Your idea is sound, pardon the pun. If you are thinking of having a dome shape, consider to construct a geodesic partial sphere/geodesic dome. This design is based on an icosahedron, a 20 face regular polyhedron in which all vertices are coincidental to a sphere. For geodesic structures, each edge is split into two, three, four and rarely five equally divided portions. The vertices which result from these splits are also coincidental to the sphere. The math could be considered daunting in earlier days (last century) but with the internet, the answers abound.
A two frequency geodesic dome/sphere (split each edge in the middle) will have a blocky appearance but fewer unique triangles.
A three-frequency dome will be more spherical, but require more triangles, of course.
An easy way to determine the frequency of a geodesic sphere or dome is to locate the points from which five struts/edges radiate, which is the center of a pentagon. Count the number of struts between one center and the next.
When constructing geodesic structures from flat panels, it is considered simpler to assemble these pentagons first, although different sites have different suggestions based on the assembly of each panel, especially if one is building from dimensional lumber and plywood or similar material.
In the case of rigid foam board, you would have to pay close attention to the edge angles and create a miter to accept the adjacent panel, or join only the inner edges and use a foam filler to secure the resulting gap.
The lower number frequency reference means larger triangles, while the higher number means smaller area triangles. This can be a factor when dealing with sheet material.
On higher frequency designs, it is common to have 1/2 dome or 5/8 domes, which provide for greater internal vertical clearance without sacrificing ground area.
You should be able to construct a lightweight rigid foam sheet geodesic dome with suitable sound damping (dampening = making wet) properties. Obviously, thicker foam panels will result in greater sound reduction.