Interpretation of the modulated pressure variable Q

Earlier a variable Q was defined from the pressure P by the equation  

$$P( \omega ) \eq Q( \omega )\ \exp \left[ \ i \omega \int_0^z {dz \over \bar v (z)} \ \right]$$ (42)

The right side is a product of two functions of $\omega$.At constant velocity (42) is expressed as  

$$P ( \omega ) \eq Q ( \omega ) \ e^{ i \omega z / v} \eq Q ( \omega )\ e^{{i} \omega t_0}$$ (43)

In the time domain $e^{{i} \omega t_0}$becomes a delta function $\delta ( t \ -\ t_0 )$.Equation (43) is a product in the frequency domain, so in the time domain it is the convolution

$$\begin{eqnarray} p ( t ) &=& q ( t ) \ {{\rm *}}\ \delta ( t\ -\ z / v ) \nonumber \\ & =& q ( t\ -\ z / v ) \nonumber \\ & =& q ( t' )\end{eqnarray}$$ (44)

This confirms that the definition of a dependent variable Q is equivalent to introducing retarded time t'.