Crossovers and dynamics

Does a 2nd or 4th order crossover really rob a speaker of dynamics, or just lower the efficiency somewhat?
They have no negative effect on dynamics or efficiency.

They do allow higher output levels than first order cross-overs without unacceptable distortion or higher cross-over points which in turn lead to the non-monotonic directivity that makes speakers sound less like live music.

Physics dictate driver displacement requirements quadruple to maintain the same SPL one octave lower. With a first order cross-over the 6dB per octave drop in SPL means that excursion continues to double for a given program SPL with each lower octave until the roll-off steepens as it must (loudspeaker drivers have an inherent second order high-pass response).
OK I will rephrase the question. I have heard that multi-ordered crossovers will reduce the dynamic capabilities of a speaker vs. having no crossover at all (full range). Is this true, or does the crossover just reduce the drivers efficiency a bit so all you need is a little more power?
>OK I will rephrase the question. I have heard that multi-ordered crossovers will reduce the dynamic capabilities of a speaker vs. having no crossover at all (full range). Is this true

No. A good sounding full range driver must be small so that its polar response and timbre are not compromised (with an acoustically large driver the resulting polar response causes different reflections to vary in spectra so you don't hear what was recorded). Excursion tends to be very limited too perhaps due to motor design which keeps inductance low and preserves high-frequency response. Total displacement, peak SPL, and dynamics are degraded compared to a multi-way speaker which didn't need to compromise design to provide extended frequency response.

A back-loaded horn can get you some bass extension using the limited displacement but only takes care of the lowest frequencies.

All else equal efficiency is proportional to cone area, so the smaller driver required for full-range use with acceptable polar response may be less efficient. Things get more complicated - physics also dictate that efficiency is proportional to box size and inversely proportional to the cube of the low frequency cut-off. If you make a speaker that extends to 40Hz instead of 80Hz it needs to have 8X the volume which is twice as big in every direction to maintain the same efficiency, and at identical size would be 9dB less efficient. With the prevalance of solid state amplifiers and dome tweeters most manufacturers opt for smaller spouse-friendly sizes with less efficiency, although if you can live with a "stand mounted monitor" which only extends to 80Hz that's 30 x 15 x 12" you can have 95dB/1W/1 meter efficiency.

>or does the crossover just reduce the drivers efficiency a bit so all you need is a little more power?

No. With even order cross-overs a better analogy is that the power is just redirected among the drivers but the physics are more interesting in the cross-over region. With passive cross-over networks the power lost is offset by efficiency gains due to having multiple sources.

Consider a text-book even-order filter operating into flat time-aligned drivers with 8 Ohm purely resistive impedances and infinite bandwidth. At the cross-over point each output is attenuated 6dB. Phase lead in the high frequency leg and lag in the low frequencies mean the two drivers are 180 degrees out of phase for every two orders, with polarity inversion on orders 2, 6, 10, etc. so they don't cancel.

Both high and low-pass filters have an 8 Ohm impedance at the cross-over point which means each leg has a 16 Ohm impedance and the two in parallel are the 8 Ohms of a single driver so amplifier power is the same as if one driver was driven.

Half the power goes into each leg.
Half the power in each leg (1/4 the total) goes into the driver, and half into the driver.
Although only half the total power is reachin the drivers at the cross-over point, the two sources together have a 3dB (double) efficiency gain so efficiency is the same as with one driver.

With an active cross-over the system would be 3dB more efficient in the cross-over region.

Odd-order cross-overs are a little different. The two drivers are 90 degrees out of phase for each order and are -3dB at the cross-over point so they add back to unity.

The filters have a 3.3 Ohm impedance at the cross-over point so the impedance with both legs in parallel is about 5.7 Ohms and it takes about 19% more power which is lost in the reactive components.

With an active system efficiency would be the same as with one driver in the cross-over region.

Since nearly all audio amplifiers are constant voltage sources this is is only of academic interest.
As a foot note inductors have some resistance associated with them but the losses there aren't significant enough to be interesting.

Inductor resistance is likely to be around .3 Ohms. That will drop the output of an ideal 8 Ohm driver wired in series with it by 0.3dB which is not enough to be perceived as a loudness difference.

Power delivered will drop 4% due to the reduced current flow for that 7% drop in acoustic power so you could say there was a 3% decrease in efficiency.

Counter-intuitively using a full range free-standing design with no cross-over doesn't necessarily get you away from them. As wave lengths grow long (at 100Hz you have 11 foot waves) they wrap around any speaker you can fit in your living room as if it was not there. SPL ultimately drops by 6dB with half the power gone by the time wave lengths are 3X baffle width or about 400Hz for a domestically friendly foot-wide speaker although you get some of that back in the form of reflections. That can be compensated for with a shelving low-pass filter that knocks ~3dB off the high frequencies to compensate with a corresponding efficiency loss when done at the speaker level.

You can avoid that in a back-loaded horn where most of your output below the baffle step is coming from the horn which can operate into the half space created by the floor.
Thank you, that was very detailed and thorough. Now when my friends tell me that my 4 way 4th order floor standing speakers are being robbed of musical dynamics I can point them to your post. You have surely "busted" that audio myth.