On Thu, Mar 18, 2010 at 10:36 PM, Edward A. Berry <[log in to unmask]> wrote:
> I have been politely reminded offline that by definition amplitudes
> cannot be negative. We could call them coefficients, but:
Hi Edward
This obviously depends on whether you're talking about the physical
entity 'amplitude' or the quantity you are calling an 'amplitude'
purely for mathematical convenience. As you say, if what is meant by
the former is the usual meaning 'peak amplitude' then by definition it
cannot be negative, but of course there's no problem for the practical
purposes of doing the calculations if the peak amplitude is a negative
number: the mathematics of amplitudes and phases works equally well if
negative numbers are substituted for amplitudes (multiplying the
amplitude by -1 obviously produces the same result as adding Pi to the
phase). Look at the MTZ file produced by the SIGMAA program & you'll
see negative amplitudes (it's been that way for ~ 25 years and no-one
has ever objected!).
The amplitude is defined as the maximum excursion of the field
variable in either the positive *or negative* direction. We are used
to dealing with sinusoidal waves which are symmetric about the time or
distance axis, so we don't need to consider the wave troughs
independently of the wave peaks, but clearly the trough amplitude is a
negative number equal to minus the peak amplitude for symmetric waves,
and not equal to minus the peak amplitude for asymmetric waves
At least I'm assuming from what you say that the objection was purely
a semantic one, and not that the calculation would produce an
incorrect result if you used a negative number as the amplitude.
Unbelievably, European mathematicians, for the most part, resisted the
concept of negative numbers until the 17th century (according to
Wikipedia!), and in AD 1759, Francis Maseres, an English
mathematician, wrote that negative numbers "darken the very whole
doctrines of the equations and make dark of the things which are in
their nature excessively obvious and simple". In the 18th century it
was common practice to ignore any negative results derived from
equations, on the assumption that they were meaningless.
The objection to the use of the word 'amplitude' (implying 'peak
amplitude') is therefore really not that it's negative, but that
you're using the wrong word to describe your quantity, and maybe
'trough amplitude' would have been more accurate. But there are
plenty of examples where apparently nonsensical values are used purely
for mathematical convenience, take negative intensities or negative B
factors as cases in point. Since an intensity is a physical entity it
cannot by definition be negative. Nevertheless everyone understands
what is meant by 'negative intensity'. Are we instead obliged to use
the term 'negative squared amplitude' henceforth?
Yours somewhat tongue in cheek,
-- Ian
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