Vibrations in Strings

Fundamental Mode (First Harmonic)

Fundamental Mode – when a string is plucked, it will vibrate in one segment with two nodes at either end.

standing waves

This is the lowest possible mode of vibration

In a guitar string, for example, it will vibrate between the fret and the tuning key. The bridge transfers the vibration to the “box” (or sound box) through the saddle.

At the fundamental frequency (f0) we have 1 loop and 2 nodes.

Note: # of nodes = # of loops +1

Also: L = ½ λ (L – length of the string, λ – wavelength)

Overtones – when strings vibrate in more than one segment.

Harmonies – frequencies which are multiples of the fundamental.

i.e.  2f0, 3f0 …

The fundamental frequency (f0) is known as the First Harmonic.

Modes of Vibrations in Strings

harmonics4)Fourth Harmonic –

harmonic

 

 

 

 

 

5 nodes, 4 loops

In general L = n/2 λ

where n is the # of loops in the vibrating string.

Example:

What is the fundamental frequency of a violin string (45 cm long) if the speed of the wave in the string is 280 m/s?

Solution:

Given:                    Required:

L = 0.45 m               f2 = ?

V = 480 Hz

L = n/2 λ     λ = 2 L/n = 2 x 0.45m/1 loop = 0.9 m

v = fλ   f = v/λ = 280m/s / 0.9m = 311Hz

∴ The frequency is 311 Hz

Modes of vibrating strings follow the same pattern and mathematic rules as STANDING WAVES.

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