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0

So you have your two normal modes, one is a sum, other is difference, and both of these change in time. If sum is zero, you have a difference and if difference is zero, you have a sum...any kind of motion can be pictured by mixing in some amount of sum and some amount of difference..in any position you have a sum of two amplitudes and a difference and they ...


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Because (as you say) the ODE is linear we have that if $\phi_i$ are all valid solutions then so is $$\sum_ia_i\phi_i$$ for any real $a_i$. You can convince yourself of this by substituting the sum into the ODE and showing it is satisfied assuming the $\phi_i$ are solutions. We use the sin and cosine functions for our decomposition because they happen to ...


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For the single slit interference problem, the slit is assumed to be made of N equally spaced oscillators, these small oscillators(Huygens Wavelets) produce disturbances/waves(electric field vector) which propagate in all radial directions. The interference of these disturbances from each oscillator produces the Diffraction pattern ~ (Huygens-Fresnel ...


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I believe the premise of the question is incorrect. If the speakers are connected in the manner specified, the sound waves for all frequencies will be out of phase, as observed at the surface of each of the speaker cones. If a single observer is located equidistant from the two speakers, he/she will also observe destructive interference for all ...


1

To quantize a classical system, start from the Poisson bracket $$\{x_i, p_j\} = \delta_{ij}.$$ This relation defines $p_i$ as the momentum canonically conjugate to $x_i$ and is equivalent to Hamilton's equations. Quantize by letting $x_i, p_j$ be Hermitian operators on a Hilbert space, with commutator $$[\hat x_i, \hat p_j] = i\delta_{ij} $$ (identity ...


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What's the mathematical process and physical logic? The Fourier transform of position space ($\vec x$ domain) is wave number space ($\vec k$ domain). This is an unambiguous, well understood mathematical result. By the De Broglie hypothesis, the momentum is $\vec p = \hbar \vec k$. This is physical hypothesis with experimental confirmation. Although ...


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A Fourier transform is the decomposition of a position space function into a basis of plane waves, each of which has a well defined momentum. $$ f(x) \sim \int \text{d}p\; F(p) e^{\text{i}px} $$ This relies on the quantum mechanical idea that waves can have a well defined momentum.


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If you let $x_i$ be the position of the $m_i$, you can write a set of coupled equations $$\begin {pmatrix} m_1\ddot{x_1}\\ m_2\ddot{x_2}\\ m_3\ddot{x_3} \end {pmatrix}=A\begin {pmatrix} x_1\\ x_2\\ x_3 \end {pmatrix}$$ where $A$ gives the forces from the springs. With more masses and springs you have more lines in the equation. If all the masses are the ...


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I hope this is useful as an answer to your question, because although this is not about your exact system it gives a helpful interpretation of what normal modes are. For the water molecule we can consider it as three masses linked by two identical springs. Similar to your system there are three normal modes, which can be represented as the following motions ...


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We sometimes forget about negative frequencies but they are always there. All real signals are composed equally of negative frequency and positive frequency components. If you take an oscillating pulse with Gaussian envelope, $$E(t) = A \exp(-\tfrac 12 t^2/\tau^2)\cos(\omega_0 t)$$ and convert it into its spectral representation, then you get two Gaussians: ...


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Bass sound-waves are really big. The lower the bass, the bigger the wave. A single wavelength can go through a window, or door and enter back in through another window in a different room. They can go around walls and corners. They can also create resonance with large objects like walls, and this helps them to pass through because the wall is ...


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The mistake is that you used the dispersion relation instead of the phase velocity. The phase velocity is actually: \begin{equation} C_p=\sqrt{\frac{g}{k}\tanh(k h)} \end{equation}


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The freespace dispersion equation is $\omega^2 = k^2\,c^2$ and this cannot change: this simply follows from considering plane wave components of propagating fields, which all fulfil the Helmholtz equation $$\nabla^2 A_j + \frac{\omega^2}{c^2} A_j = 0\tag{1}$$ which is fulfilled by all Cartesian components of the moncrhomatic EM field vectors and, for a ...


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Wavenumber k is the number of waves per metre. Frequency w is number of waves per second. The number w/k is the speed of the wave.


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Technically, $\omega^2/1^2-k^2/c^2=0$ is a degenerate hyperbola if that counts. But I don't think you can derive an equation of the form $\omega^2/a^2 - k^2/b^2 = 1$ for waves propagating in free space. You may however find something of the kind if you consider materials with fancier dispersion relations than $\omega = ck$, like e.g. plasmas.


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As has already been said: the orientation of the field drawn in your book is arbitrary. Once you have decided that the wave is say going into the page, then all that is required is that the E-field be drawn on the page (i.e. at right angles to the direction of wave propagation). You can put it up, down, left, right, diagonally, it doesn't matter, it would be ...


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There is a problem with diagrams like this Yes if that was a physical object and you looked at it end-on it would look like a plus-symbol. However it isn't intended to be interpreted physically like that. These diagrams shouldn't be interpreted as showing a vertical or horizontal displacement What is being shown is field strength and direction at a ...


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The direction of the electric field is called polarization. The direction of the electrical field in the free space lies in the plan perpendicular to the direction of propagation, and if this direction is unique for all the beam, it is said to be linear polarization. So, for linear polarization, the electric field can point in whatever direction that lies in ...


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Vertical with respect to whatever arbitrary set of coordinates you devise. If most of the pictures you see have the E-field pointing "up", that's just some kind of cultural bias. The E-field can point in any direction at all.


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This is an attempt to explain, in a purely intuitive way why sound waves reflect from the end of an open pipe, and therefore can produce a standing wave. Consider a pressure wave travelling up the pipe. I've drawn just a single maximum of the pressure wave to keep the diagram uncluttered: Call the pressure maximum $P_1$ (I haven't marked $P_1$ on the ...


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Whether or not your idea produces thrust depends on at least one of the following statements being true: The laser doesn't travel with the vehicle. The photon eventually leaves the mirror structure. Forget the structures holding the mirrors and consider each component as the photon interacts with it. In all cases, I assume each emission and reflection ...


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Yes, the speed of the soliton is $c$. You can read more about the wave equation (which governs EM waves) here.


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On axis in the 1D small angle approximation the spacing is constant. When you have point sources and you go off axis (in the direction perpendicular to the line connecting the two points) that may not be true - it is also not true when the small angle approximation breaks down (higher order fringes). You should be able to find details and diagrams in any ...


0

Provided that the electron & the atomic beams also exhibit refraction,it seems that this is a particle's property.Velocity and deflection angle depends on particle's mass/size for specific medium.Photon behaves as particle in this effect.Mass is given by de Broglie equation:m=hv/c^2 , v=frequency


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When two waves of a slightly different frequency encounter each other you will hear beats, which is nothing more than the two waves interfering with each other first constructively and then destructively, over and over as the waves go in and out of phase with each other. So now forget that there are two waves and concentrate on the sound made by the pair ...


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It is the time difference between successive maxima or successive minima. However, you cannot determine a beat period from less than 2 beats. I assume you are talking about music, because the standard scientific terminology defines the frequency of "beats per second" as Hertz.


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Simply: the ends of an organ pipe reflect the wave in the pipe. A closed end will cause a pressure antinode while an open end causes a pressure node. This information is sufficient to determine the vibrational modes of any pipe. If both ends are of the same type (open or closed) the wavelengths of allowed modes will be given by $\lambda = n \, \ell /2$; if ...


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You may find by starting from first principles, or by consulting external resources that pressure waves in air (in one dimension) are governed by the wave equation $$\frac{\partial^2 p}{\partial x^2} - \frac{1}{v^2} \frac{\partial^2 p}{\partial t^2} = 0$$ where $x$ is a position and $t$ is the time, and $p$ denotes the pressure difference away from ...


2

Note that you haven't actually found the general solution in spherical coordinates... What you have there is a solution known as a spherical wave, which describes a set of spherically symmetric wave fronts that diverges from (or converges towards) the origin $r=0$. However, in general, a wave could also be a function of the angles $\theta$ and $\phi$, which ...


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The two solutions are different because they have different boundary conditions. In the first case, the equation is indeed $$ \frac{\partial^2u}{\partial t^2} = c^2 \nabla^2 u = c^2 \left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\right) u. $$ Here though we specify $u(t,x=x_0)$ to be some value ...


2

If we try to polarize the same beam of light in two planes, or if we mix two planar polarized beams, the light will interfere. If the phases of two beams will be identical, then we get 45 degrees polarized light. If the phases of two beams will be different, then we will get so called circular polarized light In other words, any sort of polarized light ...


1

yes, you can. Actually an application is googles for 2d movies. You project on the screen two different images (that is why you see it blurred when watching without the googles), each one has a different polarization. Each plastic filter in the google is a polarizer, and each eye is tuned to a different polarization. So that each eye see only one of the two ...


4

The answer is no. The reason is that waves are the solution of the wave equation, and the wave equation cannot be derived withouth implicitly assuming Newton's third law. It is intuitive to see why in waves that propagate in a medium. For instance, the bouncing back could be up and down in a transversal wave (for instance, the atoms of a string) or back and ...


3

It is not amplification! The purpose of the guitar body is to impedance and mode match between the string and the surrounding air. Intuition When a an object vibrates it pushes on the surrounding air creating pressure waves which we hear as sound. A string vibrating alone without the body of the instrument doesn't make a very loud sound because exchange of ...


3

Your idea that a quantum fluctuation created the universe is a misinterpretation of the suggestions that I have heard. Explaining why requires introducing a few ideas, so bear with me while I do this. The object we think of as the universe is made up of two bits: a manifold equipped with a metric = spacetime some matter/energy The manifold and metric ...


1

A wave $$u(z,t) = \exp(i(\omega t-kz))$$ is a wave moving in the positive $z$ direction and $$u(z,t) = \exp(i(\omega t+kz))$$ in the negative $z$ direction. Since time always flows in the positive direction it makes sense to put it as the first argument since it looks better than $$u(z,t) = \exp(i(-kz+\omega t))$$


3

Your equation is actually a statement of either Faraday's law or Ampère's law and it only holds for plane waves, i.e. waves whose field vectors vary with position $\vec{r}$ and time as a vector of the form $\vec{X}\,\exp\left(i\,(\vec{k}\,\cdot\,\vec{r}-\omega\,t)\right)$, where $\vec{X}$ is a constant. For such a wave, it is not too hard to show that the ...


4

The statement: $$ -exp(i(kz-wt)) = exp(i(wt-kz)) $$ is obviously false as you can prove very easily by expanding the equation using Euler's formula. However they are both plane waves. So if you're studying some system where you're expanding solutions as linear combinations of plane waves then they are both valid ways to write down your plane waves.


4

John has answered this partially, however the fundamental mathematical idea is missing: If we think of a function being member of some vectorspace, then basisvectors exist. This concept will surely be familiar to you from Quantum mechanics. But also from there we remember, that problems were alot easier to handle, if we know the Eigenbasis of the Operators ...


0

The change of phase is property of waves. One way to see it, is that the waves are being transmitted and reflected at the same time, so at a hard boundary, we impose the condition that the amplitude must be zero, thus the reflected wave must be out of phase with respect to the incoming (transmitted) wave. Furthermore, there may be places where the local ...


2

There are some situations in which the plane wave ansatz is useful. It is a solution to many wave equations. Plane waves are also familiar and we have mathematical techniques to handle them, such as Fourier series. However, in some situations you can quite badly wrong using the planewave ansatz. Notably when the wave interacts with a structure that has ...


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In many cases our systems are described by linear differential equations, and these have the property that any linear combination of solutions to the differential equation is also a solution to the differential equation. This is useful because usually any arbitrary solution can be Fourier transformed to express it as a sum of plane waves. So if we can find ...


1

Not at all. Space is expanding, as in space is constantly being added. Although space might be added between the electron and nucleus, that does not effect the atom to significant degree. Its like having an atom in flatland and turning the flatland into a ball. the atom is not going to know much of a difference. If the sphere is returned to flat land the ...


1

There are different possible causes for Red Shift, e.g movement of the light source away from the detector (or vice versa), gravitational red shift, or the expansion of space. For the expansion of space, have a look at the Wikipedia page on Cosmic Microwave Background, in particular... The photons that existed at the time of photon decoupling have been ...


1

The sound velocity depends on the sound frequency (dispersion). The flow must be locally faster than the frequency of the downstream disturbances. If the latter are such that their sound velocity is small, the local flow velocity may be chosen small too. Note, that the sound velocity depends also on the void fraction. If there are bubbles (even locally - ...


1

John Rennie's Answer, that complex numbers simplify calculations with sinusoidally varying quantities by letting you do linear operations with complex exponentials and then reverting back to sinusoids at the end of your calculation, is altogether correct and a summary of what is called the phasor method for dealing with any quantity that varies sinusoidally ...


0

One thing to consider is whether the wave, by being transverse, is linearly or elliptically polarized. If it starts out as a linearly polarized wave and converts to an elliptically polarized wave (which can happen), that is different than circular to elliptical. One also needs to consider whether the wave starts in the neutral atmosphere of Earth or in the ...


1

The speed of an electromagnetic wave is indeed independent of the speed of both the source and the receiver. However, this does not mean that the relative motion between the source and the receiver has no effect on the wave's properties. The effect that is being used is called the Doppler effect, and it is the fact that the received frequency of a wave will ...


3

You're perfectly correct. Referring to Classical Electrodynamics by Jackson, we see that the index of refraction $n$ is given by: $$n=\sqrt{\frac{\mu}{\mu_{0}}\frac{\epsilon}{\epsilon_{0}}} = \sqrt{\mu_{r}\epsilon_{r}}.$$ But Jackson notes that for most optical frequencies (and non-meta-material media), $\frac{\mu}{\mu_{0}}=\mu_{r}\approx 1$, which is why ...


3

Suppose we have a drum. When you bang on the drum it will vibrate. When you look at any point on the surface of the drum head, you will see it go up and down, similar to if to take a mass hanging from a spring attached to the ceiling and watch it bob up and down. However, there is also an important difference between these two situations. In the situation ...



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