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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Eigenvalues and Normal Modes in SHM

I'm reading Symmetries part from the textbook provided by MIT OCW Physics3 8.03SC course, but have a question about the condition to find normal modes of SHM. In the book they mentioned $S$ - symmetry ...
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Finite string fixed at one end and connected to frictional massless bead [closed]

Assume we have a finite string of length $L$. The string has tension $T$ and wave speed $c$. The string is fixed at $x=0$ and connected to a massless bead at $x=L$ that can slide on a rod. The rod ...
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Vibrating string and damping

Suppose we have a string (in tension) with its ends fixed. Think of a guitar string. Suppose we start with a plucked initial position and we let the string free. If we use the wave equation: $u_{tt}=c^...
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Vibration of a continuous uniform chain and the normal modes

The question is: A vertically hanged chain with the upper end attached to a fixed point. I try to find the normal modes under the small $\theta$ condition. Consider the mass $\mathrm{d}m$ with ...
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Normal modes of three masses attached by two springs

I have the following system: I've applied Newton's 2nd Law to the system and I have found the normal modes proceeding as an eigenvalues and eigenvectors problem. I obtained the frequencies $\omega_1^...
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Why each normal mode is treated as a harmonic oscillator in Debye's calculation of specific heat?

So in Einstein's calculation of specific heat each oscillator is assumed to be vibrating with same frequency and its average energy is given by hv(n+1/2) where n is bose factor. Debye said that ...
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Equipartition of ideal gas energy in Fourier space

Consider an ideal gas of $N$ particles at temperature $T$ enclosed in a box of volume $V$. Equipartition of energy between the velocities $v_{i}$ of the particles implies that on average the magnitude ...
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How is the energy of phonon modes $\left(n+\frac{1}{2}\right)\hbar \omega_k$ when each atom in the mode has $\left(p+\frac{1}{2}\right)\hbar \omega$?

I'm having trouble understanding the quantisation of energy in normal modes of lattice vibrations. Stating Kittel : The energy of a lattice vibration is quantized. The quantum of energy is called a ...
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How is energy distributed over the normal modes?

Consider a linear chain of $N$ point masses $m$ connected by linear springs $k$ and fixed at the two ends by rigid walls separated by a distance $L=N\times l$. If I take the point mass at the center $...
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Picturing a normal mode of vibration of the monoatomic lattice in 1D that is either in-phase nor completely out-of-phase

For a monoatomic lattice of $N$ atoms in one-dimension, the ratio of the displacements of two consecutive atoms at the $(n+1)$th and the $n$th site is given by $$\frac{u_{n+1}}{u_n}=e^{ika}$$ where ...
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Degeneracy in a 2D planar cavity (box potential)

Assume a finite 2D planar cavity. One can write the energy of a photon in this cavity as $$ \begin{equation} E(k_x, k_y)=\hbar c \sqrt{k_x^2+k_y^2+k_z^2}, \end{equation} $$ where $k_z$ is fixed (hence ...
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Even or odd solutions to Maxwell/Schrodinger equations

The solutions to the source free Maxwells equations in a photonic cavity (or equally the solutions to the Schrödinger equation) can be even or odd; $u(x) = u(-x)$ or $u(x) = - u(-x)$ or neither. But ...
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Viewing String Oscillations with a Camera

In this video, a demonstrator shows normal modes on a string. If one is doing this with a high speed camera, how many frames per second does one need to view the oscillations? My intuition was that ...
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Why is the energy of the harmonics in a vibrating string not infinitesimal?

When you pluck a guitar string, initially the vibration is chaotic and complex, but the components of the vibration that aren't eigenmodes die out over time due destructive interference. This ...
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Normal modes of a circular array of interacting particles

I want to study the normal modes of an array of $N$ identical atoms placed in a circular lattice. The particles interact among them via Yukawa interaction potential, $$\phi_Y(r)=\frac{A}{r}exp(-r/r_0)....
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Normal Modes: Is the mass matrix always diagonal?

When considering normal modes we often end up with a hamiltonian of the form $$ H = \frac{1}{2} \dot{x}^T M \dot{x} + \frac{1}{2} x^T K x $$ where $x$ is a vector of degrees of freedom in the system, $...
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Calculating Fourier spectra of particle current autocorrelation functions

The particle current density can be defined as : $$\textbf j(\textbf r,t)=\sum_{i=1}^{N} \textbf v_i\delta(\textbf r-\textbf r_i(t))$$ Its spatial Fourier transform : $$\textbf j(\textbf k,t)=\sum_{i=...
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If two pendulums connected by a spring both following Simply Harmonic Motion - why do they need the same time dependence?

I was reading a section of Introduction to Mechanics by Kleppner and Kolenkow: where it talks about the same time dependence. I'm not very familiar with this term but was wondering if there was some ...
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Exciting a normal mode of $N$ coupled oscillator with driving force

Suppose we have $N$ coupled oscillator with the fixed ends. We can find the normal modes of this system by considering an infinite system and using space translation symmetry to diagonalize the ...
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Diagonalizing the tridiagonal matrix for finding the normal mode [closed]

Suppose we have n particles connected by a string tied to fixed ends. In matrix notation the equation of motion can be written as $\begin{pmatrix}2\omega_o^2&-\omega_o^2&0&0&.&.&...
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One normal mode frequency not seen in Resonance in forced coupled oscilators

The problem is that I have three charges with mass $m$ connected by two springs with the same elastic constant. The first and the last charge has charge $q$, the middle one $-q$. They are inside an ...
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Is the argument of zero tangential field components in wave guide walls a fallacy? Or: How EM waves "magically" find their correct transverse modes

I just came across a nice answer on Quora, explaining visually how wave guide modes can be constructed from the condition of zero tangential fields in the wave guide walls. In mathematical terms, this ...
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Why do systems of $n$ coupled oscillators have $n$ normal modes?

Consider a linear system of $n$ differential equations with constant coefficients corresponding to a physical scenario where I have $n$ coupled oscillators (like $n$ masses attached by springs in ...
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Initial conditions in an infinite string of masses

Assume an infinite string of masses $m$ connected by springs with constant $\kappa$. The masses in equilibrium are evenly spaced by $a$. Then the equation of motion for the $j$-th mass is $$ \ddot{\...
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Calculation of normal frequencies of a normal mode

In the Classical mechanics book by Goldstein, it is stated that if one wants to find the normal frequencies of a system, $\omega$ then the following equation has to be solved: $\left|\hat{V}-\omega^2\...
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What is the relation between Chebyshev polynomials and coupled oscillators?

I have been told that Chebyshev polynomials are key for finding the normal modes of oscillations of a linear chain of coupled oscillators, since they are the eigenmodes of the system. However, I ...
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Difference between Symmetric and Phase difference?

For a simple coupled oscillator system such as the one here, with equal spring constants and equal masses (with a displacement from equilibrium of $x_1$ and $x_2$), it follows that: $(\ddot{x}_1+\ddot{...
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How to integrate a system with a hamiltonian with non-diagonal kinetic energy

I have the following classical Hamiltonian for two coupled oscillators in the same molecule: $$H=T+V =\left(\frac{p_1^2}{2\mu}+\frac{p_2^2}{2\mu}+k_pp_1p_2\right)+ \left(\frac{1}{2}\mu\omega_1^{2}x_1^...
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Transverse Vibrations of a Cantilever beam. Why does the % change in nth natural frequency increases with n in this case?

This is the problem in detail I have a cantilever beam. Now I am making it stand upright and adding water for different heights of SUBMERGENCE I am doing this by adding an ADDED MASS term from the ...
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How to calculate the energy of a spring-mass system considering harmonic oscillation of the normal mode? [closed]

For a spring-mass system, we know that the potential and kinetic energy are $$E_p = \frac{1}{2}ku^2 \text{ and } E_k = \frac{1}{2}m\dot{u}^2.$$ where $k$, $m$ and $u$ are the spring constant, mass and ...
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Are quasi-normal modes in black hole perturbation theory the same as resonances in Quantum Mechanics?

When we evaluate quasi-normal modes in black hole perturbation theory (also here), by solving the Regge-wheeler potential, is it same as solving the Schrödinger equation for a barrier and finding ...
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Nomenclature for stationary states in the context of wave equations

Consider the Schrodinger equation $i\partial_t u=-\Delta u.$ Special solutions of the form $$u(t,x)=u_k(x)e^{ikt}$$ have many names which I've seen used, such as stationary states, solitons, standing ...
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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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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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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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