A normal mode of an oscillating system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The free motion described by the normal modes takes place at fixed frequencies. These fixed frequencies of the normal modes of a system are known as its natural frequencies or resonant frequencies. A physical object, such as a building, bridge, or molecule, has a set of normal modes and their natural frequencies that depend on its structure, materials and boundary conditions. In music, normal modes of vibrating instruments (strings, air pipes, drums, etc.) are called "harmonics" or "overtones".
The most general motion of a system is a superposition of its normal modes. The modes are normal in the sense that they can move independently, that is to say that an excitation of one mode will never cause motion of a different mode. In mathematical terms, normal modes are orthogonal to each other.
This exercise comes from Kleppner and Kolenkow, 2nd ed., problem 6-3. I'm using a solution key as a study reference, but the solution key is coming to a pretty different conclusion. Mostly the issue is in the equations of motion for this system. I'm not sure if there's something I'm...
Hello
I wrote a Matlab code to form the 8 by 8 stiffness matrix of a single, 4-noded element, for a plane strain problem for an isotropic element.
I conduct an eigenvalue analysis on this matrix
Matlabe reports 5 non-zero eigenvalue modes, and 3 zero-eigenvalue modes (as expected)
Of the 3...
On page 52 in Becker, Becker, Schwarz, there is an equation (2.148) for the number of open string excitation modes.
I tried to Tayler expand eq 2.145, but couldn't reproduce 2.148. Plus, one gets 2.145 by setting w close to 1; even if I use the 2.146 and try to analyze it around 0, I am still...
https://arxiv.org/abs/2207.02472
InAs-Al Hybrid Devices Passing the Topological Gap Protocol
"We present measurements and simulations of semiconductor-superconductor heterostructure devices that are consistent with the observation of topological superconductivity and Majorana zero modes."...
I was doing the exercise as follows:
I am not sure if you agree with me, but i disagree with the solution given.
I was expecting that the kinect energy of the mass ##m## (##T_2##) should be $$T_2 = \frac{m((\dot q+lcos(\theta)\dot \theta)^2 + (lsin(\theta) \dot \theta)^2)}{2}$$
I could be...
Homework Statement:: Find the interference function ##I(\delta)## where The emission is analyze by a Michelson interferometer.
Relevant Equations:: ##I(\delta) = \frac{1}{2} \int_{-\infty}^{\infty} G(k) r^{ik \delta} dk## ##I(\vec{r}) = I_1 + I_i + 2 \sqrt(I_1 I_i) cos (k\delta)##
I have 5...
ρ_kdk = k^2/π^2 dk is the density of field modes (what we are trying to solve for here), and as ρ_kdk = ρ_λdλ, and k=2π/λ, we can rearrange this to get ρ_λdλ = 8π/λ^4dλ
This is where my confusion lies. I am not sure what to do next. I know this equation physically means the number of modes per...
In the book "Fundamentals of photonics", the authors defined waveguide modes using the notion of linear systems, where they said:
"Every linear system is characterized by special inputs that are invariant to the system, i.e., inputs that are not altered (except for a multiplicative constant)...
Hey all,
I was citing a result from a review paper in my paper, and I think it's wrong. I would really appreciate an outside perspective if anyone has the time!
The result was for the electric field outside a metal rod (cylindrical waveguide, if you prefer) in vacuum. Here's the picture (you...
In Carroll "Spacetime and Geometry" I found the following explanation for why the analytically extended rindler modes share the same vacuum state as the Minkowski vacuum state:
I can't quite understand why the fact that the extended modes [\tex]h_k^{(1),(2)}[\tex] are analytic and bounded on...
what is fluoroscopy mode in cardiac xray imaging?. simulator shows EP,LF,MF,HI(extra low fluoroscpy, low, medium and highfluoro.what is it actually means?
I am trying to analyse the dynamics of a cluster of 79 atoms.
The system can be described with:
##\omega^2 \vec x = \tilde D\vec x##
Where ##\omega^2## (the eigenvalues) are the squares of the vibration frequencies for each mode of motion, ##\tilde D## is the "dynamical matrix" which is a...
I pretend to use the ecuation twice, once for the interior and another for the vaccum, so if I use the cilindrical coordinates for \nabla_t^2 it results in two Bessel equations, one for the interior and another fot the vaccum.
In the vaccum, the fields should experiment a exponential decay, in...
I am considering a system that includes a pn-region with N first then P, then a superconductor, but I am not interesting in the effect of the modes present due to the confinement of the p-region of the system on the conductance (from electrons to holes). How do negate these energy modes in the...
Hi,
I know there's are 2 normal modes because the system has 2 mass. I did the Newton's law for both mass.
##m\ddot x_1 = -\frac{mgx_1}{l} -k(x_1 - x_2)## (1)
##m\ddot x_2 = -\frac{mgx_2}{l} +k(x_1 - x_2)## (2)
In the pendulum mode ##x_1 = x_2## and in the breathing mode ##x_1 = -x_2##
I get...
I have studied the modes of the MOSFETs from this site https://www.electronics-tutorials.ws/transistor/tran_6.html however in many videos such as this : they don't agree with each other.Which is correct and which isnt?
In 1D Photonic crystals, a defect can be introduced to create a defect/resonance mode and enable transmission. At first considerations, the thickness of the single defect layer determines the transmission frequency. Moreover, if it is a half-wavelength layer it will enable a resonance condition...
Suppose, there is an electro-optical modulator that can couple the neighboring modes in an optical ring resonator. The Hamiltonian for the system
looks something like this^^ (see the attached image). Here we sum over all modes m and 𝜙0 is a parameter. What will be a good set of basis for the...
We need to find the normal modes of this system:
Well, this system is a little easy to deal when we put it in a system and solve the system... That's not what i want to do, i want to try my direct matrix methods.
We have springs with stiffness k1,k2,k3,k4 respectively, and block mass m1, m2...
A mass ##m## is restricted to move in the parabola ##y=ax^2##, with ##a>0##. Another mass ##M## is hanging from this first mass using a spring with constant ##k## and natural lenghth ##l_0##. The spring is restricted to be in vertical position always. The coordinates for the system are ##x##...
I saw the solution of the light propagates in cylinder.. so in every solution there is the first order Gaussain function (the slandered one) times another function which gives I think the separation, both of them gives the intensity separation.. So what does that mean?! is it as I draw on the...
This is a question from an exercise I don't have the answers to.
I have been trying to figure this out for a long time and don't know what to do after writing
mx''¨(t)=−kx(t)+mg
I figure that the frequency ω=√(k/m) since the mg term is constant and the kx term is the only term that changes.
I...
If I have three modes in an optical fiber with elliptical facet as shown in the figure, what would be the polarization direction of the modes... what I know that it is linear polarization and could have several configuration.. but what I want really to know that if could be a situation where the...
Hello!
In Optical fibers, let ##k_1## and ##k_2## be respectively the propagation constants in core and cladding, ##\beta## the propagation costant of a mode along the direction ##z##, ##a## the radius of the fiber. Using the normalized quantities ##u=a \sqrt{k_1^2 − \beta^2}## and ##w=a...
My understanding is that you can describe the complicated motion of atoms in a crystal as a sum of standing waves (normal modes). A phonon is an excitation of a normal mode in the sense that it increases the vibration amplitude of that normal mode and the energy of that mode by a quantized...
I found the equations of motion as
##m\frac{\mathrm{d}^2x_1 }{\mathrm{d} t^2} = -\frac{mg}{l}x_1 + k(x_2-x_1)##
and
##m\frac{\mathrm{d}^2x_2 }{\mathrm{d} t^2} = -\frac{mg}{l}x_2 + k(x_1-x_2)##
I think the k matrix might be
##\begin{bmatrix}
mg/l + k & -k \\
-k & mg/l + k
\end{bmatrix}##...
My thinking so far is that if the two different input mode operators of a beam splitter commute, but I can't really give any good reasoning behind it.
I defined ##\hat{A} = \frac{i\pi}{4}\hat{a}_0^{\dagger}\hat{a}_1 ##
and
##\hat{C} = \frac{-i\pi}{4}\hat{a}_0\hat{a}_1^{\dagger} ##
and am...
Dear FEA experts,
I’m trying to analyse* some finite elements model of a thin walled cylinder with variable cross-section, but I’m observing four weird issues in the buckling modes. The structure is vertically (along z-axis) and horizontally (along y-axis) loaded on top. Would you help me to...
In a step-index optical fiber, considering Bessel functions of order ##\nu = 0## and no ##\phi## dependence, it is possible to obtain two separate sets of components, which generate respectively TE and TM modes. In the former case, only ##E_{\phi}##, ##H_r##, ##H_z## are involved; in the latter...
Let's try inputting a solution of the following form into the two-dimensional wave equation: $$ \psi(x, y, t) = X(x)Y(y)T(t) $$
Solving using the method of separation of variables yields
$$ \frac {v^2} {X(x)} \frac {\partial^2 X(x)} {\partial x^2} + \frac {v^2} {Y(y)} \frac {\partial^2 Y(y)}...
I think the answer for this may be straightforward, but I don't see anywhere that explains this from the scratch:
A large resonant cavity with a small hole is used to approximate an ideal black body.
I understand the conditions for the modes inside the cavity. But there are two points that...
What are the most likely modes of decay for ##\Omega ^{-}## into 2 hadrons?
##BR_{k}=\frac{\Gamma _{k}}{\Gamma}##
##\Gamma=\frac{\hbar}{\tau }##
##\Gamma _{k}=\Gamma _{if}=2\pi \rho|<\Psi _{i}|H_{Int}|\Psi _{f}>|^{2} (E_{f})##
I took a look at the Particle Data Group, and the most likely modes...
Many papers about random lasers mention the Q-factor of random lasers. Since a random laser has multiple peaks close to each other like shown in the figure. Does each of these peaks correspond to a unique random lasing mode, or is it just a single mode?
Similarly what is the right way to...
Hi. I'm having trouble with calculating
(8.59)
from (8.58)
(## \vec H_{//} ## is simply ## \vec H ## in Jackson, but that shouldn't matter.)
in Jackson. Specifically I think I'm not sure about parallel component of H field here.
For example, shouldn't I have two terms for the...
Protium atom has two low lying excited states with long lifetimes.
These are:
2s. Decay energy would be 121 nm, but forbidden (no angular momentum difference). Fastest allowed decay is two-photon emission, lifetime 0,15 s
Triplet 1s. Decay energy 211 mm. Prevalent decay single photon emission...
Homework Statement
Show that, in the Debye theory, the number of excited vibrational modes in the frequency range ##\nu## to ##\nu+d\nu##, at temperature T, is proportional to x2e-x, where ##x=h\nu/kT##. The maximum in this function occurs at a frequency ##\nu'=2kT/h##; hence ##\nu'→0## as...
Hello,
I'd like someone to help me understand, how can I tell from available data, what is the approximate share of different decay modes, for some given nuclide activity.
Let's take 90Y for instance. It's known for being beta-emitter, but it emits gammas and X-rays as well. How to...
I am trying to solve for the theoretical relative decay rates of the various (m,n) modes of an ideal circular membrane, if that membrane is excited momentarily by an impulse or deformation.
I would ideally like the decays of the (m,n) modes in dB/s.
Imagine a simple isolated drum head being...
What actually is a mode of optical fiber propagation?Is it similar to modes which correspond to various configurations as in standing waves on a string ? Also How correct is it to consider no. of rays as no of modes?
I'm looking at what should be just a simple spring system where four identical springs are holding up a square, load-bearing pallet plate in a warehouse. Now, someone says the equation of motion for the vertical normal mode of vibration is simply d2z/dt2 = -4(k/m)z.
Right away however, I see no...
I am studying coupled oscillations and one of the refrance I'm using says that two modes can have same frequency whereas the other one says it's impossible to have same frequency for two modes. Please help.
Hi PF!
I am solving a problem for low-gravity capillary-dominated flow, where liquid rests in a rectangular (2D) channel. The text I'm reading introduces a velocity potential ##\hat u = \nabla \phi##, and then states that each oscillation mode is ##\partial_n\phi|_\Gamma = \nabla \phi \cdot...
Hello people,
I'm a little bit confused about how to define the polarization direction for TM/TE mode.
Take a look at the TE mode picture I found in some place.
Picture1
The Cartesian system of coordinate (XYZ) here is chosen by the right hand rule.
Picture2
But how we chose the direction...
I am working on some acoustic synthesis models of real world instruments. The Bessel Function zeros give the vibration modes of a circular membrane, which can be used to model a drum head or even roughly a cymbal.
However, much of a drum's sound (especially snare) comes from the "ring" of the...