Speakers convert electrical signals into soundwaves (linear motion). The larger the signal, the larger the deflection of the speaker cone, which translates to more sound.
Efficiency vs size: a large speaker potentially has a higher efficiency.
Frequency range: larger speakers usually hit lower notes (frequencies). High frequencies couple better with air, resulting in a higher efficiency and requiring smaller cone deflections, and therefore a smaller speaker. This has to do with the viscosity of air.
Xmax: lower frequencies need larger cone deflections, since more air needs to be displaced to produce low frequency sounds. From 150 Hz downwards, this effect will play a role.
Low frequencies couple badly with air, therefore a large cone deflection is necessary.
Impedance: very important to know is that a 4 ohm speaker behaves differently than a 16 Ohm speaker. If the impedance of the speaker is low, the current that flows is large, possibly damaging driver transistors or amplifiers. Always check if the circuit you use to drive the speaker can deliver the desired amount of current to the speaker. High impedance speakers are less prone to damage the connected circuit, but need a larger drive voltage.
Everywhere you need to produce sound, you can use a speaker.
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A piezo can also be used to produce sound, although they are only suitable from 250 Hz upwards.