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11

$y(t)=A \sin(\omega t+d)=A \cos(\omega t+d-\left(\frac\pi2\right))$ 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 ...

6

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 ...

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 ...

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

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 ...

2

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

1

The classical string equation that you are referring to, is formulated by making a number of assumptions, which include that the vibration of the string does not affect its tension. This makes Young's modulus irrelevant for results calculated from the idealized equation. In the real world, materials with low moduli of elasticity will follow the ideal ...

1

I'm not qualified to answer this question in detail...but I'd like to point out some things I've learned lately that may be helpful. Hilbert space is useful when you need an infinite dimensional space to characterize what you are studying and where each mode is orthogonal to the others. Great for Quantum Mechanics... Where I work, images, that is 2D ...

1

You can send sound waves of any wavelength into your instrument, but the trouble is that only specific wavelengths correspond to standing waves. The other waves will just die out because of destructive interference. When your sound wave collides with the closed end of a pipe for instance, it gets reflected back. If the wavelength is not right, the ...

1

Modern keyboards are tuned with each of the 12 half-step intervals between one octave and the next being equal, so a half-step is a ratio of the 12th root of 2 (about 1.059), and a whole-step is a ratio of the 6th root of 2 (about 1.122). But the ratios that "sound nice" musically tend to be ratios of small integers. 2:1 is an octave, of course. 3:2 is a ...

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