In general, conservation laws can result in shock waves, i.e. locations where a variable changes quickly and a discontinuity arises. One of the most common examples that is always mentioned is the shock wave that forms when fluid or an object travels at supersonic speed, which creates a shock wave. However, are there other physical systems outside of fluid dynamics that also shock waves? This reference mentions an example of shocks in traffic. But I am particularly interested if shock waves occur in electromagnetic or optical systems for example.
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1For electromagnetism, maybe you can have something like $A'^a=A^a+\theta \delta A^a$ , so that $F_{ab}$ changes by Delta function. Similar construction also holds for gravity, where you replace $A^a$ with the metric. Those are called impulsive waves. – KP99 Apr 24 '23 at 11:31
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2https://en.wikipedia.org/wiki/Cherenkov_radiation – hyportnex Apr 24 '23 at 11:44
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@KP99 I think your comment is useful, but I want to point out that this isn't really the same thing as a shock wave. Shock waves in fluid dynamics can arise in an initial condition that doesn't contain a shock wave. A sufficiently big but continuous pressure gradient can evolve over time into a shock wave. And there are actually physical ways of producing a shock wave (things actually can move faster than the speed of sound in a fluid). Cherenkov radiation in the presence of a constant index of refraction (not possible), I think, is a perfect answer. – AXensen Apr 24 '23 at 11:57
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Solid high explosives are detonated to activate the fission primary of a nuclear weapon. – John Darby Apr 24 '23 at 12:29
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@AXensen Thank you for the comment. You are correct..I thought different as title of this question mentions shockwaves outside fluid dynamics. So I wasn't thinking in terms of shock waves as the one travelling faster than local velocity of sound – KP99 Apr 24 '23 at 12:31
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In general, conservation laws can result in shock waves, i.e. locations where a variable changes quickly and the differential equation no longer holds.
(emphasis mine)
This statement is somewhat misleading and is a poor definition for shock waves. These are usually defined as the waves that have velocity greater than the phase speed of own excitation in an environment, e.g. see Shock wave:
In physics, a shock wave (also spelled shockwave), or shock, is a type of propagating disturbance that moves faster than the local speed of sound in the medium. Like an ordinary wave, a shock wave carries energy and can propagate through a medium but is characterized by an abrupt, nearly discontinuous, change in pressure, temperature, and density of the medium.
In this sense, Cherenkov radiation, already mentioned in the comments, is the electromagnetic equivalent of the shock wave created by a supersonic jet:
Cherenkov radiation is electromagnetic radiation emitted when a charged particle (such as an electron) passes through a dielectric medium at a speed greater than the phase velocity (speed of propagation of a wavefront in a medium) of light in that medium. [...] Its cause is similar to the cause of a sonic boom, the sharp sound heard when faster-than-sound movement occurs.
As far as the equations are concerned - the macroscopic Maxwell equations are not violated in Cherenkov radiation, and I suspect that the equations of hydrodynamics are not violated by a sonic boom. However, these equations are different in the sense that, while wave equation follows from Maxwell equations without approximations, this is not the case in hydrodynamics, where the wave equation for sound waves is approximate and likely do not hold in the case of supersonic boom.
More importantly, the solution for both Cherenkov radiation and supersonic boom is discontinuous, but this can be seen as propagation of a very sharp wave front. Indeed, given a wave equation $$ \frac{\partial^2 u(x,t)}{\partial t^2}-\frac{1}{c^2}\frac{\partial^2 u(x,t)}{\partial x^2}=0,$$ we have that any function $f(x\pm ct)$ is its solution. This technically fails for functions that have discontinuities or lack derivatives at certain points, such as Heaviside function $$ f(x,t)=\theta(x\pm ct) $$ or something like $$f(x,t)=|x\pm ct|. $$ However, this is the problem of of solution, imposed by boundary conditions, rather than the failure of equation - which still can be claimed to hold in terms of generalized functions.
Roger V.
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Your first paragraph is correct, I have rephased the question. Is there a partial differential equation associated with Cherenkov radiation? I guess the Maxwell equations? – Thomas Wagenaar Apr 24 '23 at 12:22
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@ThomasWagenaar macroscopic Maxwell equations - that is equations with dielectric permittivity and permeability (which are meaningless on the microscopic level.) The phase velocity of light in a medium might be smaller than the velocity of light in vacuum, $c<c_0$. We can now imagine such a medium with a small hole - a particle enters into a hole with speed $c<v<c_0$. The particle generates electromagnetic waves, but they propagate slower than the particle, and there is a "supersonic cone" formed behind it. – Roger V. Apr 24 '23 at 12:34
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@ThomasWagenaar what is a bit confusing in the case of fluid dynamics is that one cannot fully dissociate the speed of the waves (speed of sound) from the movement of the medium itself. Within the jet cone there is no medium to speak of, whereas in case of Cherenkov radiation we focus only on the propagation of the wave front. – Roger V. Apr 24 '23 at 12:34
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While it is not strictly a shock wave in the most technical sense, the generation of magnetic fields by moving charge has many characteristics of a vector shock.
When a charged particle is at rest, its electric field extends straight radially outward in all directions. If it is moving, then the radially-outward field produced now must connect to the radially-outward field produced some time t ago when it was in a different place. This results in a discontinuity with field lines connecting at right angles. See this answer for some good images. This discontinuity propagates at the speed of light and is responsible for the magnetic field generated by moving charges.
Even if the charge moves at steady speed less than the speed of light in a material the "shockwave" is still produced.
thegreatemu
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