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16

Usually, the (sinusoidal) driven harmonic oscillator is damped, and the first two parts of your solution (which depend on the initial conditions, while the third term does not) are transient, i.e. not relevant after a short time. That the solution $$ x(t) = \frac{F_0\sin(\omega t)}{m(\omega_0^2 - \omega^2)}$$ cannot be the "full" solution to the equation of ...


15

$y(t)=A \sin(\omega t+d)=A \cos(\omega t+d-\left(\frac\pi2\right))$ The two forms are the same except for the phase term, so which one you use depends on when you started your stopwatch. If you started off at max displacement at t=0, then you use the cosine form, if you start out with max velocity at t=0, then you use the sine form. If you are deducing ...


10

When you pluck a string or hit a drum or sound a not on a flute, the instrument and the air in and around it vibrate and this vibration propagates as sound waves in the air to your hear drum. When you hear an instrument being played, what you recognise as the note is the base frequency. 'C' corresponds to $261.6$ Hz and is the same for a piano or a guitar. ...


7

Let's look at frequency instead of notes. Let's say the string has a natural frequency of $100 Hz$ and that harmonics are present when you pluck it. Then, the frequency content of the sound will be of the form: $a_1 \cdot 100 Hz + a_2 \cdot 200 Hz + a_3 \cdot 300 Hz + ... $ Now, let's say you fret this string halfway such that the natural frequency ...


6

A guitar string produces harmonics because it vibrates in a non-linear fashion. An electronic oscillator can be made to generate a much purer form of vibration (near sinusoidal) than a mechanical device such as the guitar string. Hence its harmonic level, while not zero, is much lower. For example, the harmonic distortion of a guitar string is probably on ...


5

Frequency is just a way of analyzing a time dependent motion. Consider plucking a string by first pulling one point on the string away from its equilibrium. The string shape will be like a triangle, two straight bits of string coming away from where your finger is holding the string, but meeting at a slight angle where your finger holds the string. That ...


4

Clearly the motion of the mass can not be described by a single sine! What's going on here? The general solution to the simple harmonic oscillator is the sum of the unforced response (homogeneous solution) and the forced response. The homogeneous solution is $$x_h(t) = x_h(0) \cos(\omega_0 t) + \frac{\dot x_h(0)}{\omega_0}\sin(\omega_0 t)$$ Thus, ...


4

It seems that the harmonic (integer multiple) overtones of a sound usually all have the same phase. Is this true...? No, I don't think this is generally true, although it may be true for certain instruments. What led you to believe this? In trumpet tones, for example, the different harmonics come up at different times during the attack, so it seems ...


3

When you pluck a string it does not start out like the fundamental above. The string is pulled into a bent shape of two straight lines and an angle and it may not be bent at the middle. Releasing the bent string causes a bunch of harmonics of various amplitudes depending on how far off-center it was bent. (It can not return to the bent angle shape and the ...


3

There is a technique called flageolet where you damp the string with a finger laid lightly onto the site at the node of a higher harmonic. You do not press the string to the fretboard but just damp the string at a position, where there is a node of the specific harmonic. When you now pluck the string all harmonics, which do not have a node at the specified ...


3

Pitch, in music, is equivalent to frequency. How often the wavefore cycles. This is usually defined by length, i.e. how long the string is, how long the pipe is, etc. It can also be affected by the tension (how tight the string is.) Timbre, the sound of a specific instrument, is defined by the "shape" of the wavefore, whether spikes, round, square, or ...


3

Open organ pipe is the one with two open ends, and instead of the formula you mention you need to use $$L=n\frac{v}{2f_n}$$ where $f_n$ is the frequency of the ${n^{th}}$ mode, and $n=1,2,3,...$ your formula is for a closed organ pipe (with one open and one closed end). EDIT Because the number of half-wavelengths ($\lambda /2$) need to be an integral ...


3

$x_{max}$ is the amplitude of the oscillations, and yes, ${\omega}t - \varphi$ is the phase. We know that the period $T$, is the reciprocal of the frequency $f$, or $$T = 1/f$$ We also know that $\omega$, the angular frequency, is equal to $2\pi$ times the frequency, or $$\omega = 2{\pi}f$$ From here, we can use the initial conditions to find the ...


3

The physiology of human ear (and perhaps brain) makes sounds with frequency ~3000 Hz sound louder than higher and lower frequencies, for same sound wave pressure perturbation; see https://en.wikipedia.org/wiki/Equal-loudness_contour


3

A few observations. First - if you record sound for a short time, the bandwidth of the sample will result in a smearing of the peaks. This only really matters if the sample is very short - with a 1 second sample you would have 1 Hz resolution, but if you sample for 0.01 second, the bandwidth is 100 Hz. Second, you are using a scale that is quite compressed ...


3

When you release the plucked string, its shape is momentarily triangular: tied down at the ends and pointed at the location of your finger. But the solutions to the wave equation are not triangle functions, but sinusoidal functions, whose displacements from rest obey $$y_n(x) \propto \sin \frac{2\pi x}{\lambda_0 / n},$$ where $\lambda_0$ is twice the ...


2

Edit after providing the plot: Please note, that the simple model prediction of harmonics frequency position does not say anything about its strength in actual sound. It is typical feature of brass instruments in middle and lower registers that the fundamental frequency is not the strongest. Original answer: I am sorry guys, but that's well-known and ...


2

Sound waves are made of alternation of compression (higher density) and rarefaction (lower density) regions in the air. However, this can be somewhat difficult to visualize. Because of this, textbooks often show the wave like it's a string in the organ pipe. Really what the curves are showing you in the amplitude of this compression wave. It's also drawn ...


2

Like Jarosław Komar commented, you are using the wrong value for $n$. It is also easy to visualize this by looking at what the longest standing wave would look like in an air column with only one open end: Where the wavelength, $\lambda$, is defined as: $$ \lambda=\frac{v}{f} $$ So the fundamental frequency would require a $n=0$, since the length of the air ...


2

The easiest way to determine maximum and minumums of a function is to set the derivative equal to zero. Thus, in this case, setting the equation for acceleration equal to zero and solving for the variables of interest will give you what you want. Thus, in this case we have the equation for position: $$ x = a \sin(\omega t + \phi) $$ One way to see the ...


2

Hopefully David Bar Moshe can give a more rigorous explanation in terms of cohomology, but I have the following intuitive understanding of the difference between the two situations. In the Aharonov-Bohm effect, the particle is constrained to move around an (effectively) infinite solenoid. It then suffices to consider a problem on a plane, but with a hole in ...


2

Yes, it most certainly can. It's much easier to visualize if you consider a length of flexible steel or plastic. You can shake it a bit, then toss it in the air so it's not constrained, and it will (if properly initiated) vibrate at a resonant wavelength. I think the confusion most people will get from your question is that everyday string is "floppy," ...


2

For your example of a violin string, you can immediately determine that it is not simple harmonic motion by listening to it. Simple harmonic motion is a pure tone of a single frequency. Violins don't sound like that so you immediately know there are harmonics and it therefore is not a simple harmonic oscillator. As some other people have mentioned, a tuning ...


2

Consider the following diagram: A U tube contains a fluid with higher level in the right hand side. This could be achieved as you suggested or more simply by applying a partial vacuum on the right hand side: the higher (atmospheric) pressure in the left tube then pushes the fluid up into the right side, until a level difference of $y$ is achieved. At ...


2

This is the classical treatment to model vibrations in solids, using the analogy with vibrations of a one-or-two dimensional monatomic or diatomic chains. Which basically boils down to writing Newton's equation of motion to find out the force on each mass when the whole system constitutes of masses attached by Hookean springs, i.e. for our purposes the ...


2

The response can be derived mathematically. Let $u(x,t)$ denote the displacement of a point along the string at $x$ at time $t$. The function obeys the wave equation in flat $d=2$ Minkowski space, $$\frac{\partial^2 u(x,t)}{\partial t^2} - v^2 \frac{\partial^2 u(x,t)}{\partial x^2}=0$$ If we pinch the string at the middle, this corresponds to a condition ...


1

When you pluck a guitar string the potential you apply to the string is approximately a Dirac delta function. That is to say, the release of the string is a near instantaneous kick. One of the beautiful properties of the delta function is that its Fourier transform is unity. This means that it is made up of equal components of all frequencies. So, when ...



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