Cardioid bass coupled with quarter wave line?

I want to create a acoustical system with both the cardioid radiation of a resistance enclosure, coupled with a quarter wave tube tuned to the driver Fs. Which one of the two models, 1 and 2, correspond better to the design intention?
 

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Cardioid tells something about dispersion, but does not tell you how it is obtained. A quarter wave line is an enclosure style. These are unrelated things.

Usually cardioid dispersion is obtained with two sources, separated in space, which are driven such that they cancel each other when measured (far) behind the speaker. They must cancel at all frequencies within the passband. Therefore their frequency responses must match. If one of the sources contains a resonator (like a 1/4 wave line), the other one must have one as well.
 
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