The Neper

When discussing the attenuation of sound in Intermediate Physics for Medicine and Biology, Russ Hobbie and I write

In acoustics, the attenuation is usually expressed in decibels per meter.

At the bottom of the page is this footnote:

Sometimes the attenuation coefficient is expressed in nepers m⁻¹, in which case the natural logarithm of the intensity or pressure ratio is used.

The neper? What’s that?

First, let’s review the decibel. If the pressure amplitude of two sound waves are p₁ and p₂, their relative pressure can be written as

            20 log₁₀(p₂/p₁).

When expressed in this way, the pressure difference is said to be in decibels (dB). If p₂ is ten times p₁, then the pressure difference is 20 log₁₀(10) or 20 dB.

Often we express sound in terms of intensity rather than pressure. The intensity is proportional to the pressure squared, so the relative intensity difference of I₁ and I₂ is

            10 log₁₀(I₂/I₁).

The leading factor of 20 in the expression containing the pressures is replaced by 10 in the expression containing intensities. If you don’t like that factor of ten out front, you could not use it, in which case your intensity difference is expressed in the rarely used unit of bels rather than decibels.

Notice that the decibel is defined using of a logarithm with base ten, also known as the common logarithm. Alternatively, we could use the natural logarithm, with base e = 2.718..., which leads to the neper. However—and this is the confusing part—instead of having the leading factor of 20 in the expression for decibels in terms of pressure, the expression for nepers has no leading factor at all. A factor of 2 is removed because the neper is defined in terms of the pressure and not intensity, and a factor of 10 is removed because nepers are like bels and not decibels. So,

            ln(p₂/p₁)

is the pressure difference in nepers. If you insist on using intensity rather than pressure, you must use the ugly-looking expression

            ½ ln(I₂/I₁) .

A ten-fold difference in intensity is 1.15 nepers (Np), so 10 dB is the same as 1.15 Np, or 1 Np = 8.7 dB. If a sound wave attenuates at a rate of 1 neper per meter that means for every meter traveled the pressure falls by a factor of e and the intensity falls by a factor of 7.4. In tissue, attenuation is usually proportional to frequency, so as a rule of thumb the attenuation is about 100 dB per meter per megahertz or roughly 12 neper per meter per megahertz.

Asimov's Biographical Encyclopedia
of Science & Technology,
by Isaac Asimov.

Where does the strange name “neper” come from? It honors the inventor logarithms, John Napier. Here is a excerpt about Napier from Asimov's Biographical Encyclopedia of Science & Technology.

NAPIER, John (nay’pee-ur) 

Scottish mathematician

Born: Merchiston Castle, near Edinburgh, 1550 

Died: Merchiston Castle, near Edinburgh, April 4, 1617

... Napier’s solid reputation rests upon a new method of calculation that first occurred to him in 1594… It occurred to Napier that all numbers could be expressed in exponential form. That is, 4 can be written as 2², while 8 can be written as 2³, and 5, 6, and 7 can be written as 2 to some fractional power between 2 and 3. Once numbers were written in such exponential form, multiplication could be carried out by adding exponents, and division by subtracting exponents. Multiplication and division would at once become no more complicated than addition and subtraction.

Napier spent twenty years working out rather complicated formulas for obtaining exponential expressions for various numbers. He was particularly interested in the exponential forms of the trigonometric functions, for these were used in astronomical calculations and it was these which Napier wanted to simplify. His process of computing the exponential expressions led him to call them logarithms (“proportionate numbers”) and that is the word still used.

Finally, in 1614, Napier published his tables of logarithms, which were not improved on for a century, and they were seized on with avidity. Their impact on the science of the day was something like that of computers on the science of our own time. Logarithms then, like the computers now, simplified routine calculations to an amazing extent and relieved working scientists of a large part of the noncreative mental drudgery to which they were subjected.