The "pilot wave" is the same as the multi-particle wavefunction in quantum mechanics, so it evolves according to the Schrodinger equation. Such a wavefunction is not a field in three-dimensional space, it is a field in 3N-dimensional space where N is the number of particles.
The equation of motion for the particles in three-dimensional space depends on the gradient of the complex phase in that 3N-dimensional space. This part is exactly the same as classical Hamilton-Jacobi theory, which is an alternative representation of the forces of classical mechanics. But building trajectories from the quantum-mechanical wavefunction adds an extra nonlocal force.
You can't send nonlocal signals in Bohmian mechanics because you can't do that in quantum mechanics, and Bohmian mechanics is just quantum mechanics plus particle trajectories. The reason you can't do it in quantum mechanics is because the nonlocal correlations it induces aren't strong enough, they are only strong enough to add a nonlocal extra to a local signal (see quantum teleportation, which requires a local signal to be performed).
Pilot waves have no more and no less mass than Schroedinger wavefuction waves do, since their mathematics is identical.
To bypass ghastly ontological quandaries, recall that the math of the waves cares not about what the particles they guide ("pilot") do or don't. The math of the waves is just the math of the Schroedinger equation in polar (Madelung) language.
So, for a free particle, of mass m, you have
$$\psi = R \; \exp \left( \frac{i \, S}{\hbar} \right), ~~~~~ R^2 = |\psi|^2, ~~~ \leadsto $$
$$ \frac{\, \partial R^2 \,}{\, \partial t \,} + \nabla \cdot \left( R^2 \vec{v} \right) = 0 ~ , \tag {1} $$
where the particle velocity field is determined by the “guidance equation”,
$\vec{v}\left(\,\vec{r},\,t\,\right) = \frac{1}{\,m\,} \, \nabla S\left(\, \vec{r},\, t \,\right) $,
and also a further equation, a modified
Hamilton–Jacobi equation,
$$- \frac{\partial S}{\partial t} = \frac{\;\left|\, \nabla S \,\right|^2\,}{\,2m\,} + Q ~ ,\tag {2} $$
where Q is the quantum potential, an additional piece lacking in the classical equation,
$$ Q = - \frac{\hbar^2}{\,2m\,} \frac{\nabla^2 R }{ R } ~, $$
responsible for all the "mysterious" effects of quantum mechanics.
The above guidance equation then amounts to
$$ m \, \frac{d}{dt} \, \vec{v} = - \nabla Q ~ , ~~~~\hbox{where} ~~~\frac{d}{dt} \equiv \frac{ \partial }{\, \partial t \,} + \vec{v} \cdot \nabla ~ . $$
But notice that, interpretations aside, except for the fact we used the velocity field as a mere shorthand for the gradient of S in (1), the probability continuity equation, the two PDEs (1), (2), don't know or care about the particle: they are really a transcription of the two components (real and imaginary) of Schroedinger's equation describing probability amplitude waves.
- As such, the waves they describe obviously don't have a mass, unless you bought the destructive interpretational blather about the particle being the same as its probability wave. They just dictate how a particle moves as they pilot it.
They are also dispersive, i.e., they have a group velocity different than the phase velocity, but this is not thought of as a mass.
Best Answer
In the Faraday pilot-wave fluid droplet dynamics, the fluid wave is meant as an analogy for the wavefunction. More specifically, the experiments are constructed as physical implementations analogous to the de Broglie-Bohm theory, where a particle with discrete coordinates is 'guided' by a pilot wave which follows the Schrödinger equation.
To be a bit more explicit, the de Broglie-Bohm theory works with a standard wavefunction $\psi(\mathbf q,t)$ on a single- or multi-particle configuration space $\{\mathbf q\}$, which obeys the Schrödinger equation $$ i\hbar\frac{\partial}{\partial t}\psi(\mathbf q,t)=-\frac{\hbar^2}{2m}\sum_i\nabla_i^2\psi(\mathbf q,t)+V(\mathbf q)\psi(\mathbf q,t). $$ This wavefunction then guides an actual 'particle' with coordinates $\mathbf q(t)$ on the configuration space (so it may represent the coordinates of multiple particles) by matching its momentum to the local momentum of the wavefunction, understood as $$ m_k\frac{\mathrm d\mathbf q_k}{\mathrm dt}=\hbar \operatorname{Im}\left(\frac{\nabla_k\psi(\mathbf q,t)}{\psi(\mathbf q,t)}\right) $$ for the $k$th particle. The particle is then distributed on the configuration space according to $|\psi(\mathbf q,t_0)|^2$ at an initial time $t_0$, and it retains that distribution; upon measurement it is the configuration-space particle that gets detected.
The water droplets behave similarly but not exactly in this way. The fluid surface acts in a wave-like way, and it influences the particle-like droplet, exchanging energy and momentum with it. The analogy is, however, nowhere near exact, and it is described in considerable (but still readable) detail in
In short, the droplet moves balistically between bounces, and during the bounces its horizontal component $\mathbf X(t)$ obeys the Newton equation $$ m\frac{\mathrm d^2\mathbf X}{\mathrm dt^2}+\operatorname{drag}(t)\frac{\mathrm d\mathbf X}{\mathrm dt}=-F(t) \nabla \overline\eta $$ where $\overline\eta=\overline\eta(x,y)$ is the fluid surface height you'd have if the droplet wasn't bouncing.
This means, to begin with, that there's two big differences between the Faraday-wave droplets and the de Broglie-Bohm theory. For one, the wave's influence on the mechanical system is of a rather different character. To tack on to this, the droplet can influence the Faraday wave, which is unthinkable in Bohmian mechanics.
So, to summarize: the liquid wave is an (imperfect) physical analogue for the Bohmian wavefunction. What is the fluid surface itself an analogue for? Nothing. That's an over-reading of the analogy. It is just an analogy, and not all elements of the analogy need to mean something on the other side. The status of the analogy is well summarized by J.W.M. Bush in the introduction of
The bouncing-droplet experiments are indeed important. They show that particle-like system can indeed display wave-like behaviour, like two-slit interference or mode quantization, from more fundamental interactions which we might have missed at first. In this sense, they are an encouragement to keep looking for a Bohmian-like explanation for quantum mechanics.
On the other hand, bouncing-droplet experiments are fundamentally limited. They have a hard time simulating the three-dimensional motion of a single-particle, and they mostly cannot simulate the quantum mechanics of multiple particles (even two particles on one dimension) since the wavefunction is a wave on the (#particles)$\times$(#dimensions)-dimensional configuration space. In other cases, such as Mandel dips, the quantum wave interference occurs over even more abstract spaces.
The bouncing-droplet experiments, like other quantum analogues such as elastic rods, are therefore not proof of anything, and they have no new implications on the foundations of quantum theory. They are an interesting testing ground, and it would be good to see them take on interesting foundations issues, like Bell inequalities and contextuality, where they could produce interesting new questions and takes which could then be mirrored on the quantum side. As they are now, though, they're simply an interesting curiosity, I'm afraid.