This is related to another question I just asked, but they are different enough I thought it deserved its own spot. Speaker elements seem to always be shaped like a cone with a portion of a sphere at the center. What is the physics reasoning behind this shape being so standard? Is it a precise shape that all speaker elements share (e.g. parabaloid + sphere), or just kind of a general shape that "works well"? Does it work just as well for any light solid, or is it actually tuned to the speed at which sound propogates through the speaker element?
I believe it's to do with the fact that the speaker's function is to propagate pressure waves through the medium (air).
So, it's mainly a mechanical concern: you want something to push air, and you do not wish to expend much energy. So it has to be light and rigid, which the cone manages to fulfil due to its shape. A plane sheet, for instance would undergo buckling. 
The size of the cone depends on the wavelength you wish the speaker to transmit. Deep cone for low frequencies, shallow ones for higher frequencies, and curvilinear to cover a range .
As for the spherical portion, it's just a dust cap, and I have never come across literature claiming a 'significant' effect due to its shape.
Note: alls of this applies to dynamic speakers. Others, such as electrostatic speakers use different principles.
Paper cones were originally chosen for their rigidity and lightness, so they can move air quickly without deforming and couple to a motor easily at the center while also being easy to suspend from the basket by their perimeter with a simple corrugation or foam/rubber surround. Physics only played a major part in the ease of construction and performance was basically terrible.
Over time, materials technology removed the need for paper and other shapes became practical.
'Dynamic' speaker paper cones are not the only form of diaphragm used in speakers. Plenty of speakers use flat diaphragms, especially when needing to have a shallow profile to fit in a small enclosure. Some of these are actually square around the perimeter, and there are also speakers with 'cones' that have a square perimeter, pyramid-style.
Originally, speakers were constructed of treated paper because it is light and can move air efficiently. They needed a contact point around the perimeter to keep them from wobbling and another contact point at the center for the voice coil (motor) to drive them. Only turning them into a cone made them rigid enough to move air with fidelity to the electrical signal.
Other common features are pressed concentric ribs in the paper to increase rigidity and intermediate 'suspension' rings to decouple bass in the outer portion from treble in the inner portion, or 'whizzer' cones that do the same by putting a separate, steeper cone in the center, with no outer suspension, and relying on the inherent deformations of the larger cone to decouple bass from treble instead. All of these techniques are primitive by today's standards.
The spherical cap on the center of the cone is a dust cover to prevent magnetic metals and other debris from getting sucked into the magnetic gap between the central pole piece of the motor assembly and the voice coil former. Its shape was originally immaterial but a spherical cap was easy to glue on and looked good. It might help with high frequency reproduction. Dome tweeters do away with the cone and just use the cap, being both driven and suspended from the outer circumference alone.
Some retro 'audiophile' speakers use a single 6" Fostex driver and do away with the dust cover. They replace it with a nosecone-shaped waveguide on the end of the pole piece instead, but they are speciality speakers, mostly DIY modification, and susceptible to contamination.
The reasoning behind this modification is that the dust cap causes air pressure buildup with movement of the voice coil in the motor assembly, reducing sensitivity and causing distortion. Removing the cap allows the pressure to escape from the gap between the pole piece of the motor and the voice coil former. It is doubtful that this modification changes anything in the perceived sound and the improvement is almost certainly placebo. These 'full range' speakers have limited frequency response and power handling anyway.
Large speakers usually have a screen-covered vent on the back of the motor to allow trapped air pressure and dissipated heat to escape from the motor into the speaker cabinet whereas smaller speakers might have only a small or no vent, relying on seepage through the treated fabric 'spider' corrugated suspension from the basket that holds the voice coil in the center of the gap.
Speaker design is a non-trivial topic. Hidden details include the 'shorting ring' around the voice coil former that improves performance using eddy currents in some way that I forgot how it works. If you really want to find out more you need to go to web sites that discuss driver design. AVS forum is a good place to check for references. Try Wikipedia too, and Google. There is tons of information out there but not on science sites. All this development takes place in the audiophile technology marketplace and most of the latest developments are proprietary and patented, as well as basically useless IMO with only a few notable exceptions such as Thiele-Small ported cabinet designs.