# Tag Info

156

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 $\sin(kx - \omega t)$, ...

105

You have created a rather poor pinhole camera (camera obscura). You can see an "image" of the sky, a green space (trees) and even a reddish brown blur that is your driveway. This is not diffraction or refraction - it's geometrical (classical) optics. Because the hole is pretty big, you see a very blurry image. But basically, the light from the sky falling ...

31

like even when light gets on the moon why does the space appears dark from the moon? For the same reason it appears dark from the Earth (when flying at an altitude of 80,000 feet or so): Image credit: View from the SR-71 Blackbird. The fact is, we can't 'see space' from the Earth's surface during the day because the atmosphere is 'in the way'- the ...

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 ...

19

As you have probably noticed, the moon is tidally locked with the earth so that we always see the same side. You can look up in the sky and watch sunlight move across the moon's face. From the surface of the moon this change in illumination would look just like the day/night cycle on Earth ... except that it's roughly a month long. Until the advent of radar ...

17

Lorentz came with a nice model for light matter interaction that describes dispersion quite effectively. If we assume that an electron oscillates around some equilibrium position and is driven by an external electric field $\mathbf{E}$ (i.e., light), its movement can be described by the equation $$m\frac{\mathrm{d}^2\mathbf{x}}{\mathrm{d}t^2}+m\gamma\frac{\... 14 Dispersion of sound in air, with constant temperature and pressure, is very slight, increasing for very short wavelengths, and for very loud noises. Why? Because the rapid sequence of weak compression/decompression steps as the sound propagates are adiabatic, or energy-conserving, for the normal ranges of sound. This leaves the local pressure, temperature ... 9 Electromagnetic radiation in a medium propagates according to the law$$ \mathbf E,\mathbf B \propto e^{\imath(\pm k_xx-\omega t)} $$where$$ k_x^2 = \frac{n^2\omega^2}{c^2}\;. $$The refractive index n can also be complex, in which case its imaginary part describes the absorption of the EM wave in the medium. But the oscillating part is in any case$$ \...

8

In non-relativistic systems both $E\sim k$ and $E\sim k^2$ are possible. Quadratic dispersion relations occur if $\langle 0|[Q_i,Q_j]|\rangle\neq 0$ for some of the generators. This occurs in a ferromagnet because rotational invariance is broken and $J_z$ has an expectation value. In terms of effective lagrangians the difference between ferromagnets and ...

8

What you want to do is change the wave equation into a Klein-Gordon equation: $$\frac {1}{c^2} \frac{\partial^2 \psi}{\partial t^2} - \nabla^2 \psi + \alpha^2 \psi = 0,$$ where $\alpha$ is a constant of appropriate dimension and usually (in quantum theory) given by $$\alpha=\frac {m c}{\hbar}.$$ Inserting an ansatz of the form $$\psi=e^{i(kx-\omega t)... 8 Suppose you have an infinite plane wave. To find the momentum of this wave you Fourier transform it. Because it's an infinite wave the Fourier transform is a delta function and the wave has a well defined single value for the momentum. Now take a wave packet i.e. the same infinite plane wave but now multipled by some envelope function. When you Fourier ... 8 Carrying out the Fourier transform, I get a slightly different result for the frequency spectrum than 'knzouh'. I used u instead of y and c instead of v, so the PDE becomes:$$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(...

7

The speed of sound is constant in the same sense that the mass of an object is constant. In the typical audible range, at frequencies below, like, $100\:\mathrm{kHz}$ and sound pressures much less than atmospheric pressure, the behaviour of air is very well described by a simple linear wave equation that's purely second order in both space and time. As a ...

7

An easy way to make this intuitively plausible is by remarking that the Schroedinger equation in the absence of a potential is as follows $${\partial\over\partial t}\Psi = \nabla^2\Psi$$ up to constants, which is the heat equation if we ignore the fact that the omitted constants are complex numbers rather than real and of the right sign. If you consider ...

7

Short Intro The nonlinear term or steepening term, $\left( \mathbf{V} \cdot \nabla \right) \mathbf{V}$, determines the rate of steepening of a wave. This can be balanced/offset by loss terms like dispersion (e.g., $\propto \ \beta \ \partial_{x}^{3} v$), diffusion, viscosity (e.g., $\propto \ \nu \ \partial_{x}^{2} v$), resistivity, friction (e.g., $\... 7 The simple explanation given in Hewitt's Conceptual Physics is that atoms in condensed matter have a high-frequency resonance, and the index of refraction for most substances is strongest at the blue end of the spectrum because that's the high-freqency end, which is closest to the resonance. The following is my attempt to flesh this out with a little more ... 6 Two media can have equal indices of refraction. For example, you could pick the densities of two different gases so as to make their indices of refraction equal. You could do the same thing with different liquids containing properly chosen concentrations of solutes. 6 The term dispersion refers to the speed of light in a material having a dependence on frequency (or equivalently wavelength). The refraction angle's dependence on frequency is caused by the material dispersion, not the other way around. In all materials the refractive index will have dispersion but it's often the case that certain materials in certain ... 6 Take a look at Griffiths Introduction to Electrodynamics, particularly the section called "The Frequency Dependence of Permittivity". Dispersion can arise from the constraints, or bound nature, of the constituent particles in a given medium. For the example of optical dispersion in a dielectric medium, we could picture the electrons as bound, damped ... 6 Dispersion in waves arises from both material property variation with frequency and from the geometry of the fields in question. That wave dispersion will arise from material property variation is obvious. But wave geometry and boundary conditions also matter. Simple example: a conductive waveguide with rectangular cross-section with sidelengths$a$and$b$... 5 Depending on the physics underlying the particular wave equation in question, the three most fundamental limitations on dispersion are causality, stability and holomorphicity. These are most readily converted to mathematical statements about the operators in a wave equation if the wave equation is linear. I'll confine the following mainly to optics; ... 4 In a linear wave equation, there is nothing to pull a pulse or envelope of running waves apart. But there is nothing to hold it together, either. A minor disturbance such as a small obstacle or some dispersion, will change the waveshape, or break it up, such as losing some of its energy to outward spherical waves from the obstacle. Two or more pulses in ... 4 This is the generic form of a dispersion relation with a low frequency cutoff. It is a good model for the dispersion relation of electromagnetic waves in a plasma, i.e. the ionosphere, where: $$\omega^2 = \omega_0^2 + c^2 k^2$$ Which is the same as your form, though I've put back in$c$for the nondispersive phase velocity and changed$k_0 \to \omega_0$... 4 The whole point of Snell's law is about taking into account wavelength! Remember that the fundamental property of the light is its frequency. Wavelength, on the other hand, is not a fundamental property of a beam of light since the wavelength changes all the time as it passes through various media. The index of refraction of a medium is a dimensionless ... 4 It is indeed possible. This was a famous experiment by Isaac Newton (published in 1672). Place a lens of focal length$f$a distance$2f$from the first prism. Add a second identical prism$2f$past the lens and rotate it round until white light emerges. The lens is required to bring the rays back together. It creates an image of the exit of prism one on ... 4 Because by expanding the sinus term into a taylor expansion, you get$\sin(x)\approx x - \frac{x^3}{6} +\cdots$So, for small values of k you are allowed to take just the linear term. 4 The result$\omega^2=\frac{c_1+c_2}{M} \pm \frac{1}{M}\sqrt{c_1^2+c_2^2+2c_1c_2 \cos ka}$leads to two real solutions for$\omega^2$, since$ -1 \lt \cos ka \lt 1$and the square root lies between$|c_1 - c_2|$(for$\cos ka = -1$) and$c_1 + c_2$(for$\cos ka = +1$), so that$\omega^2$is always positive. The frequency$\omega\$ is taken as a positive ...

3

The speed of sound depends primarily on the properties of the medium: density (atomic weight), modulus (for solids), and adiabatic index (for gases). This means that it changes with composition of air (which is one reason why you sound funny when you speak after inhaling helium) and for a given gas, with temperature (see http://www.sengpielaudio.com/...

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