How does the direction of sound propagation affect perceived loudness?

[Ryker]

Homework Statement

The speaker receives 10W of electrical power from the amplifier, 4% of which it transforms into sound flow, propagating into a half-space (in front of the speaker) indiscriminately in all directions. How loud is the sound 10m from the speaker?

Homework Equations

$$L_\mathrm{I}=10\, \log_{10}\left(\frac{I_1}{I_0}\right)\ \mathrm{dB} \,$$
$$I = \frac{W}{S}$$

The Attempt at a Solution

The solution is pretty easy to get if you just plug the data into the equations, taking

$$S = 4\pi r^{2}$$

But that's the thing I don't quite get. If all of the power that the speaker emits is transformed into sound that only propagates in a half-space, why isn't $$S = 2\pi r^{2}$$? I mean, isn't that the point of a speaker, so as to curtail the drawback of sound propagating in all directions and directing its power into only the desired one, thus making it appear louder to our ears?

 

[lippyka]

I'm guessing that the reason for the 4\pi r^{2} is that sound does still propagate in all directions and the 2\pi r^{2} only accounts for the directions in which one can actually hear it. But this is just a guess and I'm not sure if I'm right. Can someone please explain? Thanks!