metal ball near the ebonite rod to have +ve charges on end next to rod and a -ve charge on the end touching the other ball. What confuses me is how charges separate on the second ball. The only way these balls can...

Do I need to use formula to answer this question? Can't I just divided the horizontal distance in the picture by 2. so the horizontal separation of the thread is 54.8 / 2 = 27.4 mm? Thanks

Given a probability density distribution ##P(\vec{x})##, for what named distributions is the following true: \begin{equation} \begin{split} P(\vec{x}) &= P_1(x_1) P_2(x_2) ... P_n(x_n) \end{split} \end{equation}

Each different boundary condition means a different charge configuration, how can this problem be solved using superposition?

I'm having troubles setting up this problem. I know we are to use boundary conditions to determine An and Bn since in this case (a<r<b) neither can be set to 0. I don't know how the given potentials translate into boundary conditions, especially the V3 disk.

I have no idea how to solve this problem. The solution says that the component parallel to the plane of separation is conserved, i am not sure why. Seems to me that in the problem was assumed a special field, but not a generic field.

If the boundary condition is not provided in the form of electric potential, how do we solve such problem? In this case, I want to use ##V = - \int \vec{E} \cdot{d\vec{l}}##, but I don't know how to choose an appropriate reference point.

If two events A and B are spacelike separated, is it then true that there can be no causal relationship between the two?

$\tiny{s8.2.6.3}$ Find y' $\sqrt{x}+\sqrt{y}=1$ \begin{array}{lll} \textit{isolate }y &\sqrt{y}=1-\sqrt{x} &(1)\\ \\ \textit{square both sides} &y=(1-\sqrt{x})^2 &(2)\\ \\ \textit{differentiate both sides} &y'=2\left(1-\sqrt{x}\right)\left(-\dfrac{1}{2\sqrt{x}}\right)&...

I was solving the van't Hoff equation over an interval ##[T_1 , T_2]##: The van't Hoff equation ## \frac{\mathrm{d} \ln K}{\mathrm{d} T} = \frac{\Delta_r H^{\circ}}{RT^2} ## which can be solved with separation of variables: ## d \ln K = \frac{\Delta_rH^\circ}{RT^2}dT ## ##\Updownarrow##...

Is it possible to use separation of variables on this equation? au_{xx} + bu_{yy} + c u_{xy} = u + k Where u is a function of x and y, abck are constant. I tried the u(x,y) = X(x)Y(y) type of separation but I think something more clever is needed. Thank you.

2d sin q = nl d = nl / (2 sin q) d = (1 x 0.16) / (2 sin 30o) = 0.16 nm But 0.16 is not in the option. And how to find other value of d? Thanks

Consider the static field configuration shown in the image. There are three layers: 0 = vacuum, 1 = magneto-optic fluid and 2 = covering shell. Each of these layers have their own permittivity and permeability (ε_i,μ_i) (isotrope). A uniform electric field H_0 = H_0/sqrt(2) * (e_x + e_y) is...

I have tried this question and have gotten to an answer from the following steps So my angular separation is 2.85 millidegrees. Have I done this right with the formula I have made use of? Any help would be great, thanks!

Good Morning I have a very vague memory of having read (about 40 years ago) that there are only 11 coordinate systems in which the field equations of physics can be separated. I can no longer be sure if my memory has failed me. But this issue has been in my head for all these years. (Gotta do...

I got as far as: $$[\hat \phi(x), \hat \phi(y) ] = \int \frac{d^3p}{(2\pi)^{3}(2E_p)}(\exp(-ip.(x-y) - \exp(-ip.(y-x))$$ Then I simplified the problem by taking one of the four-vectors to be the origin: $$[\hat \phi(0), \hat \phi(y) ] = \int \frac{d^3p}{(2\pi)^{3}(2E_p)}(\exp(ip.y) -...

My view is that it is possible to separate any solute from any solvent by centrifuge. Likewise it is possible to separate heavy water from H1 water by centrifuge. Molecular disassociation is probably a borderline possibility, especially if we consider the molecular disassociation of your...

Hey there! I am current taking an introductory course on PDE's, and our professor hasn't really emphasized last part of solutions from separation of variables. Now its not strictly going to be on the exam, however I remember doing this with ease a few years back, but for some reason now I...

In the given problem, i can understand that after placing the two blocks in equilibrium it oscillates with an amplitude of The answer for (b) is given as To my knowledge, m2 separate from m1 when the acceleration is greater than gsinø and so they should be separating only at max displacement...

Proton is going towards the ##\alpha## particle. So, I am thinking of using the conservation of energy as the initial kinetic energy of the proton is known and initial interaction potential energy is zero. But, we don't know the kinetic energies of proton and ##\alpha## particle when they are at...

I have a PDE which I have solved numerically using a guass-seidel method, but I want to compare it to the analytical solution. I have used separation of variables to get the general solution, but I need help applying it. The PDE is (1/s)⋅(∂/∂s)[(s/ρ)(∂ψ/∂s)] + (1/s2)⋅(∂/∂Φ)[(1/ρ)(∂ψ/∂Φ)] - 2Ω...

Attempted rewriting acceleration, a, in terms of dv/dt and then separating variables to integrate. This didn’t work... So then I remembered that my gamma factor is also a function of v (!), but then I think I would be required to play around with integration by parts, which seems overly...

Hi all, A solution contains ions, positive and negative ones (i.e k+ cl-). Is there a simple but passive way (without adding any kind of energy) to separate by exemple the potassium ions from the solution? (IMHO, I do not think so)

Distance is d=1/0.07 = 14.3 parsecs The Doppler shift of one star is, Δλ = 512 - 512.04 = -0.04 So the radical of the velocity of the star is = (-0.04/512) x (3.00 x 10^5 km/s) = 23.4km/s which is the same for both stars because they have the same mass. This is as far as I've got.

When using the separation of variable for partial differential equations, we assume the solution takes the form u(x,t) = v(x)*g(t). What is the justification for this?

Homework Statement In a convoy on a long straight level road, 50 identical cars are at rest in a queue at equal separation ## 10 m ## from each other. Engine of a car can provide a constant acceleration of ##2 m/s^2##. And brakes can provide a maximum deceleration of ##4m/s^2##. When an order...

I wrote my first LaTeX like this: ... it was easy to calculate that $$ ê^1=\hat x - \frac { \hat y } {\tan \alpha} $$ and $$ ê^2=\frac { \hat y } {\sin \alpha} $$ where ... Is it possible to make the vertical separation smaller? The first example in the latex primer does not seem to suffer...