The Example schematic only has 100uF + 0.1uF + 1000pF across the rails.
Have you considered the "mF" may be a typo? In one place, the datasheet says typically 0.1 mF to 1 uF. I wonder if they meant to type n, and accidentally hit m. Also, I copied the u symbol out of the PDF, and it got printed as m when it was pasted. Cut&Paste may be at fault here, it certainly seems to be used within TI's various datasheets.
Also, millifarads have come to be an almost unused unit. It's generally Farads -> microFarads -> (nanoFarads - somewhat uncommon) -> picoFarads.
Furthermore, looking at the TPA3111 Evaluation Kit is informative:
The device is bypassed with two 100uF electrolytics (along with 0.1uF and 1000pF ceramics).
Also, looking at similar parts from the same line is informative. The TPA3110 (15W vs TPA3111's 10W) merely says a larger aluminum electrolytic capacitor of 220 uF or greater placed near the audio power amplifier is recommended. It's worth noting that the same datasheet's example schematics only use two 100uF caps for bypassing.
The same note as in the TPA3111 is present in the TPA3112 datasheet.
It's also worth noting that the TPA3110 and TPA3113 have identical "Power Supply Decoupling" paragraphs, despite the fact that one is half the power of the other (15W vs 6W), which further inclines me to think typo.
The 25W TPA3123 only recommends 470uF of bulk capacitance.
The 100W TAS5121 only recommends 1000uF.
Edit: We will see if it is a typo: "Below is what you submitted to tis-doc-errors@list.ti.com on Tuesday, July 26, 2011 at 04:13:23; E-mail: tis-doc-errors@list.ti.com Lit Number: SLOS618BB Part Number: TPA3111D1 Error Page No: 19 Error Description: Please see this thread: HUGE capacitor recommended in datasheet for Audio Amp"
Your fundamental problem is likely that the impedance (primarily resistance) of this homemade "speaker" is far too low for the limited current drive capability of the Arduino pin. Your substitution of copper tape for the typically somewhat resistive conductive thread likely makes this even worse (at least in terms of presenting an overload to the driver).
As others have mentioned, using a relay as an audio amplifier isn't really workable. Actually a relay is fairly close to being an electromechanical buzzer; drive it with a signal of appropriate frequency and you will likely get more sound output from the relay itself than from this speaker of yours. Under the right conditions you might even get intelligible voice when using a relay as a speaker, but it will not be pleasant.
It may be possible to get sound out of your homemade speaker in one of two fairly simple ways:
you could try to use an audio output transformer to change the comparatively high voltage (nearly 5v) but low current arduino output, into a lower voltage, high current output to drive your low impedance speaker. I forget if radio shack still carries these, but you can salvage one out of an old battery powered radio. The higher turns count, typically 3-terminal side would go to the arduino (ignore the center terminal) while the lower turns count side would go to the speaker.
Since your Arduino output is digital all-or-nothing, you don't actually need a linear audio amplifier. Instead, you can make a switching circuit with an npn transistor switching the negative supply to the coil, and a resistor between the transistor's base and the arduino output pin. Not surprisingly, this is the exact same circuit you would typically build to drive a relay with an arduino output, as that is also a coil. Given your likely very low coil resistance, you will likely want to add a series resistor to limit the current to what the supply and a cheap plastic case transistor can handle. Such a circuit will likely have sufficient bandwidth to attempt pulse width modulation if you wish to try to generate quasi-analog output from your arduino software. Since you are digital all the way to the speaker, you will need the PWM frequency to be above the audible range if you do not want to hear the switching.
While it is true the bipolar drive might be more beneficial for this type of crude speaker than for traditional ones, the complexity of building that (while avoiding the shoot through issue) is probably not something you want to tackle in a first attempt. Mounting the speaker substrate tensioned in something like a small embroidery hoop (or gluing it to a carrier ring) to increase its restoring force could be a worthwhile experiment.
Best Answer
Capacitors don't have power ratings because, ideally, they don't dissipate any power. They store energy unlike resistors which consume energy, giving it off as heat.
Instead, you need to consider the following: