Transverse Displacement of Stretched String: Derivation of Poisson Eq.

[UAR]

Please how would one derive the Poisson Equation model,

\nabla^{2}\psi(x) = \frac{F(x)}{T},

for Transverse displacement \psi(x) of a stretched string under constant non-zero tension T and an externally applied transverse force F(x) . Assuming small angle with the horizontal (i.e sin(\theta)\approx\theta) ?

Thanks

 

[gabbagabbahey]

Start by by drawing a picture and labeling all the forces acting on the string. Then apply Newton's second law.