Propagation of gravitational waves The mechanical waves requires medium to travel .
The electromagnetic waves can travel without matter medium with the help of fields .
But how do other types of waves , that don't come in either category , traverse ?
For example - gravitational waves .
 A: You can think of gravitational waves as being a waves in a field similar to electromagnetic waves. So if you are happy with electromagnetic waves, gravitational waves should not provide any additional conceptual difficulty, at least as far as your question is concerned.
More precisely, if you write the metric as a background geometry plus a small perturbation, that small perturbation will obey a linear wave equation. You can think of the perturbation as a field "living on" the background geometry.
A: Gravitational waves do not need any medium to propagate. They are perturbations of the metric, they do not need any material to cover distances.
In general, waves can be described by the D'Alembert operator which is nothing but
$$  \Box \phi  =[-\partial_t^2 + \partial_x^2]\phi = 0$$
This equation is solved by a simple exponential as $\phi = \exp[i(t-x)]$.
For gravitational waves the situation is more complicated, since the wave is no longer a scalar, but a tensor perturbation around some background metric. In practical terms this means that it carries spacetime indices.
This perturbation is usually written as $h_{\mu\nu}$.
For the wave equation of gravitational waves, you can consider Eq. 2.16 of the document here. In vacuum, the energy momentum tensor $T_{\mu\nu}$ is zero, and you can see that the metric perturbation $h_{\mu\nu}$ follows the wave equation, I have given above. There is no need of any medium here.
A: The universe may a certain character at position $x$ which implies the universe must have a certain associated character at position $x+A$.
The universal constant $c$ is the rate at which "must be" becomes "is".
Light does this in an iterating process. Given the changing magnetic field at position $x$, for an appropriately chosen distance $A$, there must be a changing electric field at position $x+1A$, which means there must be a changing magnetic field at position $x+2A$ and so on. $c$ is the speed of must be turning into is, hence it is the translation speed of this iterating process, which we call light.
Gravity does this without any iteration but the mechanism is otherwise identical: must turns into is at $c$. For example, if there is a mass $m$ at position $x$, the acceleration due to gravity at position $x+A$ must be $Gm/A^2$. Suppose we move the mass form position $x$ to position $x+B$. Then (in the Newtonian approximation) the acceleration due to gravity at $x+A$ must be $Gm/(A-B)^2$. $c$ is the speed of must be turning into is, so it takes $A/c$ time before a particle at $x+A$ will experience the new acceleration due to gravity.
