Digression: more on sound fields

The pressure field. Fields derived from the pressure.

 ‐a scalar field
 ‐our ears measure pressure
 ‐there’s more to the sound field

Three dimensions plus time. Sections and evolution.

 ‐the pressure field as a static three dimensional scalar one
 ‐as a constant time section of a four dimensional space

The velocity field. Common implicit assumptions.

 ‐needed for directional propagation
 ‐in combination with pressure suffices for a complete description of the sound field
 ‐often assumed to be the gradient (or not? 90° shift?) of the pressure field
 ‐this assumption is bad!

Time phenomena via differential equations

 ‐mixing partial derivatives in space and time dimensions
 ‐mixing in other differential operators
 ‐differential equations as a condition on a solution
 ‐time evolution flattened into the structure of a four dimensional function

The wave equation. The sound field. Wave motion in a compressible fluid.

 ‐reactive sound fields
 ‐e.g. energy propagation in a standing wave: knots call for attention to velocity
 ‐parallel in electromagnetics: magnetic and electric fields
 ‐at least closed spaces cause reactive fields

Initial and boundary conditions in sound transmission. Connection to modes.

 ‐if we assume zero initial velocity
 ‐the pressure field uniquely determines what happens next?
 ‐where does this come from: velocity equals pressure gradient?

Easy cases: plane waves and point sources

 ‐plane waves are almost too easy
 ‐point sources as well
 ‐but: the velocity field of a point source is quite difficult already

In general: the Navier‐Stokes equation

 ‐all the approximations involved and what they imply
 ‐dropping the viscous losses gives us a non‐turbulent field
 ‐assuming small density fluctuations gives us the normal description of the sound field