An exact form of Lilley's equation with a velocity quadrupole/temperature dipole source term

There have been several attempts to introduce approximations into the exact form of Lilley's equation in order to express the source term as the sum of a quadrupole whose strength is quadratic in the fluctuating velocities and a dipole whose strength is proportional to the temperature fluctuations. The purpose of this note is to show that it is possible to choose the dependent (i.e. the pressure) variable so that this type of result can be derived directly from the Euler equations without introducing any additional approximations.