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62

The answer to this question has significant overlap with my answer on piano tuning. There, I discuss how a thick wire has an extra restoring force, in addition to its tension, from its resistance to bending. This modifies the usual wave equation to $$v^2 \frac{\partial^2 y}{\partial x^2} - A \frac{\partial^4 y}{\partial x^4} = \frac{\partial^2 y}{\partial t^... 18 This source shows that for free beams like the bars of the glockenspiel, angular frequency \omega=2\pi f and wavenumber k=\frac{2\pi}{\lambda} are related by$$\omega^2=\frac{YI}{\rho A} k^4$$where I=\frac{1}{12}bh^3 is 2nd moment of cross-sectional area about a horizontal axis through the centre, and A=hb is the cross-sectional area. In the ... 1 If you've got an older web browser kicking around that still runs Java applets you should check out Paul Falstad's Loaded string simulation. You can add harmonics to your heart's content. 2 Basically Oscars answers says it all, but I just want to add a few more things. When a string is plucked its motion need to follow the wave equation$$ \frac{d^2}{dt^2}y(x,t) - c^2 \frac{d^2}{dx^2}y(x,t) = 0 $$with Dirichlet boundary conditions (the ends of the string are fixed). c is the speed of sound of the string's medium. The function y_n(x,t) = \... 2 If a string has multiple waves expressed in it, this is done by adding the waves individually. Each frequency in the harmonic series can be expressed by a wave, a guitar string is the sum of these waves in different proportions. The resulting wave is significantly different than the others. See below for the sum of the first three frequencies in the ... 3 Dielectric materials have a loss tangent : at a given frequency, the response to EM radiation will have a phase lag relative to the incident wave. For pure dielectric materials, the phase angle is zero; when there are losses in the material, the loss angle will be non-zero. For a given loss angle (usually given as the tangent of the angle) the amplitude of ... 2 He is referring to the second paragraph of 31-3: However, we shall discuss the formula we have obtained, in various possible circumstances. First of all, for most ordinary gases (for instance, for air, most colorless gases, hydrogen, helium, and so on) the natural frequencies of the electron oscillators correspond to ultraviolet light. These ... 0 Intensity is energy per unit area; over short distances, the intensity can be considered constant. In a time period dt, the sound wave travels a distance c.dt, so the total energy passing through an area A will be equal to the sound energy present in a volume cAdt. (think of water flowing through a pipe, if it travels at 10 m/s through a pipe with a 0.1 m ... 3 In this type of problem one has to take great care in defining intensities. In this case there are 4 different intensities: 1. I_{ss}, the intensity received by the static observer as perceived by himself, 2. I_{ms}, the intensity received by the moving observer as perceived by a static observer. 3 I_sm, the intensity received by the static observer as ... 3 The maximum and minimum are "local" values. As you move closer to A (at 0.2 m you are MUCH closer to A than to B) the amplitude of A is much larger - so although there may be destructive interference between A and B at that point, this is by no means perfect interference, and the resulting amplitude is still quite large (lot of A minus a little of B). ... 0 Mechanical oscillators have elasticity, which causes them to "spring back" after being disturbed from their rest position and inertia, which allows the oscillation to cycle due to continual overshoot of the rest position. For a mass m attached to a spring with stiffness k, the frequency of oscillation \omega is given by:$$\omega =\ \sqrt\frac{...

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With respect to the air the wavelength will be $\lambda = 343/f$ everywhere. With respect to the ground the wavelength will be $\lambda = v_{sound}/f$ but $v_{sound}$ will depend on position with respect to the source - picture your classic Doppler shift diagram with a set of not quite concentric circles. Downwind from the source the wavelength, measured ...

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To add to the existing answer, I think there is a nomenclature issue. When you say "bass" people understand "low frequencies" but what you probably mean is "beat". Rapid changes in amplitude, like a beat, carry a lot of high frequencies. You do hear mostly the beat from other peoples' headphones, ans it's annoying. You can think about the extreme case: the ...

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Resonance is the process of driving a system to oscillate at a resonant frequency. The property of these frequencies is that they can store vibrational energy or vibrational quanta depending on the description you choose. This means that driving the system at these frequencies can create much larger amplitude oscillations than other frequencies, which ...

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Resonance is the name for the phenomenon. The resonant frequency is the frequency at which this phenomenon occurs. The frequency of the oscillation when you observe resonance. Frequency is just one of the characteristics of a resonance. Another one is Q factor. Others might be the mode structure of the resonant wave.

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I would choose as notation conventions: $\Delta X$ for a constant increment, ie if $X_{n}-X_{n-1}=\Delta X$ for all n and with underscript otherwise: $\Delta X_n=X_{n}-X_{n-1}$ I prefer the increment to have the index of the value it produces (adding increment $\Delta X_n$ gives value $X_{n}$) but that's a matter of taste. It's not clear to me what the A ...

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I would name the period at $n$ as $P_n$, and to the change in period as $\Delta P_n$. In such a case you have the rate of change of the period, that is the change in period per unit of time is $\Delta P/t_A=\left(\frac{1}{f_{n+1}}-\frac{1}{f_n}\right)/t_A$. However, to define the angular acceleration, $\alpha$, you do not need the period, the definition is ...

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You typically cannot write subscripts or superscripts (easily) as comments in code, so I'll assume you are rather interested in writing this into a specification or some kind of documentation that does support such notation. When notation becomes cumbersome, here's what I'll generally do: 1) for discrete increment always use $(n), or (k)$ or the like ...

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Yes I agree, the solution in the manual misses out some terms in the equation of motion, as well as the factor $\frac12$ in front of $k$. Probably some approximations have been made without explanation. Your equation seems correct to me. When the ball arm is vertical, and there is no rotational motion in the vertical plane, then balancing moments about O ...

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