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A linear system is a physical system responding to an external stimulation in a manner which is proportional to the amplitude of said stimulation. Stated otherwise, it is the study of a class of systems characterized by the fact that their behavior can be modeled as a linear function: $$f(x) = k·x.$$ Graphically, this means that if one plots how such a ...


8

Typically, pressure waves of different wavelengths travel at different speeds in any medium, which we call dispersion, as is described in the text you posted. But, in most everyday situations, the differences are so small that we don't notice them. One example where it can be noticed, however, is thunder. When lightning strikes, it produces sound at a lot of ...


8

Imagine you have a force $F$ acting on your system, and it responds in some way. You can write equation of motion and find the response to $F$; in case of a pendulum it would be the displacement $d$ as a function of $F$ (and time, of course). Now, a linear system is such that if it has two forces $F=F_1+F_2$ acting on it, its total response to it will be the ...


4

LIGO works essentially by monitoring the separation of two large mirrors. The zig-zag lines could be thought of as by how much the separation of the mirrors changes over the course of the 0.2 seconds represented along the x-axis. The separation oscillates in response to the passage of the gravitational wave. Confusingly, there are no indications of the size ...


3

The other answers give good physical descriptions. Let me here add the mathematical definition as well for reference. Mathematically, there are two requirements for a linear system - if both are fulfilled, then the system is linear, and vice versa if the system is linear then both apply: $$f(a+b) =f(a) +f(b)$$ $$k\cdot f(a) =f(a\cdot k)\quad, k\in \mathbb{R},...


3

E&M field creation operators are labeled by a momentum $k$ and a helicity $\{ +, - \}$: $$ a^\dagger_{\pm, k}. $$ Let's denote a single particle photon state via $$ |k, \pm \rangle \equiv a^\dagger_{\pm, k}| 0 \rangle. $$ We can now "fuzz out" the momentum and make a positive helicity wave packet state as, say, $$ \int d^3 k' f(k - k') |k', + \...


2

Time averages over a finite time span $T$ do depend on $T$. However, as already noticed in another answer, if $T$ coincides with the period the average is zero. Even more important, since $$ \left<f\right>= \frac{1}{T}\int_0^Tf(t)dt $$ provided the integral on the right-hand side of the previous formula is bounded, the average goes to zero when $T \...


2

The time average of $\sin$ and $\cos$ depends on the time interval you average those functions over. The time average over a period (or multiples of it) is zero. This is because over a period for every positive value of those functions there always is an equal but negative value as well. You don't even need to average actually, they integrate to zero over a ...


2

When you go up by one octave, the frequency doubles. When you go up by another octave, the frequency doubles again. So that means that $f$, $2f$, $4f$, $8f$, etc. are all the same note in a different octave. If we look at that list, we immediately see that $3f$ is not present! Lets say that $f$ corresponds to $A_4$ or something, such that $2f$ corresponds to ...


2

There are several well known proofs and designs (Brune, Bott-Duffing, Bode, etc.,) methods exist that can synthesize an arbitrary positive real impedance function $Z(p)$, ie., denote the complex variable by $p=\sigma + \mathfrak j \omega$ then the function is holomorphic on $\sigma >0$, satisfies $\Re {Z(\mathfrak j \omega)} = 0 $ and $Z(\sigma) >0$ ...


2

One way to look at it is the following : In theory, ideally, you could use one single speaker on your hi-fi, just the way it’s done on a cheap radio. However, you would face two problems if using one single speaker instead of a medium, plus a woofer plus a tweeter: Efficiency (bandwidth): When you send an electric signal of a given frequency f to a speaker,...


2

Welcome to stackexchange. You totally can postulate that the Planck length is a quantized minimal element of light, and you might or might not be correct about that. Truth is, we have no validated theory to describe what happens at such tiny lengths as the Planck length. The same is true for spacetime: Is there a minimum length or volume? Is there a minimum ...


1

Photons are not really particles as we imagine them (with our classical intuition), but quanta of excitations of electromagnetic field. A mode of EM field with frequency $\omega$ can be excited multiple times, e.g., $n$ times, in which case the energy of this mode is $n\hbar\omega$, and we say that it contains $n$ photons. There are of course also modes of ...


1

The irradiance of light (Watt per square meter) is directly proportional to the square of the amplitude of the electromagnetic wave. In contrast, the energy of a single photon (the wave consists of) only depends on the frequency. The link is the number of photons, which is frequency-dependent. Meaning an electromagnetic wave of given amplitude (energy) ...


1

The resonant frequency of an object depends on its internal stress distribution. For example, when you tune a guitar, you change the resonant frequency of the strings by changing their tension. The resonant frequency also depends on Young's Modulus, which changes with temperature for most materials.


1

subwoofers need to be bigger than tweeters because as the frequency is reduced, the radiation resistance of the speaker cone goes down because the cone impedance is mismatched relative to the air it is radiating into. To remedy this, the cone diameter needs to be increased for best efficiency at low frequencies. This "best diameter" scales more or ...


1

Is it possible that you are simply looking for a capacitor? A capacitor of capacitance $C$ has the (complex) impedance at frequency $\omega$ given by $$Z(\omega) = \frac{1}{i \, \omega C}.$$


1

You are correct, the angle of the transmitted beam depends on the order of diffraction and the frequency shifted. The effect can be significant, for example, we use AOMs in our lab to shift the frequency of a beam to image $^{87}$Rb atoms, we see a sudden drop in intensity after some change in frequency due to the shifted beam not coupling into the fiber ...


1

The Fourier transform is usually complex and thus multiplied by the conjugate complex to yield the square of the absolute. However, often just the absolute of the Fourier transform is shown, which is real. The square of the absolute has also the property that it describes the power that the signal has within each frequency interval, which is called Parseval'...


1

Presumably you know that experiments demonstrated the falsehood of the Classical model. The model was based on the simple idea that the more energy you hit the electrode with, the more energy it would give each electron. Light was treated as a wave of such energy. In that model, it follows that greater wave amplitude (light intensity) would impart more ...


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