The first plans to secure the Confederacy had two major defects. President Jefferson Davis supported the idea of placing troops on the perimeter of the Confederate borders. This left the South weak almost everywhere as the South simply did not have enough troops to effectively defend the entire all of the South's borders. Davis made an additional error by establishing what can be termed a departmental system. The purpose of this was to create self supporting areas to carry out the defense of the South. This policy created a situation whereby the effective coordination of Confederate troops was more difficult. With military areas under a sort of separate responsibilities, this caused confusion.
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The initial military strategy of the Confederacy was ineffective due to overconfidence.
Keep them from advancing, until we had the warships to begin pushing them back.
With the Confederacy cut in two parts by the Union's control of the Mississippi, due to the fall of Vicksburg, General Halleck had several actions he could take against the South. He believed that assaulting Arkansas and Louisiana would reap both military and political dividends. Halleck believed that if Confederate forces could be driven out of those two states, there was an excellent chance they would rejoin the Union. With these actions, Halleck believed that Texas would on its own accord also quit the war.
Superior reources, strategy and tactics plus staying power to outlast and overcome Carthage's initial strength.
Any siege is by its nature a strategy of Attrition. Grant's real stratagem was to cross the river below the city at an isolated point and encircle the city from the landward side, investing the city from an unexpected point; and even tough the initial assaults failed, the succeeding siege succeeded.
Follow that initial(!)- with a period.