Wave Speed & Power on a String

Tighten a guitar string and its note rises; the thick bass strings sound low. Both come from how fast waves travel on the string, which depends on just two things.

What should the speed depend on?#

When part of the string is disturbed, its elasticity makes it pull on the neighbouring part (measured by the tension F), and the neighbour responds according to its inertia (measured by the mass per unit length μ). Stronger pull, faster response; heavier string, slower response.

Deriving v = √(F/μ)#

Ride alongside the wave at its speed v. In your frame the wave shape stands still and the string slides backward through it at speed v, like a snake through a fixed curve. Look at a short element of length Δl at the top of a crest. Any small curve is nearly a circular arc; let its radius be R. The element is moving round this arc at speed v.

The tension F pulls on each end of the element along the string, each tilted down by a small angle θ from the horizontal, so the net downward force is $2F\sin\theta \approx 2F\theta = 2F\dfrac{\Delta l/2}{R} = \dfrac{F\,\Delta l}{R}$. The element’s mass is μΔl, and it needs a centripetal acceleration v²/R:

Power carried by a sine wave#

At position x the string to the left pulls on the string to the right with the tension F, along the string. Its vertical component is $F_y = -F\sin\theta \approx -F\,\partial y/\partial x$, and it pushes the string to the right up and down at speed $\partial y/\partial t$. The power delivered across x is therefore

The cos² term averages to ½ over a cycle. Using F = μv² and ω = 2πν: