I predict that under steady acceleration, the barometric pressure at the back will increase, but stop increasing, and stay slightly higher than the pressure at the front. the force between air molecules under acceleration will balance with the force the bus is applying to the air and the pressure will stop increasing. Make the bus accelerate more, and the pressure will increase again, but then stop increasing fairly quickly, as the forces balance once again. Have the bus brake suddenly, and the pressure at the front will rise and exceed the pressure at the back, since the direction of acceleration as reversed.
Sorry, but your "predictions" are totally incorrect.
The bus is not sealed form the outside air, therefore the pressure inside the bus cannot vary from normal atmospheric air pressure outside of the bus. Additionally, you couldn't have two areas inside the bus—even if it were sealed—registering two different pressure levels.
Yes you can but it's so quick , people misconstrue it as gravity and inertia, two made up words to describe the real reality of what is happening.
Everything that happens on earth...every movement by anything, creates high versus low pressure, it cannot happen any other way.
Density against pressure in movement, will always create a lower pressure that immediately tries to equalize in super quick time. It's no different to diving in water.
As soon as your head hits the water, your body moves it our of the way and the water immediately fills the area you pushed aside. You created a high pressure with your head and shoulders and that pressure is pushed around you to fill the gap and so on until your feet disappear into the water and it closes up.
Now think of that as a bus going at speed of ACCELERATION through it and still going. It's mass displaces the water it is in and also forces the water in front to push around it.
I'm waiting for the " but water doesn't compress" answer. Please don't, it just makes you look desperate.