[epicwhaleburger]

Electrical engineer here. Sounds waves are called transverse waves because their displacement moves in the positive/negative y directions, which appears to give the impression that the wave is a single traveling particle. Instead the sum of the displacement creates the "traveling wave", which is similar to a wave an audience might do at a sporting event. I don't think this article did a great job describing what they meant by mass. I believe they blow the word "Phonon" a bit out of proportion. High energy physics is an incredibly hard topics where very few people are smart enough to read such a paper with confidence so these are just my humble thoughts.

[whatshisface]

I think you might have got transverse swapped with longitudinal, transverse is perpendicular to the axis of propagation (string waves and EM waves), longitudinal is parallel (sound waves in fluid), and mechanical solids can support both (you can have a transverse sound wave in an iron bar but not in the air).