Questions tagged [normal-modes]

Normal modes refer to fundamental patterns of motion of a system which oscillate at fixed, well defined frequencies. They may be used as building blocks for more complicated motions.

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Terminology question: in-phase or out-of-phase?

Suppose that in a chain of many coupled oscillators, the displacements of two consecutive particles, in a normal mode of oscillation with frequency $\omega$, are given by $$x_p(t)=A_pe^{i\omega t}$$ ...
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Why is any arbitrary motion of a coupled oscillator writable as a linear combination of its normal modes? [duplicate]

Consider the following example of a coupled oscillator. Let two identical pendulums, each of length $\ell$ and mass $m$ be connected by a spring of force constant $k$. The system has two normal modes ...
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Coupled Oscillator Period [closed]

I was studying an example of a coupled oscillator the other day, namely two identical masses attached to three springs, the lateral ones of which with the same elastic constant, when I came across the ...
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Normal modes of coupled oscillators

For two pendulums of mass $m_1$ and $m_2$, coupled by a spring of constant k, both suspended by strings of length $l$, the following matrix equality results from their equations of motion: $$ \omega^2 ...
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Why dielectric waveguides support hybrid modes and metallic waveguides don't?

Can anyone explain me qualitatively, why dielectric waveguides (core and infinite cladding) support hybrid modes ($E_z$ and $H_z$ components in the guided wave) while metallic waveguides cannot. I am ...
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Which normal mode/combination of normal modes actually ensue?

For a classic problem consisting 2 coupled oscillators, I did the usual and found the normal mode frequencies, and constructed a general solution of the form: $\begin{bmatrix} x_1(t) \\ x_2(t)\end{...
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How to move atom coordinates along a normal mode given the eigenvector?

I need to displace atoms along a complicated normal mode involving many atoms. I have the normal mode eigenvector, I just don't know how to turn the eigenvector into an algorithm to translate the ...
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Center of mass position for open string theory with Dirichlet typo in textbook

In the string theory textbook Basic Concepts of String Theory by Lust, Blumenhagen, in equation (2.100) is presented the mode expansion for an open string subjected to Dirichlet boundary conditions: $...
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What are normal modes?

In normal modes analysis the differential equations of the system are Fourier transformed and the Fourier monochromatics are found. I think these monochromatics are usually called normal modes of the ...
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How to calculate mode shapes for 2D membranes?

I have a rectangular membrane which is fixed at the edges and undergoes vibrations. I measure the signals (in time domain, so time vs voltage) travelling through this material, through which I can ...
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Minimal frequency of unspecified drum

I was given the following wave function, related to an unspecified drum: $$\frac{\partial ^2Ψ}{\partial t^2}=c^2 \left ( \frac{\partial ^2Ψ}{\partial x^2} + \frac{\partial ^2Ψ}{\partial y^2} \right )-\...
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Why do guitar strings behave so nicely?

To explain the harmonics on a guitar string, we use 2D models of the string. For example we assume that the string can only go up and down. But the string is inherently a 3D object and it could ...
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Do normal modes need same amplitude?

Suppose we have a coupled pendulum of 2 masses. I understand that the first normal mode they oscillate together, in phase, with the same frequency. However, do they need to have the same amplitudes? ...
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Deriving the parameter $\exp(\eta)$ in eq. (2.9) from "Illustrative example of Feynman’s rest of the universe"

I'm working on research on the Entanglement in Coupled Harmonic Oscillators when I stumble upon the research paper "Illustrative example of Feynman’s rest of the universe" https://doi.org/10....
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Lagrangians for Non-Interacting Scalar Fields in QFT

I am currently taking a QFT class and we are using both canonical and path integral quantization to solve non-interacting scalar fields. We have seen the real scalar field with Lagrangian $$\mathcal{L}...
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Normal Mode in a Vibration?

What exactly is the Normal Mode? According to me, it forms a basis functions (where Max transverse amplitude is fixed wrt position), and all arbitrary vibration in the string can be written in the ...
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Intuition for normal modes of a beaded string

These questions are inspired by the following the paper http://www.soton.ac.uk/~stefano/courses/PHYS2006/chapter7.pdf on 'Normal Modes of a Beaded String'. Problem Statement Given a recurrence ...
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Modes of Motion of A Lennard-Jones Cluster

I am trying to analyse the dynamics of a cluster of 79 atoms. The context of the question is that each atom in the cluster (which has Lennard-Jones interactions between the atoms) is displaced by a ...
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Phonons vs Normal mode

What is the difference between Phonons and Normal mode? My professor told me that they are the same and that one can get the other from some derivation (I'm a bit unsure if he really said the '...
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Are there non-orthogonal "normal" modes for non-identical coupled oscillators?

The question is broad, I will specify an example to elaborate what I'm asking. Suppose I have two different LC circuits inductively coupled (or capacitively, but the question I have will be relevant ...
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Plasma modes analysis with Laplace and Fourier methods

I've seen that in order to investigate plasma's modes it's possible to linearize the system and then Fourier transform it (for longitudinal modes with kinetic description the Laplace transform in time ...
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Modes of vibration of triatomic molecules

$$\omega=0,\sqrt{\frac Km},\sqrt{\frac{K(2m+M)}{Mm}}$$ There are three modes here but actually triatomic linear molecule have 4 vibrational modes (e.g. CO2). So where does that remaining one mode? ...
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2 masses, 3 spring system [closed]

I came across this problem in normal modes of oscillations. Now I tried doing it this way: The potential energy of the system should be $$ V=\frac{1}{2}K(x^2+y^2)+\frac{1}{2}K'(x-y)^2$$ and the ...
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String vibration patterns

Sometimes when you pluck a string on a stringed instrument, the string turns into what looks like a nearly stationary loop with a semi-transparent middle. Sometimes, the loop doesn't appear stationary,...
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What are 'internal' and 'external' modes of vibration?

I was reading about Raman spectroscopy for crystals. I have come across these two terms: internal and external modes of vibrations in crystal. I have no clue what they might mean. I am looking for a ...
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Interpreting physical meaning of normal modes

What really is a normal mode? Maybe it's because of my teachers but I find it really abstract. I know that "numerically" corresponds to the eigenvectors of the equation $\ddot{X}= -M^{-1}KX$ ...
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Phonons and Modes

I am trying to understand how normal modes, optical and acoustical cases and longitudinal/transversal propagation of waves are all together related. Lets say we are have a clystal chain (for ...
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What is a mode in quantum optics?

I am studying quantum optics and it is often cited the word "mode", in particular there are spatial and temporal modes. I really don't know what they are. I know the general definition of ...
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Double eigenfrequencies of normal modes

If I have 3 masses displayed along a ring connected by springs, the frequencies I found were: $$\omega^2=\frac{3k}{m},0$$ I don´t understand why I have 2 double eigenfrequencies. Is this possible, if ...
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Why do we study black hole quasi-normal modes for different fields?

In Zhidenko's paper on quasi-normal modes of a Schwarzschild de Sitter black hole, he considers the damped oscillations in the background space-time (quasi-normal modes) for fields of different spin (...
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Simultaneous diagonalization of potential and kinetic energy

I am trying to prove that the matrix expression of the potential energy (Hessian matrix from a Taylor expansion in several variables of the potential) is diagonal considering small oscillations, when ...
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Confusion in Deriving Formula for Energy in Terms of Normal Modes

If we have an equation of motion of the form $$ \mathbf{\ddot{y}} = -A\cdot\mathbf{y}$$ for a real symmetric matrix $A$, we can dot product both sides of the equation by $\mathbf{\dot{y}}$ to obtain ...
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Physical Interpretation of Normal Modes

For a simple case of frictionless coupled oscillators shown in the figure below: (Image: two pendula of equal length and equal masses suspended from a level ceiling and connected by a spring) (and ...
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String vibration $\rightarrow$ particle properties

The vibrating of the strings is identified with some properties attributable to the particles. How can I find the equations that show how the mode of vibration of a string (in the Calabi-Yau manifolds)...
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Understanding quasi normal modes of black holes

I think I am missing some basic understanding of QNMs of black holes: for example, for the Kerr black hole Leaver has calculated the first few QNMs and has found that they are complex conjugated. I do ...
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What is half width of gain profile? why is it used as delta lamda while calculating number of longitudinal modes? (Correct me if wrong on this)

How many longitudinal modes can be excited for an He-Ne laser in a cavity of length 30 cm and having half width of gain profile of laser material 2 ×〖10〗^(-3) nm? The emission wavelength is 6328 Å.
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Interpretation of normal modes from the mathematical formula

In the topic of small oscillations, the system below has a normal mode described by: $$n_{1} = \frac{x1+x2}{2}.$$ This normal mode is represented as the symmetric mode: In that case, the center of ...
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Stability around an equilibrium point in a normal mode

So, I was reading the chapter of small oscillations in Landau and Lifshitz's book of Mechanics. We assume solutions of the equations of motion that are in the form of $X_a=Ae^{iω_at}$ where $A$ is an ...
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Propagator in normal modes

I started with the Hamiltonian of coupled oscillators in a circular lattice(with $m=\hbar=1$ and $x_{a+N}=x_{a}$) $$H=\frac{1}{2}\sum_{a=0}^{N-1}\left[p_a^2+\omega^2 x_a^2+\Omega^2\left(x_a-a_{a+1}\...
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The notion of TEM, TE and TM modes in an arbitrarily shaped cavity

In order to define the usual modes of EM waves in a confined space, TEM, TE and TM, one must have a well defined notion of "transverse" and "longitudinal" in the system. In the ...
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What are the normal modes of a velocity-dependent equation of motion?

I'm trying to find the normal models of a particle with charge $q$ and mass $m$ in a $3$-dimensional harmonic oscillator potential with an applied uniform magnetic field $B=B_0 \hat{z}$. The potential ...
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Generalised coordinates

I am working on a scientific project for my university and I am reading a german paper (Karas: "Platten unter seitlichem Stoß") which makes use of generalised coordinates. It's about an analytical ...
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Initial values for oscillations about a point (normal modes problem) [closed]

I have the Lagrangian $L=\frac{1}{2}(\dot{x}^2+\dot{y}^2) -7x^2+2xy-\frac{11}{2}y^2$. So the two equations of motion are $\ddot{x}+14x-2y=0$ and $\ddot{y}-2x+11y=0$. Hence the general solution (by ...
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Propagation modes of a wave

I'm studying Debye's Theory and I am really confused about what really is a propagation mode. The energy of a $3$-dimensional object is given by $$ \xi_{\alpha , k} = \hbar v_{\alpha} |k|$$ where $...
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What is the function of the amplitude of a plucked string depending on where it is plucked?

Here is the image of the situation: Where L is the lenght of the string, A, the amplitude and p is a fraction of the string's lenght (where it is plucked = pL). So, here it is, How do I find an ...
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Landau mechanics - Normal modes of oscillation

In Landau's Mechanics book there's a section in which he explains small oscillations in systems with $s \geq 1$ degrees of freedom. He writes the kinetic and potential energies as $$ T = \sum_{i, k} \...
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Why does stepping on the floor produce sound?

I've been thinking about this question for some time now, and couldn't get to any conculsion. My understanding is the following: 1) when my feet and the floor are far apart (a few centimeters), they ...
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3 votes
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Symmetries of quasinormal modes of Kerr black hole

I have been reading this following paper on numerical evolution of the Teukolsky equation (see e.g. Eq 1 in their paper) for spin -2 fields about a spinning black hole (Kerr) solution. As the ...
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Lattice vibrations in one-dimensional monatomic crystals vs. diatomic crystals

A one-dimensional diatomic crystal (with two distinct atoms A and B arranged in a line) can exhibit two types of collective motions. In one type, the consecutive atoms move in-phase and in the other, ...
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What is the limit of validity for WKB expansion for black hole Quasi-normal modes?

Which is the limit of validity for WKB expansion for black hole Quasi-normal modes? In many papers I see that the authors only report the overtone $n=1$. Is WKB expansion valid only for small $n$? As ...
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