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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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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Counting number of Antinodes for Modes in 2D

In 1D cases for standing waves, if we have the 3rd harmonic on a string of length L, then there are 3 antinodes and 2 nodes in between. This means, for each wavelength, we can establish a relationship ...
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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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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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Normal modes of vibration of a plate vs a membrane

I have been studying Chladni patterns but recently I have stumbled on some conceptual questions that I seem to not have an answer. At first I thought that the theory would be the same of a vibrating ...
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Doubt about finding normal modes in Molecular vibrations

In the book introduction to classical Mechanics by Kleppner and Kolenkow, while dealing with the analysis of molecular vibrations in a poliatomic molecule, they propose the following method, in order ...
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Quasinormal modes for certain black holes

Quasinormal modes for black holes are an important topic today. What would be the properties of a black hole whose quasinormal modes (all of them) have equal real part? And if all the imaginary parts ...
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Eigenvalue for complex variable

I was trying to reproduce the results of an exercise where they calculate the normal modes of oscillation. $$\begin{pmatrix} \dfrac{d}{dt}C \\ \dfrac{d}{dt}C^{*} \end{pmatrix}= - \dfrac{1}{i} \...
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Two-coupled oscillator: Doubt in finding normal modes and natural frequency

I want to find the natural frequency of a two coupled oscillator system like this- My book does it this way but I don't really get it. The equations of motion for the pendula are- $$I\frac{d^...
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Derivation of driven force on a string. How to prove maximum amplitud is achieved at resonant frequency?

I know if I have a driven oscillator of natural frequency $\omega$, applying a driven force $F_0 \cos (\Omega t)$ will result in a motion equation like this one (steady state/particular solution): \...
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280 views

Coupled oscillators in Hamiltonian formalism - problem with diagonalization

I have a problem with simple coupled oscillator system. I tried to solve single oscillator with Hamiltonian, and then coupled system of two, but when I try to put coupling constant $k^\prime=0$ in my ...
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Energy Eigenvalue for SHO Classical and Quantum

Let's assume we are given a potential for coupled harmonic oscillator: $$U = \frac{k_1(x_1^2 +x_3^2)+k_2 x^2+k_3 (x_1x_2 + x_2x_3)}{2}$$ If I solve the normal modes of the oscillator I get the ...
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How many linear combinations of harmonics or normal modes can describe the same periodic function as a Fourier series?

Please note that I am not asking how many terms in a linear combination can describe a specific periodic function but if given that there exist a set or linear combination of normal modes that ...
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Confusion on kinetic energy quadratic forms and eigenfrequencies

I am new to the idea of expressing kinetic energy in terms of the quadratic form. I noticed that online, people often express the kinetic energy as: $$T = \frac{1}{2} \dot q^T M \dot q \tag{1}$$ ...
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Simplified computation of matrices for normal modes?

In normal modes, we often refer the total potential energy of the system to be: $$V = q^T B q$$ where $V$ is the total potential energy, $q$ is the coordinates of the system and $B$ is just some ...
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How to calculate the normal modes from a given potential?

I have a potential of the form: $U=\frac{1}{2}\omega_{0}^{2}(x_{1}^{2}+x_{2}^{2})-\alpha x_{1}x_{2}$ How can I find the normal modes of this potencial? Should I expect solutions of harmonic oscilator ...
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At what rate is tension released from a plucked string (eg. guitar string)?

Tension in a plucked string (such as a guitar) dictates many of the important changes that occur in the sound of the string over time. eg. It causes a pitch bend, a change in inharmonicity, and ...
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Can someone please explain the meaning of the circled paragraph?

why does the off diognal elements of the matrix mediate with the coupling differential equation?
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Arbitrary motion of a normal mode

If a system having rigid boundaries is set into vibration with just right initial conditions, it would vibrate at a certain frequency (normal mode). But why we take the general motion to be ...
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Recovering symmetry in coupled oscillators

Consider a pair of LC oscillators, one with capacitance $C_1$ and inductance $L_1$ and the other with capacitance $C_2$ and inductance $L_2$. Suppose they're connected through a capacitor $C_g$. We ...
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Change of coordinates of Lagrangian

Consider the system above ($m_1$, $m_2$, and $m_3$ are connected by springs of stiffnesses $k_1$ and $k_2$, respectively. Also, $m_1 \neq m_2 \neq m_3$). The Lagrangian is $$L(x_{1},x_{2},x_{3},\dot ...
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To isolate a particular mode of vibration in a standing wave on a string

Suppose a string bound between two rigid end-points is vibrating and it is a combination of a number of normal modes of vibration, is it possible to isolate a particular mode of vibration in wave by a ...
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Coupled pendulum, energy exchange period as a function of different lengths

The energy exchange period $T_{x}$ of a coupled pendulum with coupling strength $\mu$ in the symmetric case(where the two natural frequencies with equal masses and equal lengths are the same, i.e. $\...
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Which Hamiltonian systems are intrisically linear?

What physical properties has a dynamical system whose equation of motion are linear? When does it exist a change of coordinates which turn the equation of motions in a linear system? My teacher says ...
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sound waves , standing waves,wind chimes [closed]

I don't know how to approach this problem. It's been bothering me for months. The answer seems to be the rod with shortest length
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Transverse modes in optical resonators

Im struggling with understanding transverse modes in an optical resonator or laser. Hopefully you can solve this mystery for me. Thank you very much! As far as I know, there are two types of modes: (...
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329 views

Lagrangian Mechanics - Normal Modes

System is as in the diagram shown with the hoop having mass $M$, bead having mass $m$ and the moment of inertia about the pivot point for the hoop is given by $I = 2MR^2 $. For the bead, we have $$ ...
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Confused about behaviour of spring mass system

I am writing some code that will plot the behaviour of a system consisting of 4 springs and 3 masses. They are arranged in the configuration (s:spring, m:mass) ...
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Equation of coupled springs : where does this potential come from?

In this document, we try to derive the equation of two coupled springs as in this picture. At the bottom of the page 2, they say : it would be more efficient to introduce the potential energy ...
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Probability at temperature in system has energy

Salutations, I'm starting in statistical mechanics and reviewing some related studying cases I would like to understand what occurs in small systems with normal modes of vibration, for example, a ...
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Event Horizons Vibrations

Can the event horizon of a black hole vibrate? If so, are there mechanisms that dampen the vibration? Consider a spherical, non-rotating, non-charged black hole located far away from other sources ...
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What physically determines Bessel functions' orders?

I would like a simple explanation of what, in a physical problem, determines the order of the Bessel functions that describe the solutions. What are dimensional(?) parameters of the system that induce ...
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Eigen-Energy of Vibration on a Loop

I recently started this recreational project to find the energy of the modes on a loop. To obtain the eigen-energies $E_n$, I decided to solve the 1D wave equation on a loop of circumference $2\pi l$ ...
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Calculating the decay rates for modes of an ideal circular membrane (ie. drum head) using wave equations?

I am attempting to solve for the theoretical 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 ...
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How would a layer of hot air affect the normal frequencies in a pipe?

Imagine that you have a pipe of length $L$ with one open end and one closed end. If the sound speed inside the pipe is $v_s$, then the fundamental frequency is: $$f_1=\frac{v_s}{4L}$$ and the ...
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Continuation of my previous questions on coupled harmonic oscillator

Coupled many body quantum harmonic oscillator in 3 Dimension $$H=\sum_{j}\frac{p^{2}_{j}}{2m}+\sum_{i<j}\frac{1}{2}k(R_{i}-R_{j})^{2}$$ To solve this problem I have used orthogonal jacobi ...
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How to evaluate the following Fourier calculation?

Let $\vec{k}$ and $\vec{r}$ represent coordinates in Fourier space and real space of a crystal. If there is no translational symmetry in the real space, is it possible to evaluate the following ...
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Normal modes and normal coordinates of polymer chain

Is it possible to solve to get the normal modes and normal coordinates of polymer chain (which is in 3D space) with nearest neighbour interaction (harmonic interaction). Kindly suggest some reference ....
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What is the physical meaning of the Fourier transform of the creation/annhilation operators in the nearest neighbour model?

It is possible to take the Fourier transform of the creation operator as $$a_k=\frac{1}{\sqrt N}\sum_n e^{-ik\cdot n}a_n$$ with $k=2\pi l/N$ and $l=\{-N/2+1, -N/2 +2 ... N/2\}$ but I am really ...
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Normal modes: how to get reduced masses from displacement vectors, atomic masses and vibrational frequencies

I'm calculating normal vibrational modes in a large molecular system. My goal is to obtain, for each normal mode, the vibrational frequency, the list of displacement vectors and the reduced mass. I'...
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Why are TM_10 modes not possible in a non-magnetic waveguide?

In a hollow rectangular waveguide, we may either propagate transverse electric modes (TE$_{mn}$) where we have electric standing waves quantified by m and n in each direction, or transverse magnetic ...
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What is the shape of the vibration when the system is exited at off natural/resonant frequency?

I understand that when the system is exited and left to vibrate freely many of its vibrational modes will be present as a linear combination. If the system undergoes a forced vibration at one of the ...