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.
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...
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...
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.
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!
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 |...
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...
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...
##\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##...
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...
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...
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.
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...
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...
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...
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?
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...
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)?
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>
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...
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...
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...
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.
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?
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}...
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...
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...
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 +...
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...
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...
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...
##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...
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...
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!
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 -...
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...
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 ;)
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...
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)?
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...
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...