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?

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 $\Delta m$?

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

Einstein used light waves (or photons) to prove his famous equation, by considering the situation of a light-emitting material at Origin $O$, emitting both in the +y-axis and the -y-axis, equally (total momentum = 0), and then analyzing that exact same situation naturally in a different moving frame. My intuition tells me that such a similar derivation should be possible with waves that are nothing like light, purely Newtonian waves like sound waves, waves on string/rope, water waves, etc...

Is this correct? Can anyone provide such a derivation? Was such a derivation, in principle, available even for Newton using Galilean Relativity?

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?