vibrating string

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Consider a thin string of linear density \rho\, and tension T\, stretched horizontally between two fixed points, and suppose that it is plucked in such a way that it oscillates only in the vertical direction. Let the height of the string be denoted by y(x)\,. At any point on the string, we can apply Newton's second law to an element of length ds\,:

\frac{d^2 y}{dt^2} \rho ds = F_y\,,
\frac{d^2 x}{dt^2} \rho ds = F_x\,.

The force F_y\, is due to the tension in the string, and equal to

F_y = T \sin\theta_2 - T \sin\theta_1\,,

while

F_x = T \cos\theta_2 - T \cos\theta_1\,,

where \theta_1\, and \theta_2\, are the angles the string makes with the horizontal, to the left and to the right of the element ds\,, respectively.

Small angle approxmation

For small oscillations, these angles are small, and we may use a small angle approximation:

F_y = T (\tan\theta_2 - \tan\theta_1)\,,

while

F_x =0\,.

Then

\tan\theta_2 = \frac{dy\left(x+\frac{1}{2}ds\right)}{dx}\,,

and

\tan\theta_1 = \frac{dy\left(x-\frac{1}{2}ds\right)}{dx}\,,

Therefore,

\frac{d^2 y}{dt^2} = \frac{T}{\rho} \frac{1}{ds}\left[ \frac{dy\left(x+\frac{1}{2}ds\right)}{dx} - \frac{dy\left(x-\frac{1}{2}ds\right)}{dx} \right]\,.

Taking the limit as ds \to 0\,, with ds \approx dx\,, we get

\frac{d^2 y}{dt^2} = \frac{T}{\rho} \frac{d^2 y}{dx^2}\,.

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