What is Harmonics: Definition and 281 Discussions

A harmonic is any member of the harmonic series. The term is employed in various disciplines, including music, physics, acoustics, electronic power transmission, radio technology, and other fields. It is typically applied to repeating signals, such as sinusoidal waves. A harmonic is a wave with a frequency that is a positive integer multiple of the frequency of the original wave, known as the fundamental frequency. The original wave is also called the 1st harmonic, the following harmonics are known as higher harmonics. As all harmonics are periodic at the fundamental frequency, the sum of harmonics is also periodic at that frequency. For example, if the fundamental frequency is 50 Hz, a common AC power supply frequency, the frequencies of the first three higher harmonics are 100 Hz (2nd harmonic), 150 Hz (3rd harmonic), 200 Hz (4th harmonic) and any addition of waves with these frequencies is periodic at 50 Hz.

An nth characteristic mode, for n > 1, will have nodes that are not vibrating. For example, the 3rd characteristic mode will have nodes at


{\displaystyle {\tfrac {1}{3}}}
L and


{\displaystyle {\tfrac {2}{3}}}
L, where L is the length of the string. In fact, each nth characteristic mode, for n not a multiple of 3, will not have nodes at these points. These other characteristic modes will be vibrating at the positions


{\displaystyle {\tfrac {1}{3}}}
L and


{\displaystyle {\tfrac {2}{3}}}
L. If the player gently touches one of these positions, then these other characteristic modes will be suppressed. The tonal harmonics from these other characteristic modes will then also be suppressed. Consequently, the tonal harmonics from the nth characteristic modes, where n is a multiple of 3, will be made relatively more prominent.
In music, harmonics are used on string instruments and wind instruments as a way of producing sound on the instrument, particularly to play higher notes and, with strings, obtain notes that have a unique sound quality or "tone colour". On strings, bowed harmonics have a "glassy", pure tone. On stringed instruments, harmonics are played by touching (but not fully pressing down the string) at an exact point on the string while sounding the string (plucking, bowing, etc.); this allows the harmonic to sound, a pitch which is always higher than the fundamental frequency of the string.

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  1. G

    I Do hyperbolic harmonics exist?

    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...
  2. Like Tony Stark

    Rotation of spherical harmonics

    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...
  3. Bob Walance

    Music Equal temperament vs instrument harmonics in music

    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...
  4. The Bill

    Analysis Resource(s) for introduction to spherical harmonics with exercises?

    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...
  5. T

    VFD harmonics effects on generator set

    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...
  6. P

    Commutation relations between Ladder operators and Spherical Harmonics

    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?
  7. Salmone

    I Inner products with spherical harmonics in quantum mechanics

    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...
  8. jackal123

    Finding Lagrange Point L2: Gravity and Harmonics

    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...
  9. Jason-Li

    Comp Sci Harmonic Amplitudes - 3rd & 101st

    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...
  10. R

    Expanding potential in Legendre polynomials (or spherical harmonics)

    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...
  11. P

    3 Phase Harmonics and Solid State Relay Issues

    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...
  12. docnet

    Solve Spherical Harmonics: Y_{1,1} Eigenfunction of L^2 & L_z

    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}...
  13. P

    I Second Matrices from Spherical Harmonics with Eigenvalue l+1

    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...
  14. Athenian

    I Transforming Cartesian Coordinates in terms of Spherical Harmonics

    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...
  15. P

    A Matrices from Spherical Harmonics with Eigenvalue l+1

    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\\...
  16. entropy1

    Randomizing phases of harmonics

    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...
  17. Taz

    A Normalization of the radial part of the spherical harmonics

    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...
  18. Coltrane8

    I Spherical Harmonics Expansion convergence

    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...
  19. F

    Does operator L^2 commute with spherical harmonics?

    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...
  20. M

    A What are the uses of spherical harmonics?

    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!
  21. D

    How does an antenna work with harmonics?

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

    Addition of Harmonics in a string 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...
  23. dRic2

    Are harmonics "real" in a vibrating string?

    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...
  24. M

    Engineering Adding Harmonics to Form the Final Signal

    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.
  25. B

    Harmonics and Integer Multiples

    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...
  26. M

    I Spherical Harmonics Axisymmetry

    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?
  27. F

    Standing waves on a string experiment -- Relative amplitude of harmonics

    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...
  28. H

    Harmonics in the "electrical" domain

    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...
  29. Fisherman199

    LV Harmonic Filters for Utility Customer's Power Outages?

    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...
  30. beefbrisket

    I Computing inner products of spherical harmonics

    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...
  31. J

    Why do Harmonics Decay Faster than the Fundamental?

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

    Sph Harmonics Homework: Find Potential for r>a, Contribution Using Superposition

    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...
  33. D

    Integration of Spherical Harmonics with a Gaussian (QM)

    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...
  34. G

    Frequency spectrum of a clarinet

    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...
  35. J

    Phase of a Wein Bridge Oscillator (AC Mains Harmonics Analyzer)

    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...
  36. renec112

    QM: Issues with parity of spherical harmonics and Heisenberg

    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}##...
  37. S

    Calculating frequency of the second harmonic

    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...
  38. T

    Why do radio transmission harmonics occur?

    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?
  39. A

    Relevant highest order harmonics in power systems

    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...
  40. S

    Solving Spherical Harmonics Homework

    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)...
  41. T

    Possible Line Power Conditioner hobby project

    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...
  42. N

    Power Series Equation for Amplifier and Harmonics

    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...
  43. T

    Which harmonics would be observed on an electric guitar?

    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...
  44. bahtiyar

    What factors affect the frequency of harmonics in a soda bottle?

    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...
  45. S

    What is the reason for harmonics in dc-dc converter?

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

    A Spherical Harmonics: A Primer on Barton's Relations & Addition Theorem

    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...
  47. T

    The physics behind rifle barrel harmonics

    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...
  48. G

    How harmonics are produced in a guitar string?

    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...
  49. Plat

    How are RF harmonics affected by the shape of a waveform?

    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...
  50. J

    Oscillations and harmonics issues for components

    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...