Visualizing Standing Waves on a String

When a string is fixed at both ends and set into vibration, it cannot oscillate arbitrarily. Instead, the boundary conditions restrict the motion, allowing only specific patterns of vibration known as normal modes. Each normal mode corresponds to a standing wave, formed by the superposition of two waves traveling in opposite directions.

In this interactive simulation, you will see these standing waves come to life.

Fixed-End Boundary Conditions

Because the string is fixed at both ends:

  • The displacement at each end must always be zero
  • These fixed points are nodes
  • Any allowed wave pattern must satisfy this constraint

This requirement leads to discrete normal modes, rather than a continuous range of motion.

Mathematical Description of Normal Modes

For a string of length $L$ and wave speed $v$, the allowed wavelengths and frequencies are:

\[\begin{aligned} \lambda_n = \frac{2L}{n} \qquad\qquad\qquad f_n= \frac{n v}{2L} \end{aligned}\]

where $n = 1, 2, 3, \dots $ is the mode number. Notice that higher modes correspond to shorter wavelengths and higher frequencies.

How to Read the Simulation

  • 🔵 Blue points represent nodes
    • They remain fixed in position
    • The displacement is always zero
  • 🔴 Red points represent antinodes
    • They oscillate up and down with the maximum amplitude
    • Their positions along the string are fixed, but their displacement changes with time
  • The black curve represents the instantaneous shape of the string

Interactive Simulation: Explore the Normal Modes

Use the buttons below to switch between normal modes.

Guided Exploration

As you change the mode number, consider the following questions:

  1. How many antinodes appear in each mode?
    What is the relationship between the number of antinodes and the mode number $n$?
  2. What happens to the wavelength as the mode number increases?
    Does the string appear more or less “compressed”?
  3. Do the node positions ever change with time?
    Why must this be the case physically?
  4. How does this relate to musical instruments?
    Which mode determines the pitch you hear?

Why Normal Modes Matter

Normal modes appear throughout physics and engineering:

  • Vibrations of strings and membranes
  • Resonances in bridges and buildings
  • Electromagnetic modes in waveguides and cavities
  • Quantum mechanical bound states

Understanding this simple system builds intuition that applies far beyond strings.

Standing waves are not just mathematical solutions — they are the natural language of constrained motion.