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ringenesherre

Velocity of Propagation at Audio Frequencies

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Starting Point

A commonly stated "scientific" reason for alledged audible differences between (speaker) cables is the fact that electrical signals travel at different speeds for different frequencies.

Background
The characteristics of a cable are often simplified to its impedance (Z), its capicatance (C), and its inductance (L) where Z = sqrt(L/C). This is a simplification of the more complete formula Z = sqrt((R+j*2*pi*f*L)/(G+j*2*pi*f*C)) where R is the resistance and G the conductance.

For high frequencies, i.e., for very large f, R and G are much smaller than 2*pi*f*L and 2*pi*f*C, respectively. Thus, the formula simplifies to Z= sqrt((j*2*pi*f*L)/(j*2*pi*f*C)) = sqrt(L/C).

Unfortunately, audio frequencies are not very large by any measure (which is why skin effects and similar high frequency effects do not effectively matter for audio cables).

Thus, in the range of 20 Hz to 20 KHz, the signal propagates at greatly varying speeds from 5 million m/s to aprox. 125 million m/s. That's a factor of 25 in speed!

Luckily, audio cables for home use are very short. Let us consider a typical run of 10 metres. A signal of 20 Hz needs 10 m / (5*10^6 m/s) = 2*10^-6 s for this distance, i.e., 2 microseconds. A signal of 20 KHz needs 10 m / (1.25*10^8 m/s) = 0.08*10^-6 s for this distance, i.e., 0.08 microseconds. That is, the time difference for these two extreme audible frequences is 2-0.08 = 1.92 microseconds.

A further factor is the dielectric constant of the insulation, which determines slowdown of the propagation (the so-called velocity factor). For typical insulations used, the velocity factor is around 0.66. Thus, the difference will be boosted to 2.86 microseconds.

Conclusion
Even for extreme audio frequencies (20 Hz vs 20.000 Hz), the "time smearing" is below the audible range (>= 5-10 microseconds). Any reasonably constructed (speaker) cable will not exhibit audible problems at runs up to 35 feet.

[b]References:[b]
  • Wikipedia: Transmission Line, available from http://en.wikipedia.org/wiki/Transmission_line

  • Jim Brown: Transmission Lines at Audio Frequencies, and a Bit of History, available from http://audiosystemsgroup.com/TransLines-LowFreq.pdf
  • Wikipedia: Speakerwire, available from http://en.wikipedia.org/wiki/Speaker_wire
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Comments

  1. esldude's Avatar
    Excellent blog. Well done.
  2. mitchco's Avatar
    Agreed, excellent blog. I love the math/science! I would be interested to get your thoughts on an article Nelson Pass wrote several years ago: http://passdiy.com/pdf/spkrcabl.pdf
  3. The Computer Audiophile's Avatar
    Very cool ringenesharre. One can't argue with the numbers but the conclusions drawn based on the numbers are debatable of course.