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1

Most sounds (even the sound of a "single note") contain multiple frequencies. For pure sounds, there is the fundamental frequency and its harmonics, but almost any "real" sound contains some additional components - due to the envelope of the sound (e.g. the fact that a string must be plucked, then decays) or due to sampling (your sample is finite - so there ...


3

The Fourier series has terms like $\sin \left(\frac {2\pi n t}P \right)$ After time $P$ all the terms will repeat, so the Fourier series can only represent functions that have period $P$. Many functions in nature are periodic, so these series can represent a lot of things we are interested in. The fixed period makes the various terms orthogonal, so ...


4

This is a lot more subtle problem than is indicated in any of the comments. The problem is not just the issue of how the sum of non-causal signals can approximate a causal one, but how is it possible that while all real-life signals must start and stop at some time they must also be band-limited beyond some frequency, but as we know these two are ...


6

I just wanted to add to a previous (very accurate) answer: you can think of it as an Fourier expansion of the actual (physical) wave profile. It is not a real life process, it is a mathematical approximation. The wave pulse can be thought of as a superposition of plane waves, which happens to interfere destructively in entire space, except for the localized ...


18

In this case it's probably best to be pragmatic. A pulse can be described as a superposition of sine waves that extend infinitely into space and time. But it's just that: a mathematical description that is useful for your purposes. There is not necessarily a physical meaning connected to it. Nevertheless, in quantum mechanics the wave-description of ...


3

A measurement typically involves the convolution of the thing being measured with the response function of the instrument. Now if the Fourier Transform of your response function has zeros in it, the convolution theorem tells you that information at the corresponding frequencies will be destroyed by the measurement process There are many possible examples of ...


1

You ask about laboratory experiments involving convolution. A common example is measuring the time profile of some event when the process by which the event is triggered has a time profile similar to the duration of the event itself. For example measuring the decay of an exited species when excited by a laser pulse. A common experiment in biophysics and ...


2

A process that is linear and shift invariant can be described by a convolution integral. As an example, consider the scalar diffraction of light, which can be computed with the Fresnel diffraction integral. Scalar diffraction is a linear shift invariant process. That's why the Fresnel diffraction integral is a convolution integral.


4

Fourier transforms occur very often in most fields of physics. Products of functions occur very often in most fields of physics. As a consequence of points 1 and 2, it is common to encounter Fourier transforms of products when manipulating an algebraic expression. To move forward algebraically, you would need to apply the convolution rule.


1

You should understand the difference between two related quantities: the wavevector $\vec{k}$ and the momentum $\vec{p}$. The wavevector has dimensions $\frac{1}{length}$ whereas the momentum has dimensions $\frac{mass*length}{time}$. In quantum mechanics, these are linked by the equation $\vec{p} = \hbar \vec{k}$, where Plank's constant $\hbar$ has ...


0

The confusion is why there the term $E(\omega_2)E^∗(\omega_2)E(\omega_1)$ has factor of $6$ and $E(\omega_1)E^∗(\omega_1)E(\omega_1)$ has factor of $3$ I understand this confusion as follows. In first term there are two fields and in second term there is only one field. Hence if $E(\omega_1)=E(\omega_2)$ you are actually pumping double power in first case ...


2

The Heisenberg Uncertainty Principle has two distinct aspects: One is the identification of matter as a wave and, in particular, the relationship between a particle's momentum $p$ and its wavelength $\lambda$ through de Broglie's relationship $p=h/\lambda$. This is the crucial bit of physical input. The second one is purely mathematical, and it's the ...



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