3D spherical vs 2D radial waves

[jjustinn]

The Green's functions for a 3d wave are like δ(r - ct)/r -- so if you have static source at the origin that is turned on at t=0, you get an expanding ball around it of radius ct, with strength 1/r. If you look just at the XY plane, you see an expanding disc of value 1/r.

Similarly, if you turned the source on at t = 0 and off at t=1, you would get a an expanding spherical shell of radius ct and thickness c, or an expanding annulus in the XY plane.

However, for 2d waves, there is an "afterglow" -- I don't recall the exact Green function, but rather than a δ-function, it's a lograthmic or exponential decay, and there's something similar for 1-D waves -- e.g. If someone turns a light on for one second, you will still see lit for longer than 1s, though it will get dimmer as time goes on.

This is often discussed around Huygens' principle -- IIRC the principle is the "no afterglow" rule for 3D, which holds in odd dimensions > 1.

However, because of symmetry, the spherical wave equation satisfies the one-dimensional wave equation: (rV),tt = (rV),rr -- where r is the distance from the origin (√xx+yy+zz), V(r, t) is the amplitude a distance r from the origin, and t is the time.

So, then, shouldn't our point source at the origin - which is obviously spherically-symmetrical -- exhibit the afterglow,and therefore *not* give the simple constant 1/r dependence? E.g. Since there is an afterglow, the value at r is not just affected by the source at t=r/c, but also all previous times, leading to (apparently) V -> infinity as t-> infinity?

Similarly, it would seem th XY plane (or any plane through the origin) would satisfy a 2D wave equation, by symmetry...but I'm not so sure there.

 

[vanhees71]

The wave equation in any spatial dimension reads (setting the phase speed of the wave to unity)

[tex](\partial_t^2-\Delta_D) \Phi(t,\vec{x})=0,[/tex]

where [itex]\Delta_D[/itex] is the Laplace(-Beltrami) operator of Euclidean space in [itex]D[/itex] dimensions. For isotropic problems, we have [itex]\Phi(t,\vec{x})=\Phi(t,r)[/itex] with [itex]r=|\vec{x}|[/itex], and the equation reads

[tex]\left [\partial_t^2 -\frac{1}{r^{D-1}}\frac{\partial}{\partial r} \left (r^{D-1} \frac{\partial}{\partial r} \right ) \right ] \Phi(t,r)=0.[/tex]

This shows that the equations are different for 2 or 3 dimensions. The equation also holds for the Green's function of the wave operator (except at the origin),

[tex](\partial_t^2-\Delta_{D}) G(t,\vec{x})=\delta(t) \delta^{(D)}(\vec{x}).[/tex]

This explains why the Green's functions are different for different space dimensions.

 

[haruspex]

However, for 2d waves, there is an "afterglow" -- e.g. If someone turns a light on for one second, you will still see lit for longer than 1s, though it will get dimmer as time goes on.

This is often discussed around Huygens' principle -- IIRC the principle is the "no afterglow" rule for 3D, which holds in odd dimensions > 1.

A light is in 3D, so why do you expect afterglow?
The afterglow from a doused incandescent lamp is because it takes a while to cool down.
You won't see it with a LED.

However, because of symmetry, the spherical wave equation satisfies the one-dimensional wave equation: (rV),tt = (rV),rr -- where r is the distance from the origin (√xx+yy+zz), V(r, t) is the amplitude a distance r from the origin, and t is the time.

So, then, shouldn't our point source at the origin - which is obviously spherically-symmetrical -- exhibit the afterglow,and therefore *not* give the simple constant 1/r dependence? E.g. Since there is an afterglow, the value at r is not just affected by the source at t=r/c, but also all previous times, leading to (apparently) V -> infinity as t-> infinity?

You've lost me. There's no afterglow in one or three dimensions, right? So why do you think there should be afterglow?
And the reason the spherical wave equation can be transformed into a 1-D equation does not follow from symmetry (or it would happen in all dimensions). Read http://bigbro.biophys.cornell.edu/~toombes/Science_Education/Laser_Diffraction/Resources/Huygens_Principle.htm .

 

[jjustinn]

A light is in 3D, so why do you expect afterglow?
The afterglow from a doused incandescent lamp is because it takes a while to cool down.
You won't see it with a LED.

I didn't mean to imply that the was any observed evidence of the "afterglow" in 3D -- there's clearly not (to take the incandescent lamp, the source itself is what has a finite turn-off time).

You've lost me. There's no afterglow in one or three dimensions, right? So why do you think there should be afterglow?

For some reason, I recalled reading that there was an afterglow in 1D / that Huygens' principle held in 2n+1, n > 0 dimensions. If I'm recalling correctly, isn't the "afterglow" actually equal to the original impulse in 1D? E.g. a unit origin impulse at t=0 is felt at x for *all* t > x/c? If I'm retroactively hallucinating, forgive me.

And the reason the spherical wave equation can be transformed into a 1-D equation does not follow from symmetry (or it would happen in all dimensions).

I think that's what was confusing me -- why it doesn't happen in all dimensions. Ill have to check out the links you and vanhees posted, and see if those clear anything up.

Thanks,
Justin

 

[PJC01]

I would like to provide a response to the above content by first addressing the difference between 3D spherical and 2D radial waves. The Green's functions for these two types of waves are different, resulting in different behaviors. In the case of a static source at the origin that is turned on at t=0, a 3D spherical wave would result in an expanding ball around it with a radius of ct and a strength of 1/r. On the other hand, a 2D radial wave would result in an expanding disc with a value of 1/r when looking at just the XY plane.

Furthermore, the behavior of the waves also differs when the source is turned on and off at different times. For a 3D spherical wave, turning the source on at t=0 and off at t=1 would result in an expanding spherical shell with a radius of ct and thickness of c. In the case of a 2D radial wave, an expanding annulus would be observed in the XY plane.

It is also important to note that the Green's function for 2D waves exhibits an "afterglow" effect, which is not present in the Green's function for 3D waves. This afterglow effect is a logarithmic or exponential decay, which means that even after the source is turned off, the wave will still be observed for a longer period of time. This can be seen in everyday examples such as a light being turned on for one second, but still being visible for a longer period of time.

This phenomenon is often discussed in relation to Huygens' principle, which states that every point on a wavefront can be considered as a source of secondary spherical waves. This principle holds true for 3D waves, but not for odd dimensions greater than 1. Therefore, it is expected that the point source at the origin, which is spherically symmetrical, would exhibit an afterglow effect and not a simple 1/r dependence.

In regards to the 2D wave equation, it is true that a plane through the origin would satisfy a 2D wave equation by symmetry. However, the afterglow effect in the Green's function for 2D waves may complicate this and further investigation is needed to fully understand the behavior of waves in this scenario.

In conclusion, the difference between 3D spherical and 2D radial waves lies in their Green's functions, resulting