Standing waves on a violin string problem with solution

Problem statement:

The G string on a violin has a length L = 33 cm and a fundamental frequency ν0 = 196 Hz. Give an expression for the position at which the fingers must press the string to play the different notes.

Find the position at which fingers 1, 2, 3 and 4 must be placed to play the notes A (220 Hz), B (247 Hz), C (262 Hz) y D (294 Hz).

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Solution:

When a string of length L is fixed at both ends, standing waves will only be produced at specific frequencies of two interfering harmonic waves (of the same amplitude, frequency and wavelength) traveling in opposite directions.

The relationship between these frequencies and the length of the string is:

where n is a positive integer and v is the speed of the waves propagating along the string.

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When the string of a violin is plucked we can produce standing waves of different frequencies depending on the position where the fingers press the string; that is so because pressing firmly on the string, effectively changes its length Ln (see figure below).

We will call xn the distance from the end of the rope to the position where the finger press the string.

The G string of a violin without any fingers pressed down (also known as open string) produces a sound of fundamental frequency ν0. Using the previous relationship between the frequency and the length of the string, and substituting n = 1 we can find the speed of the waves traveling along the string:

When the string is plucked, the different lengths Ln are:

And the distance xn will therefore be the total length of the string L minus Ln:

And simplifying we get:

In the following table are summarized the different values of xn corresponding to the different notes:

Finger number
Note
υn (Hz)
xn (cm)
0 (Open string)
G
196
1
A
220
3.6
2
B
247
6.8
3
C
262
8.3
4
D
294
11
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