$E=mc^2$ derivation using waves other than light

Question

Can $E=mc^2$ be derived using waves other than light?

Einstein's derivation of his famous equation $E=mc^2$ relies on light waves (or photons). He considered a scenario with a light-emitting material at the origin $O$ that emits light equally in both the $+y$ and $-y$ directions, resulting in a total momentum of zero. He then analyzed this same situation from a different moving frame.

This leads me to wonder: could a similar derivation be possible using waves that aren't light-based—such as sound waves, waves on strings, or water waves? Is this a valid line of thought? Can anyone provide a derivation using such waves? Furthermore, was a derivation of this nature theoretically accessible to Newton through Galilean relativity?

Edit:

After reflecting on @ACuriousMind's insights, I've come to understand that we likely can't derive the precise formula $E=mc^2$ solely within the confines of Newtonian mechanics and Galilean relativity. However, is it possible to derive an approximate relationship between the energy $E$ lost by the material and the corresponding change in mass $\Delta m$?

Answer 1

Let me propose another angle:
Note that $E=mc^2$ is the size of the gap for the spectrum obtained from the Dirac equation. Of course, the Dirac equation is based on $E= mc^2+p^2c^4$, but this indight essentially means that we can define the rest energy for any gapped spectrum - electrons in semiconductors, phonons, magnons, etc.

Answer 2

Let us derive mass-energy-relation if momentum of any sound-wave is zero:

Two identical blocks move side-by-side along y-axis. An energetic sound-wave is is emitted by the left block and absorbed by the right block.

Before and after both blocks have momentum $mv$, and velocity $v$, so before and after they must have mass $m$.

So, as energies of blocks changed, but masses did not change, it follows that energy is massless.

---

Let us derive mass-energy-relation if momentum of sound-wave is:

$ E/c^2 * v$ where E is energy and v is velocity vector of the sound-wave.

Two identical blocks move slowly side-by-side along y-axis. A sound-wave with energy $1J$ is emitted by the left block and absorbed by the right block.

As y-momentums of the blocks changed by $1J/c^2*v$, and y-velocities of blocks did not change, it follows that $1J$ of energy moving at velocity $v$ must have momentum $1J/c^2 * v$.

So $1J$ of energy must have mass $1J/c^2$.