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This is an interesting question and the answers which have been given show that the $v$ in your equation should be called the magnitude of the velocity or just the speed of the wave. The mixing of the terms speed and velocity happens all the time. Now there is an equation for wave velocity but in comes about in a somewhat convoluted way. Suppose that you ...


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What I assert is that instead of the line being formed because it 'follows the path of an electron or muon', it is a chain of interactions. That is, there IS a disturbance, but the path begins with an electron-atom (or even an atom-atom, doesn't matter) interaction, and instead follows like a row of billiard balls (atoms) hitting each other and revealing ...


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Since the wavelength is the distance between the two consecutive crests or troughs it isn't a vector quantity. It is scalar. Simillarly, Frequency is a scalar quantity, since it just number of crest and trough per second. How can such number have direction.


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I built a small cloud chamber at home years ago, powered with the Americium taken from a smoke detector. Dry ice, alcohol, a container, and an electric field ... didn't work very well, but you could see little ion trails. I should make a better one ... And yes, very little energy is lost in forming the ion trails. For the cloud chamber to work the ...


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No. The general relation is given by $$v = \lambda\nu$$ Where $v$ is the velocity of the considered wave and $\lambda$ and $\nu$ its wavelength and frequency. Of course in the case of an electromagnetic wave which is traveling at the speed of light you gain $$c = \lambda\nu$$ If you're treating instead some massive particle, then thou have $$p = ...


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Here some details \begin{equation*} \partial _{t}\mathbf{w}(x,t)+\mathbf{A}\cdot \partial _{x}\mathbf{w}(x,t)=0 \end{equation*} Let $\mathbf{A}$ be an $n\times n$ matrix. Then $\mathbf{w}$ must be $n$ -dimensional. Let us assume that $\mathbf{A}$ has real entries and $\mathbf{w }$ has real components. \begin{eqnarray*} \mathbf{w}(x,t) &=&\exp ...


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I find Pulliam's notes for the Euler equations to be a pretty good introduction to this topic using the equations of fluid motion. The idea is that you start with a conservation law: $$ \frac{\partial \vec{Q}}{\partial t} + \frac{\partial \vec{F}\left(\vec{Q}\right)}{\partial x} = 0$$ where $Q$ is your variable vector and $F$ is your flux function. You can ...


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Here is a series of images from two point objects (stars) to illustrate the idea of resolving power. The objective lens of the telescope produces the diffraction. If the distant stars are closer together the diffraction patterns come closer to one another. Diagram (a) shows to diffraction patterns (the intermediate image in a telescope) well spaced ...


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Yes it would depending on how thick the smoke was. Photons intercepted by the smoke would be scattered or absorbed but the photons that still have a clear shot to the screen would contribute to the original fringe pattern. The pattern would still be the same but not as clear.


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EM waves are formed when an electric field couples with a magnetic field. The magnetic & electric fields of an EM wave are perpendicular to each other & to the direction of the wave. The wavelength is just that--the length of the wave through one frequency cycle.


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The issue with massive waves on a 1 meter deep ocean is that the waves cannot propagate fast enough on a planetary size object. We get fast moving shallow tsunami waves in the open ocean over a thousand meters deep. The tsunami piles up when the wave slows down due to contact with a shallow shoreline. Hundred meter high waves could never propagate fast ...


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A permanent magnet has a fixed north/south polarity - in this example, lets say north is facing up and south is facing down. This magnet has a membrane of some kind attached to its north face. An electromagnet beneath the permanent magnet can switch the direction of its north/south polarities by changing the direction of the electric current running ...


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It does not. Neglecting huge gravitational fields (e.g. black holes), which distort even the traveling path of light, a light wave propagates in a straight line. The "wave" part is expressed in the electric and magnetic field of the light beam/pulse, but these two fields oscillate in the plane transverse to the propagation direction. Lastly, the ...


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There are two ways to describe a sound wave. One is in terms of displacement of the medium and the other is in terms of pressure. This simple diagram shows that tthe two descriptions are $90^\circ$ out of phase with one another. Note that at a compression $C$ where the pressure is a maximum the displacement of the particle is zero and the same is true ...


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These curves show acoustic displacement or acoustic velocity. For acoustic pressure they would be "inverted" (nodes at the open end, antinodes at the closed end). In presented 1D case are all of them actually scalars (or can be treated as such). The curves show just the magnitude. I know, these graphics are confusing. Nowadays it could be easily done by ...


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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 ...


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The problem in answering this question is that the term wave itself is so loosely used to define that which we observe. But fundamentally a wave is just an expression of the flow of energy in time and space. It's observable evidence that energy is flowing. It's clear that traveling waves expresses the flow of energy. And the interference or superposition ...


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$f(x-vt)$ isn't the general formulation of a wave, but is the general expression for a travelling wave. Also, a standing wave can be transverse or longitudinal (on a guitar string or inside a laser cavity it is transverse, inside a flute it is longitudinal). The reason you can hear the sound of a flute is because some of the energy coming from the standing ...


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Assuming (sigma)^2 is >>wavelength, then is is a superposition of several plane waves. Essentially the spatial frequency spectrum is the fourier transform of the gaussian term. So, the spectrum would be centered at "k," but with a width ~sigma. The smaller sigma, the larger the spread in wave vectors. Similarly if sigma were very large, such that the ...


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Although it can be generalized to accommodate an n-dimensional non-linear state space system, I'll describe for you a 2 dimensional linear state space system as you have posed in your specific example. This way to make matters simpler and more illustrative. For this system, the $B$ vector is a 2 X 1 vector and $b(t)$ is a scalar. The $B$ vector is properly ...


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There is an equation that translate properly the situation of a string under a pulse of frequency $\omega$ at the point $x_0$ of the string. $$ \rho\frac{d^2y}{dt^2}=T\frac{d^2y}{dx^2}+\kappa \delta(x-x_0)\sin(\omega t) $$ this equation is nothing more than an application of the Newton's second law. The first term is the $ma$ part of the $f=ma$. Here ...


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I would say it has to do with energy/information conservation laws. You can imagine a wave (be it electromagnetic or vibrations on a string) is propagating through a medium. The energy of such a wave is given by: Classical harmonic oscillator (pendulum or vibrating string) $$ U = \frac{1}{2}kx^2 $$ where $x$ is displacement from the steady-state rest ...


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The addition of two simple harmonic motions at right angles to one another produces what is called a Lissajous figure. If search for Lissajous Figures Simulation you will find a number of simulators. Make the x and y frequencies the same and the phase difference 90 degrees and you get your circle produced. I have not found a good one but here are two ...


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The field given by components $E_x = \cos(wt)$ and $E_y = \cos(wt-\pi/2) = \sin (wt)$ is always a rotating field, unless $w= 0$, and it will always have amplitude $$ \sqrt{E_x^2 +E_y^2} = \sqrt{\cos^2(wt)+\sin^2(wt)} = 1. $$ If $w>0$, the field will rotate counterclockwise, whereas if $w < 0$, the field will rotate clockwise. If $w = 0$, the field ...


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There shouldn't be any relationship if you do the modeling properly. Could it be that your simulation is not using fine enough time steps to see it? You need multiple steps per rotation - probably at least 20 - to see the rotation. It is possible that your choice of time step and $\omega$ makes it that the field appears to be "frozen" at a particular ...


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That's almost duplicate of this question. You should definitely read it. It's caused by the change of propagation media. If you shout, the sound is produced by the disturbances of the airflow done by your vocal folds. Again: the airflow. Then it must overcome the huge impedance change to the water (and there comes the loss of power). And from the water to ...


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If you have a rod, there are three different ways you can try to distort it which can give rise to a wave traveling along the rod. you can give the end a quick twist: this will give rise to a torsional wave you can push on the end: this will give rise to a pressure ("sound") wave you can move it from side to side: this will give rise to a transverse wave ...


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Consider the following experiment: A regularly shaped metal rod is fixed at its top to a ceiling (or such like), so it cannot move. At the bottom of the rod we rigidly attach two strong and heavy handle bars, which allow to exert a torque $\tau$ to the rod. As a result of this torque the rod now torsionally deforms, so that the handle bars are now at an ...


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When you meet this the first time it can be a difficult concept to understand. To get fit two friends decide to walk up and down a flight of stairs. Let the steps be labelled $0$ to $10$. So to go up and come down they have to ascend 10 steps and then descend 10 steps a total of 20 steps. Call this one repetition or cycle of the exercise. After a time ...


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I don't understand why are you basing you r definition as either SISO or MIMO on the dimensionality of $B$. The same physical system (viberating string in your case), can be either SISO or MIMO depending on your configuration. The question of classifying a system as SISO or MIMO depends on your control parameters, and the parameters which you "read out" or ...


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Perhaps a better way to put this question is how is energy concentrated within a resonant system? As others have already stated energy is always conserved if one considers the universe in tallying where energy comes and goes. But if you are considering a system, defined within spatial boundaries, the system can lose or gain energy through its boundaries. In ...


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I will answer this question in two parts: first comparing plane waves, spherical waves and cylindrical waves (which are really the same thing as I will explain). how this relates to ray optics (which is completely different). Plane waves, spherical waves and cylindrical waves are 3 different examples of doing a decomposition of a wavefield. The method is ...


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In spherical coordinates, you can write a spherical wave as $F(x,t) = A \exp (ikr - i\omega t)$. To say that two spherical waves are in phase at a certain point $(x,t)$ in space and time is to say that at this particular point, the argument inside the exponential is the same for both of them, up to a $2 \pi$ phase factor. Imagine you have two sources that ...


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Huygens's Wave Theory is what you call a first order scalar diffraction theory of light. So what does it describe and what does it fail to describe? First order means that electromagnetic effects like induced currents in surfaces etc. are ignored. These can be described by solving Maxwell's equations for the same system instead of working with the wave ...


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From the center measure out any direction to a certain radius and all the waves at that distance will be at the same phase somewhere between a positive or negative amplitude.


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Certain phosphors will work in that manner. They will absorb 2 photons of visible light for every 1 photon they re-emit in the ultraviolet.


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See Greg Egan's applet. This will clear up all confusion, and ought to be viewed by any reporter who covers a "scientists break light speed" report. Subluminal shows how a wave composed of a multitude of frequencies moving at different velocities — all less than or equal to c, the speed of light in a vacuum — can appear to have features moving faster ...


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Point a laser to the sky, track where it lands very far away from the Earth, and then move your arm. The point you are tracking in space is moving faster tha the speed of light, even though light itself isn't. This isn't the same as group velocity, but it is very similar. There is no problem in having arbitrary points you imagine going faster than light, ...


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You have a good question. The only answer that I can come up with at the moment is not very satisfying. I'll keep your question in the back of my head for a few days and update my answer if I come up with something. In the mean time, you can use dimensional analysis! You are interested in the difference in between the phase and group velocities, ...


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If we take your case of amplitude modulating two frequencies with pulses which are a second wide - then these are not pure frequencies any more - they will have a bandwidth of approx 2Hz (depends on pulse shape) [because there will be two sidebands at +/- 1Hz from carrier]. These pulse therefore have a group velocity. Only if you emit a wave at a single ...


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You have to be careful to interpret your statements 1 and 2 correctly. In statement 1 you are talking about an infinitely long wave train which is made up of only one single frequency. As soon as you have a wave train of finite length then the wave train is made up of the sum many frequencies. So in statement 2 what you think of as a pulse of a wave with a ...


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This is a common misconception. The function above can be interpreted as follows. Sound of frequency $\dfrac{\omega_1+\omega_2}{2}$ with amplitude modulated by the cos function of frequency $\dfrac{\omega_1-\omega_2}{2}$. The cosine function becomes zero twice every cycle as well as reaching a maximum magnitude twice every cycle. So the intensity of the ...


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The transitions between two frequencies will not move with the phase velocity! Nor will the beginning of transmission! Only in the steady plane wave parts in the middle, the wave crests will move faster than light. And that's just "appearance" in the sense that bewteen any two times, you can find a maximum of the electric field and pretend the wave moved ...


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When a receiving antenna picks up a signal, a current flows. This current acts as a secondary "transmission", and this will partially cancel out the electromagnetic field that is incident - this is how you get power from the EM field into the antenna. If you look "slightly downstream" from the antenna, you will see a reduction in the EM field (assuming for a ...


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A radio wave may "bend" due to diffraction or scattering off objects, but it will not "bend" just because a receiving antenna exists.


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When trying to understand light, there are two components to consider: the amplitude, and the polarization, and you can assign both these properties to all points in space. This is very difficult to visualize, so most of these images trying to visualize light are bound to be inaccurate. The first image you show in your post is accurate in showing how the ...


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Any electromagnetic radiation - and in special case light as a small and visible for us part of EM radiation - is composed of photons. This is right for the process of emission as well as for the absorption of EM radiation. Any photon has a electric E and a magnetic B field component, both perpendicular to each other and to the direction of propagation v ...


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Note that $k>0$ so in shallow water $\omega \approx \sqrt{gh}k$. These are pure surface waves whose amplitudes are much smaller than the depth, and is also assumed that no other waves propagate, thus reflections from the bottom are completely ignored.


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Edit: I think you'll find all the details you need at this question. As Asher commented, when a wave is described as sinusoidal, or triangular, or square, that's its amplitude profile. When a wave is described as plane or spherical, that's the spatial profile perpendicular to the direction of propagation. For example, a plane wave of sinusoidal ...


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The first problem you might have with the wave equation is to visualise the graph. This is because the displacement of a particle in the medium through which the wave is travelling $y$ depends on both position $x$ and time $t$. So you need to draw a 2D graph on a 2D computer screen and then this not even take care of a 3d wave with position $(x,y,z)$ and ...



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