Standing Wave String Simulator (2D)

String

Length L 1.00 m

Tension T 40 N

Linear density μ 2.0 g/m

Harmonic

Harmonic number n 1

Fundamental (n=1): 1 antinode, 2 nodes.

Live Readout

Wave speed v

0 m/s

Frequency f_n

0 Hz

Wavelength λ_n

0 m

Nodes / Antinodes

2 / 1

How it works

A string of length L fixed at both ends supports transverse waves that travel at speed v = √(T/μ), where T is the tension and μ is the linear mass density. Only shapes that satisfy the fixed-end boundary condition at both x = 0 and x = L survive as standing waves: y(x,t) = A sin(nπx/L) cos(ωt), for harmonic number n = 1, 2, 3…

Each harmonic n has n+1 nodes (motionless points, evenly spaced at x = kL/n, including both fixed ends) and n antinodes (points of maximum swing, at x = (k+½)L/n). The corresponding wavelength is λ_n = 2L/n and frequency f_n = n·v/(2L) = v/λ_n.

Raising tension or lowering linear density increases v and therefore every f_n; shortening the string raises every f_n as well, since f_n ∝ 1/L for a fixed n — exactly the same relations used in the guitar-string case, generalized to any harmonic.

Node — string motionless Antinode — maximum swing