What is Harmonic: Definition and 1000 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

1
3

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

2
3

{\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

1
3

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

2
3

{\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.

View More On Wikipedia.org
1. Continuity Equation for a Dimensionless Harmonic Oscillator

I've tried to solve this problem (Zettili, Exercise 3.5) four times at this point. I believe my equation for the wave function at a later time ##t## is correct. The problem is my continuity equation is not satisfied; it does not equal zero. It's close but I'm off by a factor of ##m## and...
2. Thermodynamics and harmonic motion

TL;DR Summary: Problem said that the ball moves in a harmonic motion and asked to prove it. The process is adiabatic Problem said that the ball moves in a harmonic motion and asked to prove it. The process is adiabatic. I did the development, but at certain point I'm having a problem. The...
3. B I'm trying to find a general formula for a harmonic(ish) series

Im looking for a general formula for the partial sum of a series where the nth term is in the form of a_n = 1/(c+kn), where c and k are arbitrary constants. it "looks" like a harmonic series but not in the form I'm capable of figuring out. help.
4. Solve Disk & Coil Spring Motion: Get Amplitude Help

I was able to solve part 1 but I am not not getting how to find new amplitude of the motion . Please help me
5. Solve First Year Harmonic Motion Problem: Cylinder Rolling

Thank you guys for taking the time to read this - I'm decently struggling with first year and need some tips on how to properly conceptualize problems and learn what the right approach is on certain problems. Have a wonderful day, again thank you for checking this post out!
6. I Translating the harmonic oscillator

Let's say I know the position space wavefunctions of the 1d harmonic oscillator ##\psi_n(x)## corresponding to the state ##| n \rangle## are known. I want to write ##\psi_m(x + a)##, for fixed ##m = 1,2,...##, in terms of all of the ##\psi_n(x)##. I know \begin{align*} \psi_n(x+a) = \langle x |...
7. I Thoughts about coupled harmonic oscillator system

Same instruction was given while finding value of 'g' by a bar pendulum. In the former case,does the spring obeys hooke's law while it forms a coupled harmonic oscillator system?Does the bar pendulum somehow breaks the simple harmonic motion(such that we can't apply the law for sumple harmonic...
8. B How is the harmonic mean affected by additional data points?

We have a collection of 8 discrete data points. They are: 10, 20, 30, 20, 30, 40, 30, 40 In increasing order: 10, 20*2, 30*3, 40*2 The harmonic mean of this data series is 22.86 I read on Wikipedia that the harmonic mean is skewed towards the smaller values i.e. smaller values will affect the...
9. Superposition of two one-dimensional harmonic waves

##\mathbf {Homework ~Statement:}## Consider the superposition of two one-dimensional harmonic waves $$s_1(x,t)=3.5 cm \cdot cos(27.5s^{-1} \cdot t - 5.65m^{-1} \cdot x)$$ $$s_2(x,t)=3.5 cm \cdot cos(27.5s^{-1} \cdot t - 5.5m^{-1} \cdot x)$$ ##\mathbf {a)}## Calculate the wavelength ##\lambda##...
10. Damped harmonic oscillation of a swingboat

Hi, so of course Φ0 = 15° and after solving after solving Φ(t=5*T = 5/f) I found γ = 0.012 I need help with b). If I do 2° = 15° * exp(-0.012t)*cos(2πf*t), I'm not able to find t so I did something else by assuming that the amplitude decreases at a constant rate: After 5*T = 5*1/f = 18.52 s...
11. How to prove that motion is periodic but not simple harmonic?

TL;DR Summary: Prove that a sum of trigonometric ratios is periodic but not not simple harmonic. We need to prove that ##x = sin{\omega t} + sin{2\omega t} + sin{4\omega t}## where ##x## is the displacement from the equilibrium position at time ##t##. I can see that each term is a SHM, but...
12. Solving for Simple Harmonic Motion: A Picture Problem

Here is a picture of the problem It is not clear to me how to really prove that the equation for ##\theta(t)## is simple harmonic motion, and what the period of this motion is.
13. Bungee jump | simple harmonic motion

As you all know, a bungee jump is where a person is tied to a cord and the person jumps off and bounces up again. The natural length of a cord is 75 metres. Then when a person is attached onto the cord, the length becomes 83 metres when the person is at rest. I am sure that the person is not...
14. Modification to the simple harmonic oscillator

I was assuming there could be something via perturbation theory? I am unsure.
15. Study of harmonic motion of a liquid in a V shaped tube

A V-shaped tube with a cross-section A contains a perfect liquid with mass density and length L plus and the angles between the horizontal plane and the tube arms as shown in the attached figure. We displace the liquid from its equilibrium position with a distance and without any initial...
16. I Can the Inverse Tan Function Justify Phase Lag in a Driven Harmonic Oscillator?

This is an equation I found for the delta phase lag of a driven oscillator. W is the driving angular frequency and Wo is the natural angular frequency of the driven system. Of course this is just a small part of the solution to the differential equation. Now ... 1) when W is much smaller than Wo...
17. X^4 perturbative energy eigenvalues for harmonic oscillator

The book(Schaum) says the above is the solution but after two hours of tedious checking and rechecking I get 2n^2 in place or the 3n^2. Am I missing something or is this just a typo?
18. I How to interpret complex solutions to simple harmonic oscillator?

Consider the equation of motion for a simple harmonic oscillator: ##m\ddot {x}(t)=-kx(t).## The solutions are ##x(t)=Ae^{i\omega t}+Be^{-i\omega t},## where ##\omega=\sqrt{\frac{k}{m}}##, and constants ##A## and ##B##. Physically, what does it mean for a solution to be complex? Is it only the...
19. I The speed of a waves on a string in Simple harmonic motion

The speed of a wave in simple harmonic motion on a string is $$v= \sqrt{\frac{F}{\mu}}$$ where v= the horizontal velocity of the wave on a string. Is the F the horizontal force or the resultant force (combination of Fy and Fx)?
20. B Is simple harmonic motion also a pure translatory motion?

Is simple harmonic motion also a pure translatory motion?"A rigid body moves in pure translation if each particle of the body undergoes the same displacement as every other particle in any given time interval" [Halliday and Resnick, Physics].If not,then how does shm deviate from this definition>
21. B Harmonic oscillator and simple pendulum time period

Hi, I have been thinking about pendulums a bit and discovered that a HO(harmonic Oscillator) will take the same time to complete one period T no matter which amplitude A/length l it has, if stiffness k and mass m are the same. But moving on to a simple pendulum suddenly the time period for one...
22. I Doubt on Morse potential and harmonic oscillator

I have a little doubt about Morse potential used for vibration levels of diatomic molecules. With regard to the image below, if the diatomic molecule is in the vibrational ground state, when the oscillation reaches the maximum amplitude for that state the velocity of the molecule must be zero so...
23. I Constructive interference of harmonic electromagnetic waves

Hello ! I have a doubt as to how is this case, if it occurs, of the constructive interference of two harmonic electromagnetic waves but of different wavelengths or frequencies between them. That is, if between the two electromagnetic waves a new and unique electromagnetic wave is created and...
24. Question on Intro QM pertaining to Harmonic Oscillator

Hi. I have attached a neatly done solution to the above question. I request someone to please check my solution and help me rectify any possible mistakes that I may have made.
25. Transformer modelling for harmonic studies

Why transformer core is modeled as resistance even though it is frequency dependent.With increasing frequency current rises. Doesn't it seem capacitive reactance behaviour? Which decrease with increasing omega?
26. Proving that a "composition" is harmonic

I am learning some complex analysis as it is a prerequisite for the masters program that I was accepted into and I didn't take it yet during my bachelors. I am using some lecture notes in Slovene and I have run into a problem that has proven troublesome for me : If ##g: D \rightarrow \mathbb{C}...
27. Modelling of tranformer and MV cable for harmonic studies

For my research project, modelling is needed for dyn5 630 kVA transformer and 20 kV cable NA2XS(F)2Y to do harmonic analysis. For that i need leakage reactance, and resistance of copper losses for transformer. And following parameters for 20 kV cable are also needed. Positive, negative and...
28. I Particle on a cylinder with harmonic oscillator along z-axis

I need to know if I have solved the following problem well: A spin-less particle of mass m is confined to move on the surface of a cylinder of infinite height with a harmonic potential on the z-axis and Hamiltonian ##H=\frac{p_z^2}{2m}+\frac{L_z^2}{2mR^2}+\frac{1}{2}m\omega^2z^2## and I need to...
29. Position expectation value of 2D harmonic oscillator in magnetic field

Hello there, for the above problem the wavefunctions can be shown to be: $$\psi_{n,l}=\left[ \frac {b}{2\pi l_b^2} \frac{n!}{2^l(n+l)!}\right]^{\frac12} \exp{(-il\theta - \frac {r^2\sqrt{b}}{4l_b^2})} \left( \frac {r\sqrt{b}}{l_b}\right)^lL_n^l(\frac {r^2b}{4l_b^2})$$ Here ##b = \sqrt{1 +...
30. Quantum Harmonic Oscillator with Additional Potential

Hello there, I am trying to solve the above and I'm thinking that the solutions will be Hermite polynomials multiplied by a decaying exponential, much like the standard harmonic oscillator problem. The new Hamiltonian would be like so: H = - \frac \hbar {2m} \frac {d^2}{dx^2}\psi + \frac...
31. B Why is a simple pendulum not a perfect simple harmonic oscillator?

Khan Academy claims that a simple pendulum not a perfect simple harmonic oscillator. Why is it so?
32. Finding Spring Constant When Given Amplitude, Time, and Mass

I averaged the masses and times (which i took the time given and divided by 10 because in the problem it says you measure the time it takes to complete 10 oscillations) then plugged them directly into the T=(2(pi)((m/k)^1/2) and got the wrong answer. This is really confusing me because I don't...
33. An object oscillating in simple harmonic motion

Since it passes through the origin every ##3.6s## the period is ##T=3.6s## hence ##\omega=\frac{2\pi}{\omega}=\frac{2\pi}{3.6}\frac{rad}{s}## thus ##A=\frac{v_{max}}{\omega}=\frac{1.2}{\frac{2\pi}{3.6}}m\simeq 0.69m## and ##a_{max}=\omega^2 A=(\frac{2\pi}{T})^2 A=(\frac{2\pi}{3.6})^2 \cdot...
34. Discretizing a 1D quantum harmonic oscillator, finding eigenvalues

##x## can be discretized as ##x \rightarrow x_k ## such that ##x_{k + 1} = x_k + dx## with a positive integer ##k##. Throughout we may assume that ##dx## is finite, albeit tiny. By applying the Taylor expansion of the wavefunction ##\psi_n(x_{k+1})## and ##\psi_n(x_{k-1})##, we can quickly...
35. 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...

37. B Arithmetic progression, Geometric progression and Harmonic progression

How do I build functions by using Arithmetic Sequence, Geometric Sequence, Harmonic Sequence? Is it possible to create all the possible function by using these sequences? Thanks!

40. Weakly interacting Bosons in a 3D harmonic oscillator

1. Since N is large, ignore the kinetic energy term. ##[-\mu + V(r) + U|\Psi (r)|^2]\Psi (r) = 0## 2. Solve for the density ##|\Psi (r)|^2## ##|\Psi (r)|^2 = \frac{\mu - V(r)}{U}## 3. Integrate density times volume to get number of bosons ##\int|\Psi (r)|^2 d\tau = \int \frac{\mu -...
41. Damped harmonic motion problem

Hi ! Problem : y = 5 e^-0.25t sin (0.5.t) (m, s). Determine the deviation at a time when the amplitude has dropped to 1/5 of the original value. I tried with A=A0 e^-bt=5 e^-0.25t - Do i need to determine the time here or recreate the deviation equation when A decreased ? I don't understand...
42. Superposition of two simple harmonic motion

Hey! I am stuck in this problem, i don't know how to sum this ecuations. I remember that its possible because the direction is the same So, i try to sum like this: cos (t+5325) + 1.5 cos (t+5325) =1.5 cos (t+5325) I don't know if i fine. I thanks your help, please ;)
43. Find Ground State Energy of 3D Harmonic Oscillator

Summary:: I am trying to find the exact ground state energy of the hamiltonian.kindly help me with this
44. Simple harmonic motion (help please)

The first ecuation values i am 99% that is correct. But, in the second and three problem i don't know if my results are ok. The problem number 2 i comprobate with the teacher that te aceleration its correct, so, with this i calculate the velocity. I use like example the second problem for try...

ia
46. I Stimulated emission in harmonic oscillator

Hello! Is stimulated emission possible for a harmonic oscillator (HO) i.e. you send a quanta of light at the right energy, and you end up with 2 quantas and the HO one energy level lower (as you would have in a 2 level system, like an atom)?
47. Harmonic motion - Find the Mass held between two Springs

So first I find the energy using the eqn (1/2)kA^2. Since there are two springs with the same k I multiply it by two to get kA^2. Energy I get is 2.0475, Now I use E=(1/2)m(wA)^2 to find mass. Again since there are two springs I use E=m(wA)^2. m=E/(wA)^2. w=(2(pi))/T btw. I get the answer of...

49. I How to solve 2nd order TDSE for a Gaussian-kicked harmonic oscillator?

Consider the gaussian kick potential, ##\hat{V}(t) = \hat{x} \exp{(\frac{-t^2}{2 \tau^2})}## where ##\hat{x} = a+a^\dagger## in terms of creation and annihilation operators. Then we define the potential in the interaction picture, ##\hat{V}_I(t) = e^{i\hat{H}t}\hat{V}(t)e^{-i\hat{H}t}## I...
50. What is the maximum kinetic energy for harmonic motion with a reduced amplitude?

Solutions in a file.