Heres the math for that .4% difference in resistance: Accourding to a chart listing the resistance of 16 gauge copper wire as .004 ohms/ft: 5 ohm speakers + 10' x .004 = 5.04 ohms total. .004 ohms/ft. x 5' plus the 5 ohm speaker = 5.02 ohms. 5.04/5.02 = 100.4%.
For more fun with numbers if we use 28 volts to the speakers...98w @ 8 ohms (28v / 8 ohms = 3.5a x 28v), the difference in power is 156.2w for 10' and 155.6w for 5' cables. A 50' cable will still deliever 150.8w to the speaker. Even a 250' cable would deliever 130.7w to the speaker. A difference of less than 1 db...barely percievable.
Sorry for the long delay in responding, other things have come up. Actually, I figured you figured it that way, it was just your wording of "10' cable has .4% more resistance than the 5' cable" implies the cable resistance doesn't double with doubling the length. Had you said, "total load, as seen by the amp, is 0.4% different for 10' cable as compared to a 5' cable," there'd be no confusion.
But, let's look at the math that you were so kind to supply, per my request. It is true that 16 gauge
wire has 4milliohms per foot of resistance. But, a speaker
cable is TWO
wires in series. So, a five foot speaker
cable is actually 10 feet of
wire. Therefore, your calculations are 100% wrong. 10 feet is 80milliohms of total resistance, not 40. Five feet is 40milliohms, not 20. Thus, using your difference as seen by the amp, it is now 0.8% different.
But, let's take this thought experiment another step. You suggested a 5ohm speaker for your calculations, perhaps to exaggerate the effects of the cable, and still show that they are insignificant. Unfortunately, an overwhelming number of speakers do not provide a constant load across the audible spectrum. Let's take an old fave for an example, the Original Larger Advent, or OLA for those into nicknames. I believe it has a minimum of 5ohms, like your example, and it has a maximum (at resonance) of 32ohms. If we were to assume the amp driving these is an ideal voltage source, then we'd have diffenences of the signal being applied to the speaker terminals between the two cables at the two extremes of speaker load of:
5ohms/5ft = -0.0692dB
32ohms/5ft = -0.0109dB
5ohms/10ft = -0.1379dB
32ohms/10ft = -0.0217dB
From that we can conclude that the five foot speaker cable imparts 0.0583dB error to the frequency response, while the 10 foot cable does twice the damage with 0.1162dB peak-to-peak deviation. Now, it can certainly be argued that such small frequency response imbalances aren't audible, and I'd tend to agree. But, in my experience, when doing the final voicing on a speaker, I've been known to tweak the crossover to sub dB values (like 0.3dB), and these do indeed impart an audible difference - subtle, to be sure. It is not hard then, to come to the realization that these very small errors due to cables account for some of the diffences people hear between different speaker cables.
