6.1 Formation of standing waves

Wave Superposition and Standing Waves

When two identical waves travel in opposite directions through the same medium, they combine to produce a standing wave: a wave pattern that oscillates in place rather than propagating forward. This typically happens when a wave reflects off a boundary and interferes with the incoming wave. Because the wave doesn't travel, there's no net energy transfer along the medium.

Standing waves are central to how stringed and wind instruments produce musical tones. The fixed patterns they create determine which frequencies a vibrating string or air column can sustain.

Formation of standing waves

A standing wave forms through superposition of an incident wave and its reflection. For this to work, the two waves must share the same amplitude, frequency, and wavelength, but travel in opposite directions.

At certain points along the medium, the two waves always cancel (destructive interference). At other points, they always reinforce (constructive interference). The result is a pattern that appears to vibrate in place rather than move left or right.

A violin string is a clear example: the wave launched by the bow reflects off each fixed end, and the overlap of outgoing and returning waves produces the standing wave you see vibrating between the bridge and the nut.

Characteristics of standing waves

• Nodes are points of permanent zero displacement where destructive interference occurs. They're evenly spaced along the medium.
• Antinodes are points of maximum displacement where constructive interference occurs. Each antinode sits midway between two adjacent nodes.
• The distance between two consecutive nodes (or two consecutive antinodes) equals one half-wavelength.
• Amplitude is not uniform along a standing wave. It's greatest at the antinodes and zero at the nodes.
• All points between two adjacent nodes oscillate in phase with each other (they reach their peaks at the same time). Points on opposite sides of a node oscillate 180° out of phase.

Mathematical Analysis of Standing Waves

Equations for standing waves

For a string fixed at both ends, the boundary conditions require nodes at each endpoint. This constraint means only certain wavelengths can "fit" on the string. Specifically, the string length $L$ must equal a whole number of half-wavelengths.

The allowed wavelengths and frequencies are:

1. Wavelength for mode $n$: $\lambda_n = \frac{2L}{n}$ where $n = 1, 2, 3, \ldots$ is the mode number.

2. Frequency for mode $n$: $f_n = \frac{nv}{2L}$ where $v$ is the wave speed on the string.

3. Fundamental frequency ($n = 1$, the lowest possible mode): $f_1 = \frac{v}{2L}$

Every higher mode is an integer multiple of the fundamental: $f_n = n f_1$. Together these form the harmonic series ($f_1, 2f_1, 3f_1, \ldots$).

Relationships in standing waves

The wave speed on a string depends on two physical properties:

$v = \sqrt{\frac{T}{\mu}}$

where $T$ is the tension in the string and $\mu$ is its linear mass density (mass per unit length).

From this, several practical relationships follow:

• String length and frequency are inversely proportional. Doubling $L$ cuts the frequency of any given mode in half. This is why pressing a guitar string against a fret (shortening the vibrating length) raises the pitch.
• Frequency is proportional to the square root of tension: $f \propto \sqrt{T}$. Tightening a string raises its pitch because the wave travels faster.
• Frequency is inversely proportional to the square root of linear mass density: $f \propto \frac{1}{\sqrt{\mu}}$. Thicker, heavier strings vibrate at lower frequencies, which is why a guitar's low E string is much thicker than the high E string.
• Higher mode numbers increase frequency linearly: $f_n = n f_1$. The second harmonic is twice the fundamental, the third is three times, and so on.