All Labs

Standing Waves Lab

Observe standing waves forming on strings and inside pipes. Adjust harmonic number, length, tension, and wave speed to see how nodes and antinodes shift. Compare the harmonic spectra of open and closed pipes side by side.

Guided Experiment: Harmonic Series Investigation

If you increase the harmonic number n on a fixed string, what do you predict will happen to frequency and the number of nodes?

Write your hypothesis in the Lab Report panel, then click Next.

Standing Wave— harmonic n=1

Node (N) — zero displacementAntinode (A) — max displacement

Controls

Length (L)1.00 m
Harmonic (n)n = 1
Tension (T)100 N
Linear Density (μ)0.0100 kg/m

Results

fn=n2LTμf_n = \dfrac{n}{2L}\sqrt{\dfrac{T}{\mu}}
Frequency
50.00 Hz
Wavelength
2.000 m
Wave Speed
100.00 m/s
Period
0.02000 s
Nodes
2
Antinodes
1
λ1=2.000 m(λn=2Ln)\lambda_1 = 2.000\text{ m} \quad \text{(}\lambda_n = \dfrac{2L}{n}\text{)}

Frequency vs Harmonic Number

Data Table

(0 rows)
#TrialHarmonic (n)Frequency(Hz)Wavelength(m)Wave Speed(m/s)NodesAntinodes
0 / 500
0 / 500
0 / 500

Reference Guide

Standing Waves

A standing wave forms when two identical waves travel in opposite directions and interfere. The result is a pattern that appears stationary, with fixed nodes (zero displacement) and antinodes (maximum displacement).

y(x,t)=Asin⁡(kx)cos⁡(ωt)y(x,t) = A\sin(kx)\cos(\omega t)

Nodes occur where sin(kx) = 0. Antinodes where |sin(kx)| = 1. The wave does not travel — energy oscillates in place between kinetic and potential forms.

Harmonics

Each resonant frequency is called a harmonic. The fundamental (n=1) is the lowest. Higher harmonics have frequencies that are integer multiples of f₁.

fn=n⋅f1n=1,2,3,…f_n = n \cdot f_1 \qquad n = 1,2,3,\ldots

For a string: the nth harmonic has n+1 nodes and n antinodes. Doubling the harmonic number doubles the frequency and halves the wavelength.

Strings vs Pipes

A vibrating string (fixed at both ends) and an open pipe (open at both ends) both support all harmonics with the same formula.

fn=nv2Lf_n = \frac{nv}{2L}

A closed pipe (one closed end, one open end) supports only odd harmonics. The closed end must be a node; the open end an antinode.

fn=nv4L,n=1,3,5,…f_n = \frac{nv}{4L}, \quad n = 1,3,5,\ldots

Wave speed on a string: v=T/μv = \sqrt{T/\mu}, where T is tension and μ is linear density.

Resonance

Resonance occurs when a system is driven at one of its natural frequencies. Energy builds up and the amplitude grows large. Musical instruments exploit resonance to amplify specific harmonics.

The wavelength for each harmonic is related to length by boundary conditions. For a string or open pipe, the nth harmonic fits exactly n half-wavelengths in the length L.

L=n⋅λn2L = n \cdot \frac{\lambda_n}{2}

A closed pipe fits n quarter-wavelengths (odd n only), giving a fundamental one octave lower than an open pipe of the same length.