Harmonics Definition and 281 Threads

With the algebra so(3) are associated the spherical harmonics. I would assume that comparably with the algebra so(2,1) are associated functions that can be addressed as hyperbolic harmonics. But I nowhere found any reference to them. Do they exist and if so, where can they be found? Thank you...

I tried using the Wigner matrices: $$\sum_{m'=-2}^{2} {d^{(2)}}_{1m'} Y_{2; m'}={d^{(2)}}_{1 -2} Y_{2; -2} + {d^{(2)}}_{1 -1} Y_{2; -1} + ...= -\frac{1-\cos(\beta)}{2} \sin(\beta) \sqrt{\frac{15}{32 \pi}} \sin^2(\theta) e^{-i \phi} + ...$$ where $$\beta=\frac{\pi}{4}$$. But I don't know if...

Our brains are apparently musically happy when different individual notes are harmonically related to each other. In our music system, the fundamental frequencies of the different notes on a musical instrument such as the piano or the guitar are related to each other in such a way as to allow...

What combination of resources can you recommend for introducing people to spherical harmonics? Assume that the audience has the mathematical maturity of first-year grad students in mathematics, and will want a decent introduction to the theory and constructions. But also assume that this is part...

I would like to use VFDs for motor sizes ranging from 0.4kw to 15kw. My idea here is to reduce the generator foot print to the lowest possible value. My thinking is that, theoretically, I could use a 25kva generator set to drive a 20hp motor (80% operating capacity) and since the VFD eliminates...

I've tried figuring out commutation relations between ##L_+## and various other operators and ##L^2## could've been A, but ##L_z, L^2## commute. Can someone help me out in figuring how to actually proceed from here?

Let ##|l,m\rangle## be a simultaneous eigenstate of operators ##L^2## and ##L_z## and we want to calculate ##\langle l,m|cos(\theta)|l,m'\rangle## where ##\theta## is the angle ##[0,\pi]##. It is true that in general ##\langle l,m|cos(\theta)|l,m'\rangle=0## ##(1)## for the same ##l## even if...

So we are finding the L2 Lagrange point, specifically the distance from the earth, or d in this instance. I have used the equation above and I have come out with 1.5 * 10^9 meters as d, or L2's distance from the earth. Can anyone verify this, is the equation correct and is my final distance...

I have attempted the question using the large formula here with figures from the waveform with n as 3 & 101: however I think that the large formula is for the total waveform encompassing all harmonics as the 3rd and 101st harmonic amplitudes are very similar 10.9662222 *10^-6V for the 3rd...

Using the generating function for the legendre polynomial: $$ \sum_{n=0}^{\infty} P_{n}(x) t^{n}=\frac{1}{\sqrt{1-2 x t+t^{2}}} $$ It's possible to expand the coulomb potential in a basis of legendre polynomials (or even spherical harmonic ) like this: $$ \begin{aligned} &\frac{1}{\left.\mid...

I have a 3 phase problem that I think might be applicable to this forum. The setup is equipment which contains 5 resistive heaters (A through E). The unit is powered by 208V 3 phase and each of the heaters are hooked up line to line in an unbalanced configuration as follows: L1-L2: Heater D...

To show ##Y_{1,1}(\theta,\phi)## is an eigenfunction of ##\hat{L}^2## we operate on ##Y_{1,1}(\theta,\phi)## with ##\hat{L}^2## \begin{equation} \hat{L}^2Y_{1,1}(\theta,\phi)=\hat{L}^2\Big(-\sqrt{{\frac{3}{8\pi}}}sin\theta e^{i\phi}\Big) \end{equation} \begin{equation}...

See the first post in the previous thread ‘Matrices from Spherical Harmonics with Eigenvalue l+1’ first. Originally when I came across the Lxyz operator and the Rlm matrices I had a different question. If this had to do with something like the quantum Hydrogen atom then why did it appear that...

As the subject title states, I am wondering how would one go about transforming Cartesian coordinates in terms of spherical harmonics. To my understanding, cartesian coordinates can be transformed into spherical coordinates as shown below. $$x=\rho \sin \phi \cos \theta$$ $$y= \rho \sin \phi...

I’m New to the forum. I’m Interested if a certain set of matrices have any significance. To start out the unit vectors ##\vec i , \vec j, and ~\vec k ## are replaced with two dimensional matrices. ##\sigma r = \begin{pmatrix}1&0\\0&1\\\end{pmatrix}, ~\sigma z = \begin{pmatrix}1&0\\...

Suppose I decompose a discrete audio signal in a set of frequency components. Now, if I would add the harmonics I got, I would get the original discrete signal. My question is: if I would randomize the phases of the harmonics first, and then add them, I would get a different signal, but would it...

Im trying to solve the equation 62.7 of this numerical on mathematica. Whenever i try to normalized the function it shows function diverges. As the Bessel function contains trigonometry term so it diverges. I don't know how to solve the integral. Can i use the hydrogen atom wavefunction in exp...

In the contex of ##L^2## space, it is usually stated that any square-integrable function can be expanded as a linear combination of Spherical Harmonics: $$ f(\theta,\varphi)=\sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell f_\ell^m \, Y_\ell^m(\theta,\varphi)\tag 2 $$ where ##Y_\ell^m( \theta , \varphi...

My teacher said me this commutator is zero because the spherical harmonics are eigenfunctions of L^2. Actually, he said that any operator must commute with its eigenfunctions. I tried an example: [L^2,Y_20] expressing L^2 on spherical coordinates and I determined this commutator is not zero...

Hi PF! When solving the Laplace equation in spherical coordinates, the spherical harmonics are functions of ##\phi,\theta## but not ##r##. Why don't they include the ##r## component? Thanks!

if you fed a rectangular waves to a radio antenna, would it still produce radio waves? does it have to be a sine wave?

In basic optics, we are given the general solution of the wave equation (massless string of length L) as a linear combination of normal modes, that need to have some of the permitted frequencies due to boundary conditions. In laboratory, we observed that phenomenon. We generated a wave in a...

This question reminds me of the interpretation of the "wave packet" in QM for a free particle moving freely in the whole space; but in QM it is obvious that plane waves can't be of physical meaning. Now consider a guitar string (fixed at both ends) and suppose the solution to the wave equation...

Formula I have shows is probably for resulting harmonic? Does syntesizing mean writing the main formula? I havo no clue about effective value of signal and representation is probably done in excel or other graphing app.

Homework Statement [/B] So, this is a question I have in one of my assignments and I'm really going round in circles with it. Any pointers or links to additional reading would be gratefully received! This is a pass criteria question on a General Engineering HNC. Homework Equations [/B] Why do...

I'm expanding a function in spherical harmonics. I want to conserve axisymmetry of the function. what harmonics would respect that? Should I only include m=0 terms?

Hello forum, I am wondering why the higher order (higher harmonics) standing waves developed on a string under tension generated by an oscillating mechanical vibrator (set at the same amplitude but with variable frequency) have lower amplitude when compared to the lower harmonics (the...

There is some controversy as to why the spectral content of vinyl records contain frequency "information" above what the analog tape recorder was capable of capturing. A thought experiment: If I have a piano that has a key which plays 15khz (primary freq. of that key's piano string), what...

Seeking the guidance of the Physics forums gurus, I was told this would be a good question to contribute to the community. Perhaps it will lead to an in-depth discussion of harmonics: Their creation, implications to power quality, and how to mitigate them. I've been looking at some power...

In this video, at around 37:10 he is explaining the orthogonality of spherical harmonics. I don't understand his explanation of the \sin \theta in the integrand when taking the inner product. As I interpret this integral, we are integrating these two spherical harmonics over the surface of a...

When looking at the FFT spectrum of a sonometer, I noticed that the harmonics decayed faster than the fundamental. Why is this?

Homework Statement A sphere of radius a has V = 0 everywhere except between 0 < θ < π/2 and 0 < φ < π. Write an expression in spherical harmonics for the potential for r > a. For which values of m are there contributions? Determine the contributions through l= 2. How could you determine the...

Homework Statement I wish to solve this integral $$b_{lm}(k) = \frac{1}{2(\hbar)^{9/4}(2\pi)^{5/2}\sqrt{\sigma_{px} \sigma_{py} \sigma_{pz}}} \int_{\theta_k = 0}^{\pi}\int_{\varphi_k = 0}^{2\pi} i^l \text{exp}\left[ - \frac{1}{(2\hbar)^2}\left(\frac{(k_z - k_{z0})^2}{\sigma_{pz}^2} + \frac{(k_y...

Hi. Usually, the clarinet is presented as acting like a pipe system closed at one end, which only allows for harmonics that are odd multiples of the fundamental frequency. I used the app "SpectrumView" by OxfordWaveResearch to measure the following spectrum: Fair enough, the amplitudes of the...

Hello, I am working on a project to calculate harmonic distortion on a mains signal. The signal has already been stepped down so it oscillates between 0-5V at 50Hz. I looked into some old THD analyzers and it seemed they would generate a clean signal to mimic the fundamental deduct this from the...

Hi physics forms! I'm practicing to for an Quantum mechanics exam, and i have a problem. 1. Homework Statement I have two problems, but it's all related to the same main task. I have a state for the Hydrogen: ## \Psi = \frac{1}{\sqrt{2}}(\psi_{100} + i \psi_{211})## where ## \psi_{nlm}##...

Homework Statement The fundamental frequency of a violin string is 283 Hz. Calculate the frequency of the 2nd harmonic. Known: f = 283 Hz Homework Equations v = fλ f(n) = n*v/2L λ= L v(sound) = 343 m/s The Attempt at a Solution λ = 343/283 = 1.21 m f(2) = 2*343/2*1.21 = 283 Hz. I'm getting...

What I understand about harmonics, is when something is transmitted at high power, the antenna resonates on other frequencies besides the desired one. But Why?

i am learning about harmonic analyzers and i have a couple of questions concerning them that need some sort of practical experience to answer them ( and i have none :) ) : 1-can harmonic analyzer analyze up to any harmonic? or is there a maximum ? if there is no maximum what's the highest...

Homework Statement The spherical harmonic, Ym,l(θ,φ) is given by: Y2,3(θ,φ) = √((105/32π))*sin2θcosθe2iφ 1) Use the ladder operator, L+ = +ħeiφ(∂/∂θ+icotθ∂/∂φ) to evaluate L+Y2,3(θ,φ) 2) Use the result in 1) to calculate Y3,3(θ,φ) Homework Equations L+Ym,l(θ,φ)=Am,lYm+1,l(θ,φ)...

Hi All, [This thread has been re-started to accommodate modification of the project for the benefit of anyone that isn't a qualified Professional. Specifically the use of a Low Voltage isolated Source.] I want to get the opinion of people with more experience than me, regarding a project I was...

Hi, I keep reading in multiple sources that amplifier output can be given by Vout = a0 + a1v(t) + a2v2(t) + a3v3(t) + ... + anvn(t) I've checked in three of my textbooks and there is not a clear definition (its often just stated) why this equation is used and why it works. I am not looking...

Homework Statement The electric guitar radiates very little sound directly, but instead relies upon pick-ups (transducers) placed beneath the strings. Establish, showing clearly how you reach your answer, which harmonics of the open string you are likely to observe if the pick-ups are placed...

I saw a question "If you blow across the open end of a soda bottle and produce a tone of 250 Hz, what will be the frequency of the next harmonic heard if you blow much harder?" the answer is 750 Hz but I'm curious about "if you blow much harder" part, is it really depends on how much harder...

There is a nonlinear switch in every dc-dc converter.Is it the reason of harmonics in output voltage?

Hello. I was recently reading Barton's book. I reached the part where he proved that in spherical polar coordinates ##δ(\vec r - \vec r')=1/r^2δ(r-r')δ(cosθ-cosθ')δ(φ-φ')## ##=1/r^2δ(r-r')δ(\Omega -\Omega')## Then he said that the most fruitful presentation of ##δ(\Omega-\Omega')## stems from...

Hi all this is my first post. I am trying to do a simulation project revolving around the transient response of a rifle barrel after incurring a large pressure spike in the chamber. However, I have not found any firearms textbooks explicitly addressing this, and when I open a vibrations textbook...

My question is simply 'are all notes produced in a guitar produced by first harmonics?', but I will clarify what made me ask this question. Now, if you have a wave driver you can make several harmonics in a string by increasing the frequency of the machine. In a guitar string, however, it does...

How can the power/amplitude of a particular RF harmonic be calculated? I would assume it is some well-defined fraction of the amplitude of the main frequency? Do RF harmonics from a square-wave drive appear only on the even, odd, or both, multiples of the base frequency? How do high-frequency...

Is there a book/report(s) that can shed light on this issue? I am into making custom projects that are powered by engines and one, in particular, I am very interested in making sure I don't get any induced vibrations or oscillations. Suppose I have my engine affixed to a frame and I wish to...