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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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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33 views

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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26 views

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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36 views

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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26 views

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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74 views

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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43 views

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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50 views

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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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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51 views

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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66 views

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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1answer
91 views

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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33 views

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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234 views

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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600 views

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 ...
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Difference between a “mode” and a “state” in quantum mechanics?

I am studying the book Introductory Quantum Optics by Gerry & Knight at the moment and as a reader, I stumble upon their seemingly interchangable use of the tems "mode" and "state". As far as I ...
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252 views

Clarifying and simplifying what 'mode' means in terms of SHM

I am dealing with a string-coupled pendulum, where two pendulum are tied onto one string as seen in image 1. (Image attributed to Young-ki Cho available from pre-view at DeepDyve) The symmetrical ...
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Simple coupled quantum harmonic oscillator

I am having trouble finding the eigenvalues for the Hamiltonian $$ H = \frac{P_1^2}{2M} + \frac{P_2^2}{2m} + \frac{K}{2}x_1^2 + \frac{k}{2}(x_1 - x_2)^2$$ Even though I can find a basis where the $...
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What is a mode? Regarding electromagnetic waves in a closed cavity

This concept is very unclear to me, and the more different sources I find, the more contradicting definitions and explanations I stumble upon. Let's use this source as a basis for my question: http://...
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245 views

Why do normal modes not exchange energy between each other?

I've been looking around for an answer to this and have had trouble learning about why this is. Does it have to do with the fact that the modes are eigenvalues?
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318 views

Canonical commutation relations and a scalar interacting field

Let $\hat\phi(x)$ be a quantum scalar hermitian field and $\hat \pi(x)$ its conjugated field. IF the scalar field evolves according to a FREE theory it is possible to write down the following normal ...
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Vibrational modes of a black hole's event horizon

I understand that it is possible for the event horizon of a black hole to support damped modes of vibration called quasi-normal modes, in which it oscillates between spherical and and various oblate ...
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Normal modes of cylinder and two pendulums

I am trying to understand(falsify) interpretation of the answer to the following problem: Suppose we have cylinder of mass M which can oscillate along the horizontal line. There are two pendulums ...
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229 views

String vibration and damping

In case of a home experiment about string vibration under the boundary condition $$y(l,t)=y(0,t)=0$$ Where $y=$ the displacement of the string at spatial co-ordinate $x$ and at time $t$, I observed ...
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187 views

String vibration for inhomogeneous string

In case of the vibration of a homogeneous string under the boundary conditions that the string is fixed at both its ends, using fourier analysis we can show how the amplitudes of successive ...
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409 views

What is the difference between “normal mode” and just “mode”?

So in the oscillation problems, is there difference between "mode" and "normal mode"? I know that "normal modes" are independent and orthogonal, so one doesn't affect the other. Now I am not sure when ...
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Meaning of imaginary part of complex field amplitude for waveguide modes (e.g. TE, TM, HE, EH)

In classical waveguide analysis (e.g. for optical fibers, as in the notes Modal analysis of step-index fibers, ECE 4006/5166 Guided Wave Optics, Robert R. McLeod, University of Colorado), one can find ...
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106 views

How does scaling friction in Langevin equation effect time delayed cross correlations?

Assume we have a stochastic process $$ dX_t = V_tdt $$ $$ dV_t = -KX_tdt -MV_tdt +dB_t. $$ with $X_t \in \mathbb{R}^n$ representing position and $V_t$ velocity. This is a linear elastic model ...
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662 views

Chladni Patterns [closed]

I'm trying to make a simulation for Chladni plates the diagram above shows the patterns on rectangular plates, and relates them to 2 values (n, m) many of the explanations suggest that (n) is the ...
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1answer
126 views

Wave function normal modes in 3-torus

I'm studying the states of spin-less neutral particles and their Fock space. I upload the picture in order to make my question clearer. Talking about $(N,r)$-improper states, the expectation value ...
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151 views

Why is the sound of filling a container with liquid always the same regardless of container?

I found a similar question asking why the sound changes as fluid level changes. Foobarbeque's answer mentions Helmholtz Resonance, and that the frequency depends on cavity volume, length of neck and ...
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Transversal string with free ends question

We are given a string of length $L$ tied to massless rings at both ends so as they can move freely in the transversal direction. The linear density of the string is $ρ$ and the tension at rest (where $...
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63 views

Real Electromagnetic Waveguide Modes in loss media

Suppose a waveguide with 2 Perfect electric conductor at both boundaries. The waveguide is filled with a lossy media modelled with a conductivity $\sigma$. Solving for the following Maxwell's ...