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The most mathematically general way to write a Fourier expansion is to use complex waves with complex amplitudes. In this case the phase of the waves is represented by the complex phase of the amplitude. You can see this if you write the amplitude in polar form. Here's how it looks for your two-component wave example: $$ f(\mathbf x,t) = z_1 e^{\mathbf k_1 ...


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Let's say $\Phi$ is a delta function, $\Phi(k)=\delta(k-k_0)$. Presumably, you want this to be an eigenstate of the momentum operator with momentum $\hbar k_0$. With the convention you've chosen, we can convert this to a real-space wavefunction (I'm ignoring normalization for convenience): $$ \Psi(r)= \int dk \delta(k-k_0)e^{ikr}=e^{ik_0 r} $$ We can then ...


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Whether a 100 Hz input signal will show up as exactly 100 Hz in the FFT actually depends on the sampling frequency of your input, because the FFT is a discrete transform that operates on a finite number of samples. Because of this, the frequencies that appear in the FFT are necessarily multiples of the fundamental frequency $f_0$ which is $$f_0 = ...


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In many cases people do seem to say "Fourier frequency" when they mean "frequency". However, when dealing with data defined only on discrete time points the phrase "Fourier frequency" has important meaning. Consider a sequence of $N$ values $\{ x_n \}$ where $n \in \{1, 2, \ldots N \}$. This situation comes up all the time if we have a physical signal ...


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It is not a fundamental quality of light so much as it is a fundamental quality of anything that can be modelled as a wave. The easiest illustration of this is diffraction by a single slit (this could be diffraction of light sound or any other wave). So if we have plane waves arriving at the slit and we look at any point, P, beyond the slit we can think of ...


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I) Well, the Legendre transformation can be e.g. seen as the leading classical tree-level formula of a formal semiclassical Fourier transformation. This fact is e.g. used in QFT when relating the quantum action $S[\varphi]$, the partition function $Z[J]$, generating functional $W_c[J]$ for connected diagrams, and the effective action $\Gamma[\Phi]$. II) ...


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The propagators of massless Bosons are only zero inside the lightcone if the number of spatial dimensions is odd and larger than 1. You can determine the exact form of the propagator on the light-cone, even though it is divergent, and do so for any number of spatial dimensions, in the following way: Use this expression to derive the propagator in d spatial ...


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In my opinion, you don't want to take a Laplace transform just to prove it can be done. For mathematical systems that are the solution to 1st and 2nd order differential equations (i.e., from the process control world), in which several physical pieces of equipment are interacting with each other (e.g., the final outlet composition of a series of stirred ...


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You are not required to. Functions can be decomposed into a wide array of orthogonal basis functions, including the Bessel functions (in the Hankel transform) and the Legendre functions. The sine function just happens to be the overall simplest to deal with in the general case.


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You refer to Fourier Series. The brilliance of Fourier was to use sin to express a function.You know that you can create any vector from the sum of some unit vectors.Exactly the same think happens here. The number you multiply the unit vectors is the coefficients in F.S. To answer to your question of why we use sin and cos is that they have (mathematicly) ...



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