Editing String vibration (section)

== Wave ==

The velocity of propagation of a wave in a string (<math>v</math>) is proportional to the [[square root]] of the force of tension of the string (<math>T</math>) and inversely proportional to the square root of the linear density (<math>\mu</math>) of the string:

<math>v = \sqrt{T \over \mu}.</math>

This relationship was discovered by [[Vincenzo Galilei]] in the late 1500s. {{Citation needed|date=November 2017}}
===Derivation===
[[Image:StringParameters.svg|right|Illustration for a vibrating string]]
Source:<ref>[http://www.animations.physics.unsw.edu.au/jw/wave_equation_speed.htm The wave equation and wave speed]</ref>

Let <math>\Delta x</math> be the [[length]] of a piece of string, <math>m</math> its [[mass]], and <math>\mu</math> its [[linear density]].  If angles <math>\alpha</math> and <math>\beta</math> are small, then the horizontal components of [[Tension (mechanics)|tension]] on either side can both be approximated by a constant <math>T</math>, for which the net horizontal force is zero. Accordingly, using the [[Small-angle approximation|small angle approximation]], the horizontal tensions acting on both sides of the string segment are given by

:<math>T_{1x}=T_1 \cos(\alpha) \approx T.</math>
:<math>T_{2x}=T_2 \cos(\beta)\approx T.</math>

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, <math>a</math>, will be equal to the net force on the piece:

:<math>\Sigma F_y=T_{1y}-T_{2y}=-T_2 \sin(\beta)+T_1 \sin(\alpha)=\Delta m a\approx\mu\Delta x \frac{\partial^2 y}{\partial t^2}.</math>

Dividing this expression by <math>T</math> and substituting the first and second equations obtains (we can choose either the first or the second equation for <math>T</math>, so we conveniently choose each one with the matching angle <math>\beta</math> and <math>\alpha</math>)

:<math>-\frac{T_2 \sin(\beta)}{T_2 \cos(\beta)}+\frac{T_1 \sin(\alpha)}{T_1 \cos(\alpha)}=-\tan(\beta)+\tan(\alpha)=\frac{\mu\Delta x}{T}\frac{\partial^2 y}{\partial t^2}.</math>

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 <math>\alpha</math> and <math>\beta</math>.  Using this fact and rearranging provides

:<math>\frac{1}{\Delta x}\left(\left.\frac{\partial y}{\partial x}\right|^{x+\Delta x}-\left.\frac{\partial y}{\partial x}\right|^x\right)=\frac{\mu}{T}\frac{\partial^2 y}{\partial t^2}.</math>

In the limit that <math>\Delta x</math> approaches zero, the left hand side is the definition of the second derivative of <math>y</math>, 

:<math>\frac{\partial^2 y}{\partial x^2}=\frac{\mu}{T}\frac{\partial^2 y}{\partial t^2}.</math>

this equation is known as the [[wave equation]], and the coefficient of the second time derivative term is equal to <math>\frac{1}{v^{2}}</math>; thus

:<math>v=\sqrt{T\over\mu},</math>

Where <math>v</math> is the [[speed]] of propagation of the wave in the string.  However, this derivation is only valid for small amplitude vibrations; for those of large amplitude, <math>\Delta x</math> is not a good approximation for the length of the string piece, the horizontal component of tension is not necessarily constant. The horizontal tensions are not well approximated by <math>T</math>.