Huh, maybe a heat exchanger problem? I know that a few years ago goodman installed screens in the heat exchanger in front of burners for some emissions standard and these can cause some combustion noise. Only other thing I have seen is if a gas meter is in the bsaement and a vent line from the regulator is run outside, there sometimes is a pulsing of the gas pressure causing a pulsing of the flame. I'd be interested to see what the problem is.

 

This is not an older Gmp unit is it? They had replacement burners for this situation. If it is the newer then i would disconnect the vent and intake to make sure it is not a venting issue otherwise I would say the heat exchanger had issues.

 

if the burners have adjustable air shutters on them, they are the defective design and the updated burners mentioned above should be installed. This will get rid of the pulsing. On some furnaces it was so bad that individual burners would actually go out after lighting and sometime shut the unit down on flame failure.

Update:
I see this GMS is a current model, so my above advice does not apply. It was mostly for GMPs and other old models with adjustable shutters

 

If you've checked for the faulty burner design already, I would check for air leaks next.
I had one GMP down flow a while back making a loud burner noise. After a closer inspection I found the faceplate of the heat exchanger allowing blower air to creep into the combustion housing. Sealed the plate, replaced some screws, and the noise was eliminated.

 

Was out on routine service.Goodman GMS,this is single stage nat gas,hi eff. (sorry no Mod&Ser). First thing I noticed was,plate in front of burner brown in color,more burnt look than rust. Visually the burners seemed to be burning normal,however they have this "woop.woop.woop noise going on.Customer said he had noticed also.
I checked gas pressure,low pressure system. Had 6.4 in. 3.8 out. Adjusted to 3.5,still noisy. Noise the same,blower on or off.Have installed alot of Goodmans,I know this is not normal. Thanks ahead.
IM BACK...,long story! 6 monthes later get call to same home,seized inducer motor. Another contractor had installed a gas valve after my first visit,I assumed he had fixed "pulsating". I installed inducer fired up,ran fine till blower came on,pulsating badly.Pulled heat ex changer,hugh hole in first pipe bend.Lots of mistakes made on this one!