The speed of a wave traveling along a stretched string — a guitar string, a piano wire, a rope, or a power line — depends on just two physical properties: how tightly it's stretched and how much mass it packs per unit length. This calculator applies the classical wave-on-a-string formula, v = √(T/µ), to find the wave speed directly, or to work backward and solve for the tension or linear mass density needed to hit a target speed.
How the wave speed formula works
A wave on a string is a transverse disturbance that propagates because tension in the string pulls each displaced segment back toward equilibrium, while the string's own mass resists that acceleration (inertia). The balance between the restoring force (tension) and the resisting mass (linear density) sets how fast the disturbance travels: v = √(T/µ).
Because the relationship is a square root, wave speed does not scale linearly with tension. Quadrupling the tension only doubles the wave speed; increasing tension by 44% is what's needed to double the speed on a string with a given µ (since 1.44 ≈ √2 squared). This formula assumes an idealized string — perfectly flexible, uniform, and under uniform tension along its length — which is a good approximation for thin strings and wires but breaks down for stiff rods or cables where bending resistance matters.
Inputs and what they mean
Tension (T) is the pulling force in newtons stretching the string taut. In real instruments and structures, tension is often set by a tuning mechanism (a guitar's tuning peg, a cable's anchor tensioner) and can range from a few newtons for a loose string to hundreds of newtons for a taut instrument string or thousands for structural cables.
Linear mass density (µ) is the mass per unit length in kilograms per meter. Thin, light strings (like a high guitar E string) have a small µ; thick, wound bass strings or heavy cables have a much larger µ. If you don't know µ directly, the From Mass & Length tab derives it by dividing the string's total mass by its length — the input that has the largest effect on the result is usually whichever of tension or µ you're least certain about, since both enter the formula, but µ enters under a square root in the denominator so halving µ has the same effect as doubling tension.
Limits and edge cases
The formula assumes the string is perfectly flexible (no bending stiffness), the tension is uniform along its entire length, and the wave amplitude is small enough that the string's geometry doesn't change meaningfully as it vibrates. It does not account for damping (energy loss to air resistance or internal friction), which causes real waves to lose amplitude over distance and time even though this formula's speed value stays accurate. It also doesn't apply to stiff rods, thick cables where bending resistance is significant, or waves traveling through 2D/3D media like membranes or solids — those require different wave equations. Tension must be non-negative and linear mass density must be strictly positive for the formula to produce a physically meaningful, finite speed.