Transverse wave equation on a string

A transverse wave is a wave in which the particles of the medium vibrate perpendicular to the direction of propagation of the wave, e.g. waves on a stretched string.

Wave equation for a stretched string

Take a string of linear mass density μ (mass per unit length) under tension T, lying along the x-axis. Displacements y are small, so T is nearly constant.

          T          y
   θ1  /----\ θ2     |
 ------A--dx--B----> x

1. Consider a small element AB of length dx. Tension T pulls at each end along the tangent, making small angles θ1 at A and θ2 at B with the x-axis.
2. Horizontal components cancel (cosθ ≈ 1). The net vertical force:

F = T sinθ2 − T sinθ1 ≈ T (tanθ2 − tanθ1)

3. tanθ = ∂y/∂x, so tanθ2 − tanθ1 = (∂²y/∂x²) dx. Hence F = T (∂²y/∂x²) dx.
4. Mass of element = μ dx; its acceleration is ∂²y/∂t². Newton's second law:

μ dx ∂²y/∂t² = T (∂²y/∂x²) dx

∂²y/∂x² = (μ/T) ∂²y/∂t²

This is the differential equation of the wave on a string.

Velocity

5. Compare with the standard wave equation ∂²y/∂x² = (1/v²) ∂²y/∂t²:

1/v² = μ/T   →   v = √(T/μ)

The speed of the transverse wave depends only on the tension and the mass per unit length of the string: it increases with T and decreases with μ.