It is a curve well-known in the world of mechanics. Later, Leibniz, Johann Bernoulli and others studied the curve. The mouth size of a horn determines the lowest frequency at which there still is any significant reinforcement of the sound. This together defines the absolute available low-end cutoff and it is obvious from what is said, that a small horn cannot go very low. The throat of the horn can be as big as the transducer cone or smaller.
Maximum efficiency is achieved when they are the same size, but the bandwidth is decreased.
To increase the bandwidth, mouth must be smaller. The improved efficiency of the horn is gained at the loss of bandwidth. So, it may be louder but it will drop off in frequency at the upper end sooner. Front horns are primarly used for high, mid or higher bass freq. They can be used as bass enforcement especially at PA loudspeakers in multi-way configurations, where one unit covers only limited frequency range.