# Why energy spread between different surface areas are the same with sound waves?

I got really confused about something that I asked yesterday so I will better rephrase and explain my question. The text seems big, but wanted to write down everything so there is less space for misunderstanding. I'm not also sure why this would deserve the downvote!

Sound waves move spatially (in sphere directions - 3D). For simplicity, let's just say that no heat loss is occurring at all. So loudspeaker emits sound waves.

The notion/assumption that I have received yesterday was that if no heat loss, energy received at the surface area of $$r$$ radius sphere will be the same as the energy received at the surface area of $$r+dr$$. This means energy gets spread, but each time it spreads (dr increasing), it spreads over more particles.

My question is why the energy at the surface area of r is the same as at the surface area of r+dr ?

Let me explain why I think this is confusing to me. We know that sound waves is elastic collisions. So in order for the above to be TRUE, we must assume that while particles of r surface area hit with particles of r+dr area, they give their whole energy to them. Otherwise, if they don't, then r+dr surface area won't get the same energy spread over its particles than it was between r surface area particles. So why does the assumption hold true ? As I understand, particles of r area will collide with r+dr particles(not head-on collisions, but with some angles which will NOT cause r particles KE to become 0, so they will bounce off). This means r particles didn't give their full energy to the r+dr particles.

Where am I making a mistake ? Note that my assumption seems more correct because we know when particle 1 hits another particle 2/3/4, particle 1 doesn't stop and bounce off(we call sound wave longitudinal, so this is more logical that particle after collision comes back to equilibrum position and passes by in the opposite direction - we otherwise wouldn't get sinusoidal form of graph - This makes me believe more that energy between $$r$$ and $$r+dr$$ areas are not fully transfered (even if there was no heat loss).

• Then, where is this energy if there is no heat loss and is not transmitted?
– nasu
Jun 2, 2023 at 17:37