Cowl shake is the shaking/vibration you feel in the steering column, typically when going over rough turrain or bumps. It results from the body twisting. It is obviously more prevalent in verts and t-tops since the chassis is generally less rigid. Strut brace etc help reduce this by stiffening the chassis.
The roof of a car is an essential part that makes the chassis rigid. Rigidity in a chassis is highly desirable because it leads to less noise, better handling, better crash performance, among other things. Manufacturers are always trying to make the chassis of the car more rigid with less flex.
When a roof is chopped off in a Targa or convertible, it takes away one of the key components of chassis rigidity. You experience this in convertible/targa top cars if you go over bumps or in hard corners and you can hear little creaks and literally see the windsheild/hood area "bounce". With the top up when going over bumps, you can sometimes hear a squeaking from the weather seals as they rub against each other because the chassis is twisting.
Needless to say, it's a little disconcerting. Thankfully, my BMW is almost totally devoid of cowl shake. BTW, I was too lazy to type so I copied the above three paragraphs from an NSX forum