When analyzing a surface chart you will notice the isobars bend in the vicinity of the warm front and the cold front. The isobars do not make perfect circles around low-pressure centers because of the pressure troughs created by the fronts. Pressure can decrease in the atmosphere by:
(1) causing the air to rise
(2) decreasing the density of the air
(3) decreasing the mass of the air (i.e. upper level divergence).
Causing the air to rise counteracts some the downward force created by gravity. This lowers pressure just as if someone started pushing up on you when standing on a scale; your weight would decrease. Fronts force the air to rise. This causes the surface pressure to decrease in the vicinity of the front. Cold fronts have a more defined pressure trough than warm fronts because the average cold front has a steeper slope and stronger temperature gradient than the average warm front. A warm front raises the air gradually while a cold front lifts the air more quickly in the vertical. The faster the air rises, the more pressure will lower. A mid-latitude cyclone and a front will both cause the air to rise and pressure to lower. The stronger the front, the more well defined the pressure trough will be.
Now to shortwaves. A shortwave is an upper level front or a cool pocket aloft. Just as a surface front causes the air to rise, upper level fronts can do the same. First, let's start with a general description of a shortwave:
(1) It is smaller than a longwave trough (shortwave ranges from 1 degree to about 30 degrees in longitude (the average one is about the size of two U.S. states put together (Iowa and Missouri put together is a good example)
(2) Isotherms cross the height contours (if it is a baroclinic shortwave). This creates an upper level temperature gradient and therefore an upper level front
(3) They are best examined on the 700 and 500 millibar charts
(4) They generate positive vorticity (mainly due to the counterclockwise curvature within the shortwave). This creates positive curvature and positive shear vorticity. If PVA occurs with the shortwave then the shortwave will deepen and strengthen due to lift created by upper level divergence.
(5) They can create an environment conducive to surface based convection or elevated convection due to the cooling aloft.
It is important to see how much moisture is associated with the shortwave. A shortwave moving over a warm and moist lower troposphere has a better chance of producing precipitation and strengthening than one moving over a dry lower troposphere. If the low level dewpoint depressions are low, the instability and lift associated with the shortwave can enhance cloudiness and precipitation. In summary, a pressure trough is associated with a low-level front while a shortwave is associated with an upper level front or a cool pocket aloft. Both are associated with rising air and can add instability to the atmosphere.
The diagrams below show an upper level shortwave and a surface pressure trough. The pressure trough is the kinking along a front.