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Carrying out the Fourier transform, I get a slightly different result for the frequency spectrum than 'knzouh'. I used $u$ instead of $y$ and $c$ instead of $v$, so the PDE becomes: $$u_{tt}=c^2u_{xx}-Au_{xxxx}$$ Fourier transforming the equation: $$F\{u_{tt}\}=F\{c^2u_{xx}\}-F\{Au_{xxxx}\}$$ Transforming $x$ to $k$: $$\hat{u}(k,t)=\int_{-\infty}^{+\infty}u(... 20 In plain English, there is stiffness at the ends of the strings where they are fixed in place, which makes the string's frequency of vibration slightly higher (sharper)—effectively shortening the length of the string slightly, for all practical purposes. And the resistance to bending is dependent on the frequency. It behaves more “stiffly” with regard to ... 157 This effect is known as inharmonicity, and it is important for precision piano tuning. Ideally, waves on a string satisfy the wave equation$$v^2 \frac{\partial^2 y}{\partial x^2} = \frac{\partial^2 y}{\partial t^2}. The left-hand side is from the tension in the string acting as a restoring force. The solutions are of the form $\sin(kx - \omega t)$, ...

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I think the sound makes the particles move more because there's more interaction when the sound particles are included, but I'm probably wrong because I don't know much about physics...

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"Comfortable ride" is a tricky thing to quantify. Jerk is not the right metric to use. The reason it works for roller coaster design is the fact that in a roller coaster, you brace yourself against the rather large low-frequency acceleration. If you make a sharp turn to the left, you will want to lean left. If you then suddenly make a large turn to the right,...

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To formalize the comments (now in chat here): Associated jerk is probably what you want to calculate, as it is the measure of how violently something is shaken.$^1$ Jerk is the derivative of the acceleration with respect to time. To properly calculate this, you would use the formula $\left| a \right| = \sqrt{a_x^2+a_y^2 +a_z^2}$ (from the Pythagorean ...

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This effect is significant in debris flows, where large numbers of huge boulders move like a fluid. For a technical discussion see: https://profile.usgs.gov/myscience/upload_folder/ci2013Mar07174849246641997.Iverson.PhysicsDebrisFlows.Rev.Geophys.pdf For a spectacular demonstration: https://www.youtube.com/watch?v=51C7vEAVbxk

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