Is $\delta(r-ct)/4\pi r$, the 3D wave equation elementary solution, a transverse or longitudinal wave? Background:
https://en.wikipedia.org/wiki/Longitudinal_wave
'Longitudinal waves are waves in which the displacement of the medium is in the same direction as, or the opposite direction to, the direction of propagation of the wave. ... The other main type of wave is the transverse wave, in which the displacements of the medium are at right angles to the direction of propagation.'
Also see Derivation of Green's Function for Wave Equation where:
$ \delta(r-ct)/4\pi r$ is the time-space Green's function (or elementary solution) to the 3D wave equation.
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My question is:
Is $ \delta(r-ct)/4\pi r$ a transverse or longitudinal wave, or is it either one?  Please give references if possible. Also please explain how it can represent each type of wave, and (if possible) the differences in the required initial conditions needed by the wave equation for a transverse vs a longitudinal elementary solution.
Note that the terms "transverse" and "longitudinal" DO apply to scalar fields.  They do not only apply to vector fields.
For example see:
https://www.physicsclassroom.com/class/waves/Lesson-1/Categories-of-Waves
https://www.khanacademy.org/science/ap-physics-1/ap-mechanical-waves-and-sound/introduction-to-transverse-and-longitudinal-waves-ap/a/transverse-and-longitudinal-waves-ap1
https://www.acs.psu.edu/drussell/demos/waves/wavemotion.html
http://230nsc1.phy-astr.gsu.edu/hbase/Sound/tralon.html
and many other references--search Google for 'transverse vs longitudinal waves'
 A: The question doesn't make sense, since the terms "transverse" and "longitudinal" don't apply to scalar fields. They refer to the relationship between the polarization of a wave and its propagation direction, but a scalar wave has no polarization; its value at every point is just a number. 
On the other hand, if you're using the scalar wave equation to describe a wave with polarization, you're simply using it as a simpler incomplete description; you cannot recover the polarization information.

Note that the terms "transverse" and "longitudinal" DO apply to scalar fields. They do not only apply to vector fields.

In all the examples you gave, we started with a physical phenomenon that we knew was either transverse or longitudinal already. Then, we chose to describe the configuration by taking a scalar value at each point. (This isn't even possible outside of the simplest of cases.) These scalar values satisfy the wave equation. You can clearly see this reasoning doesn't go backward; the scalar wave equation by itself doesn't contain all the information. You might as well ask, "is the scalar wave equation actually describing sound or light?"
A: Usually transversality is discussed in the context of EM force fields. Here the question refers to a potential. For example for a Coulomb potential, the time like component of a four vector potential, the concept has no Lorentz or gauge invariant definition. Note that the Coulomb potential is not a scalar field.
Is the electric field of a point charge transversal? Almost, as $\vec \nabla \cdot \vec E = 0$ thus $\vec E$ is transversal everywhere except at the origin.
Here, the OP discusses a solution of the 3D wave equation, the source of which is dimensionless. The "force" associated with it is like the electric field of a point charge, transversal everywhere except in the origin.
A: Longitudinal and transverse 3D spherical waves obey the same wave equation. It is neither  the equation nor the boundary conditions which can tell us whether we are dealing with a longitudinal or transverse wave. Once we have the direction of propagation of a wave, it is the physical interpretation of the model which allows to classify the wave as longitudinal or transversal.
This is true for all the solutions, including the elementary solutions (Green functions of the equation). Notice that even if one would be working with vector waves, it is not the function dependence on $r-ct$ which could tell anything about the transverse/longitudinal character. 
