Consider a string element of equilibrium length

d x

. Its mass is

ρ d x

, where

ρ

is mass per unit length. Under the small-slope approximation, the tension is approximately uniform and its transverse component at an endpoint is

T \partial y / \partial x

. The two endpoint forces have net transverse component

T [\frac{\partial y}{\partial x} (x + d x, t) - \frac{\partial y}{\partial x} (x, t)] ≃ T \frac{\partial ^{2} y}{\partial x ^{2}} d x .

ρ d x y_{tt} = T d x y_{xx}

. Cancelling the element length yields

\frac{\partial ^{2} y}{\partial t ^{2}} - c^{2} \frac{\partial ^{2} y}{\partial x ^{2}} = 0, c = \frac{T}{ρ} .

This is the one-dimensional wave equation for transverse displacement. Its derivation neglects gravity, bending stiffness and nonlinear changes in tension and segment length; the given small slope makes those omissions consistent to first order in the displacement.