# 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