D'Alembert and String Tension Equations

Dear Fellow Builders,

Before I begin, please note that due to email font limitations, in the
following equations, the unique symbol ( _| ) represents the
Square Root Symbol.

A very popular equation for the mode frequencies of flexible
strings

[[ Eq. i(a) ]]: F(n) = ( n / ( 2*L )) * _| ( T / Mul )

states: The frequency of a given mode F(n), equals
the quantity of the mode number ( n ), any integer, divided by 2
times the string Length, multiplied by the quantity of the square
root of the string Tension divided by the string Mass per unit
length.

Since Mul = Pi * r^2 * rho

we also have:

[[ Eq. i(b) ]]: F(n) = ( n / ( 2*L )) * _| ( T / (Pi * r^2 * rho))

We may solve Equation i(b) for the following three variables:

Acoustically correct: T = 4 * F^2 * L^2 * Pi * r^2 * rho
Acoustically correct: L = (1 / ( F * D )) * _| ( T / ( Pi * rho ))
Acoustically correct: D = (1 / ( F * L )) * _| ( T / ( Pi * rho ))

where ( D ) is the diameter of the string.

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Only these three solutions --
which ubiquitously exclude the mode number squared ( n^2 ) --
are correct.

A given string has only one value for T; a given string has only
one value for L; and a given string has only one value for D.
In other words, for the upper modes of vibration, T does not and
cannot change, L does not and cannot change, and D also does
not and cannot change.

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In direct contrast, refer to Equation i(b), and note that although the
following solution for T is algebraically correct

Acoustically Incorrect: T = (4 * F^2 * L^2 * Pi * r^2 * rho) / n^2

it is _acoustically_ incorrect and, therefore, does not constitute a
solution, or solutions, for the variable T of a flexible string.

To understand an acoustically correct solution, consider the
following alternative for the mode frequencies of flexible strings:

[[ Eq. ii ]]: F(n) = c(t) / [ 2*L / (n) ]

where c(t) is the speed of transverse waves in a string, in
meters/second.

The following quote is from my manuscript
_Musical Mathematics: A Practice in the Mathematics of Tuning
Instruments and Analyzing Scales_, Chapter 10:

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The solution to this problem came from Jean le Rond D'Alembert
(1717-1783), who solved a second order partial differential
equation, known as the _wave equation_, which enabled him to
formulate an equation for the constant speed of transverse
traveling waves in flexible strings. (See Equation 3.11.)
D'Alembert also created a purely mathematical model to show that
standing waves in strings are caused by the superposition of
traveling waves that propagate in opposite directions.
(See Chapter 3, Figure 6.) This principle applies to all acoustic
sound-producing systems.

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D'Alembert's famous solution of the _wave equation_

c(t) = _| ( T / Mul)

states: The speed of transverse waves in a string equals the
quantity of the square root of the string Tension divided by the
string Mass per unit length.

The following quote is from -- University Physics, 7th ed. Addison-
Wesley Publishing Company, Reading, Massachusetts, 1988. --
one of the finest physics textbooks used throughout the US:

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p. 501: "The wave speed ( c ) is the same for all frequencies."

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Since c(t) is the same for all frequencies, the tension ( T ) must be
constant for all frequencies, and, therefore, does not and cannot
change; and the mass per unit length ( Mul ) must also be
constant for all frequencies, and, therefore, does not and cannot
change.

In Equation ii, the quantity [ 2*L / (n) ] describes how -- due to the
superposition of transverse traveling waves propagating in
opposite directions -- the wavelengths of the upper modes of
vibrations _decrease_ in length. Therefore, ( n ) is a variable that
exclusive modifies 2*L. It categorically never modifies: T, L, or D.

For the first mode of vibration, wavelength (lambda) = (2*L) / 1;
for the second mode of vibration, lambda = (2*L) / 2;
for the third mode of vibration, lambda = (2*L) / 3; etc.

Given the strict acoustic context of [ 2*L / (n) ],
only the mode number variable ( n )
accounts for increases in the mode frequencies of strings.

Given the string properties in my Tuning Group Message # 58811,
solutions for Equation ii are as follows:

For F(1): 392.0 Hz = _| (843.8 N / .00574 kg/m) / ((2 * .489 m) / 1);
for F(2): 784.0 Hz = _| (843.8 N / .00574 kg/m) / ((2 * .489 m) / 2);
etc.

Therefore, in my Tuning Group Message # 58798,
only the following four equations for Tension are correct:

((1)) T = F^2 * L^2 * D^2 * Pi * rho

((2)) T = 4 * F^2 * L^2 * Pi * r^2 * rho

((3)) T = 4 * F^2 * L^2 * M/u.l.

((4)) T = 4 * F^2 * L * M

These four equations exclude the mode number squared ( n^2 )
because all flexible strings have only one tension, or, equivalently,
have only one transverse wave speed for all modes of vibration.

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With respect to the speed of longitudinal waves in strings, the
following equation

[[ Eq. iii ]]: c(l) = _| (E / rho)

states: The speed of a longitudinal wave equals the
quantity of the square root of the stringing material modulus of
elasticity ( E ) divided by the stringing material mass density ( rho).
Here there exists only one longitudinal wave speed for all strings
made from the same material.

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Sincerely,

Cris Forster, Music Director
www.Chrysalis-Foundation.org

http://www.Chrysalis-Foundation.org/musical_mathematics.htm