Concept

String vibration

Summary
A vibration in a string is a wave. Resonance causes a vibrating string to produce a sound with constant frequency, i.e. constant pitch. If the length or tension of the string is correctly adjusted, the sound produced is a musical tone. Vibrating strings are the basis of string instruments such as guitars, cellos, and pianos. The velocity of propagation of a wave in a string () is proportional to the square root of the force of tension of the string () and inversely proportional to the square root of the linear density () of the string: This relationship was discovered by Vincenzo Galilei in the late 1500s. Source: Let be the length of a piece of string, its mass, and its linear density. If angles and are small, then the horizontal components of tension on either side can both be approximated by a constant , for which the net horizontal force is zero. Accordingly, using the small angle approximation, the horizontal tensions acting on both sides of the string segment are given by From Newton's second law for the vertical component, the mass (which is the product of its linear density and length) of this piece times its acceleration, , will be equal to the net force on the piece: Dividing this expression by and substituting the first and second equations obtains (we can choose either the first or the second equation for , so we conveniently choose each one with the matching angle and ) According to the small-angle approximation, the tangents of the angles at the ends of the string piece are equal to the slopes at the ends, with an additional minus sign due to the definition of and . Using this fact and rearranging provides In the limit that approaches zero, the left hand side is the definition of the second derivative of : This is the wave equation for , and the coefficient of the second time derivative term is equal to ; thus Where is the speed of propagation of the wave in the string (see the article on the wave equation for more about this).
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