Is there a phase shift $\pi$ radians when a pressure wave is reflected upon a medium having less acoustic impedance? In my text book it is written that a sound wave modeled as pressure fluctuations does not undergo a phase shift of $\pi$ radians upon reflection as there will be a pressure antinode at the interface and the pressure needs to be continuous at the interference. I wonder if a phase shift $\pi$ radians occurs when a pressure wave is reflected upon a medium having less acoustic impedance, as this would mean there is a pressure node at the interference?
 A: Short answer: Yes, for a pressure wave reflected from a surface with less acoustic impedance than the original medium the phase shift would be $\pi$ radians, neglecting absorptive processes.
Longer answer:
I have some equations, so let me provide the full context.  Consider a plane pressure wave propagating in a homogeneous fluid and is normally incident upon a flat interface to another domain.  Let the characteristic acoustic impedance of the original domain be denoted as $Z_1$ and for the second domain as $Z_2$.  The reflection coefficient may then be written as
$$
R = \frac{Z_2-Z_1}{Z_2+Z_1}.
$$
In general $Z_1$ and $Z_2$ may be complex quantities (frequency domain; they would be convolution operations in the time domain), but if we neglect absorptive processes they are real.  Thus, for $Z_2>Z_1$ (e.g., rigid surface) the reflection coefficient is positive, and the phase change is 0 radians.  If $Z_2<Z_1$ the reflection coefficient is negative, which is a phase change of $\pi$ radians.  Obviously, complex impedances lead to reflection coefficients that are not restricted to 0 or $\pi$ radians phase shifts, and in general are not those simple cases.
