Calculate Gain of Modified Band-Pass Filter

The 6.8uf electrolytic capacitor is not a suitable choice for an accurate band pass filter like this, note that the tolerance (spec sheet) is 20%. Most electrolytic caps are also polarized parts (needing a DC voltage) and are not normally a good choice for use in a series connected filter like this (even though you show a polarized voltage source as the audio driver). A much better choice for accuracy would be to use a film type capacitor (polycarbonate, polyester, polyethylene, etc). These can be found with better tolerances and are not polarized.

The wire wound resistors are not the best choice either, even though they can be tight tolerance wire wound resistors have some inductance (some types try to limit this) this would throw off your calculations a bit. A carbon film resistor with a tight tolerance is a better choice. Also using a 3w power rating for the resistor is very much over-kill for an earphone circuit. A 1/4w 1% resistor is likely fine here.

Also note that the accuracy of your stop band center is affected by the tolerance build up of all the components. Your design requirement is asking for a stop band in the range 5800-6000hz, this is only a 1.7% range at the center value of 5900hz. Note that your inductor is already at a 5% tolerance, the other component tolerances could further increase the offset error to the center frequency .

The first formula you have derived is for this circuit: -

^{simulate this circuit – Schematic created using CircuitLab}

H(s) = \$\dfrac{s\dfrac{R}{L}}{s^2 + s\dfrac{R}{L} +\dfrac{1}{LC}}\$

You have then made a mistake in assuming this formula "turns into" a TF that has no zeta (damping ratio). You should have used the standard formula of a bandpass filter to recognize that the TF has terms for both \$\omega\$ and \$\zeta\$ (zeta): -

H(s) = \$\dfrac{s2\zeta\omega_c}{s^2 + s2\zeta\omega_c +\omega_c^2}\$

You cannot equate this circuit to one that has the two cascaded TF's mentioned in the 2nd part of your question.

For a start, to multiply two single order filters together, (like you have) doesn't take into account the interaction between inductor impedance and capacitor impedance - your 2nd method makes the assumption that there is a voltage buffer between the two first order filters and this just isn't present in the series RLC band pass filter.

Another major difference in the 2nd scenario is that it can only produce a critically damped circuit - the middle term in the denominator (\$2\omega_c s\$) has no zeta term at all.

The two scenarios are different.