Andrew has done OK with his explanation.
A lot of people are left floundering with that one!
WZ's explanation of a long wave trough can be found in the "glossary" link on the WZ home page along with a lot of other brief but descriptive outlines of weather and climate phenomena.
Here it is;
A trough in the prevailing westerly flow aloft which is characterized by large length and (usually) long duration. Generally, there are no more than about five longwave troughs around the Southern Hemisphere at any given time. Their position and intensity govern general weather patterns (e.g., hot/cold, wet/dry) over periods of days, weeks, or months.
Smaller disturbances (e.g., shortwave troughs) typically move more rapidly through the broader flow of a longwave trough, producing weather changes over shorter time periods (a day or less).
For an actual analysis of the current southern hemisphere long wave patterns as they move around Antarctica, go to the BOM's numerical prediction division and have a look at the products such as this one
which gives the current long wave pattern.
Information such as this will give you a much better idea on the immense amount of data that the meteorologists have to deal with when they are drawing up forecasts, particularly for international aviation and maritime use.
Not only international aircraft use this data for flight safety reasons but also to pick winds that will favour fuel and time savings.
Ships use this info to avoid bad weather and to cut sailing times between ports by choosing sea and wind favourable routes.