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TL;DR The "leading-order" correction to the ODE is $$\frac{d^2y}{dx^2}=\frac{\mu}{T_0}g(1+2k^2x^2)^3$$ where $T_0$ is the tension in the middle of the rope, and $k=\frac{4y_{max}}{L^2}$ for an estimated maximum displacement $y_{max}$ and for a rope hanging between supports separated by a length $L$. The solution to this ODE is: $$y(x)=y_0+\frac{\mu g}{...


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I think the Heisenberg uncertainty principle, or at least the simple wave mechanics version of it (which may be the only one your students are equipped to understand at this point), really stems from the idea that most wavepackets don't have a definite frequency and wavelength. That is, in $p = h/\lambda$, there is not just one $\lambda$. Maybe this concept ...


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It seems from the way you have phrased your question that you think the frequency of the light corresponds to the number of photons per second. That's not the way it works. The number of photons per seconds would correspond to the brightness of the light for example you could have relatively few blue photons per second in the case of dim blue light or many ...


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Mathematically, you can define any dynamic law you'd like on a field. A dynamic law is simply a map $\Phi^t(C)$ mapping a system configuration $C$ - no matter what it is, could be a field, position/velocity pair, state of a game grid, etc. - to an "evolved" one after a lapse interval $t$ that satisfies the composition principle $\Phi^s \circ \Phi^t = \Phi^{s ...


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Fields are not, in general, solutions of a wave equation. They are solutions of the equations of motion derived from a particular Lagrangian density. The dynamics of a field are described by a Lagrangian density which is specified as part of the foundations of the field theory. For the free scalar field $\phi$, this Lagrangian density is: $$\mathcal{L}=\...


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That is a lot of questions, but they are all strung together sensibly, so I will try to answer all of them. Yes, two point sources if they are mutually coherent and are stationary will produce an interference pattern where their beams overlap. If you shift the phase of one of the sources (e.g., by interposing a thin sheet of glass to retard one of the ...


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One of the Maxwell equations in the vacuum has the magnetic and electric constants: $$\nabla \times \mathbf B = \mu_0 \epsilon_0 \frac{\partial \mathbf E}{\partial t}$$ so that in the wave equation, derived from the above and the other three: $$\frac{\partial^2 \mathbf E}{\partial t^2} = \frac{1}{\mu_0 \epsilon_0}\frac{\partial^2 \mathbf E}{\partial x^...


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Let's consider a more general version of the problem first. Suppose the distance between the emitter and receiver is $d(t)$; we'll allow this to be an arbitrary function of time. We'll also suppose that the amplitude of the signal emitted as a function of time is $S(t)$, again allowing it to be an arbitrary function of time. Suppose the signal has a speed $c$...


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An answer from an experimental physics physicist, who used theories to analyze data of high energy physics experiments: Fields are similar to a coordinate system on which the behavior of elementary particle interactions can be modeled with mathematical functions. In essence they replace the luminiferus aether with a Lorenz invariant "substance" that ...


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Actually, you've gotten the issue backwards! You complain that the Fourier series is illegitimate because we don't have an "actual" periodic function, so we repeat the function by "brute force". But that's not the right way to look at it. The Fourier series properly represents functions defined on the circle, i.e. functions $f(x)$ for $x \in [0, a]$ with $...


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$f$ is meant to be any arbitrary function $f:\mathbb{R} \to \mathbb{R}$. This can be understood from the wave equation $$\frac{\partial^2 y(x,t)}{\partial t^2}=v^2\frac{\partial^2 y(x,t)}{\partial x^2}$$ which describes the underlying physics of a wave. The most general solution of this differential equation is $$y(x,t)=f_1(x+vt)+f_2(x-vt)$$ where $f_1$ ...


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Writing your second equation using the same symbols of the first equation $(s\rightarrow d)$: $$ w = \frac{m \lambda D}{d}.$$ Then, from geometry, $w = D \tan\Theta$, so you end up with: $$ d\tan\Theta = m\lambda $$ for the double slit diffraction pattern. Which, as you are saying, is different from the general diffraction grating formula: $$ d\sin\Theta = ...


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The wave from an impulsive point source is the Green's function $G(\vec{r},t)$ for the inhomogenous wave equation. In other words, if we wish to solve the equation $$ - \frac{1}{c^2} \frac{\partial^2 \psi}{\partial t^2} + \nabla^2 \psi = \rho(\vec{r},t), $$ we can do so by solving the related equation $$ - \frac{1}{c^2} \frac{\partial^2 G}{\partial t^2} + \...


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The note produced by vibrating air in the instrument. Blowing air over the player's lips is what sets up the vibration. You can do this without an instrument. The instrument has a resonance frequency. Vibrations at that frequency get reinforced. The oscillating pressure acts on the lips and encourages them to vibrate at the resonance frequency. This makes ...


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Now if we think that sound wave is like an object and use relative motion than sound will approach wall with speed $v + v_{s}$... No, I think you are misunderstanding something here. A linear sound wave will always propagate at the speed of sound once emitted in a homogeneous, uniform medium. If your expression were correct, how could a shock wave form? ...


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