What is an acceptable phugoid oscillation duration?
Here is the response of a 180 passenger high-subsonic twin-jet simulator: the first period takes about 87 seconds.

Conditions:


  From the trimmed position, excite the phugoid mode by applying longitudinal control in one direction in order to change airspeed by approximately 10 kts, then release.


Initial conditions:


COG 0.2205 MAC
Pressure altitude 35,000 ft
Engine N1 rotor speed 87.47% (both engines)
Calibrated Airspeed 264.54 kts
Gross weight 66,360 kg


 



EDIT

To answer the additional question of when the resulting motion amplitude would be below 0.1 deg, I made a kind of a digital simulation according to this block diagram:



and then tuned spring stiffness C, inverse inertia 1/M, and damping constant K until the resulting response overlaid the original aircraft one. The resulting motion gets to a max. amplitude of 0.1 deg in about 1,600 sec - 27 minutes.