6.2 Resonance in strings, pipes, and cavities

Fundamentals of Resonance

Resonance occurs when an external driving force matches an object's natural frequency of vibration, causing the amplitude of vibration to increase dramatically. This principle is central to how musical instruments produce sound, how concert halls are designed, and how engineers control unwanted noise.

Concept of Resonance in Acoustics

Every physical system has one or more natural frequencies determined by its properties: mass, stiffness, and geometry. When energy is supplied at one of these frequencies, the system absorbs that energy efficiently, and vibrations grow much larger than they would at other frequencies. Energy transfers back and forth between kinetic and potential forms, building up with each cycle.

This shows up everywhere in acoustics:

• Musical instruments rely on resonance to amplify sound. A guitar string vibrates at its natural frequency, and the body of the guitar resonates in response, projecting the sound outward.
• Acoustic space design depends on controlling which frequencies resonate in a room. Concert halls and recording studios are shaped and treated so that resonances enhance clarity rather than create muddiness.
• Noise control uses resonance principles in reverse. Mufflers and sound barriers are engineered to absorb or redirect energy at specific frequencies, reducing unwanted sound in buildings and vehicles.

Conditions for Resonance

For resonance to occur, two things must be true: the driving force must be at or near the system's natural frequency, and the system's damping (energy loss per cycle) must be low enough to allow vibrations to build up.

Beyond that, each type of resonator has its own physical requirements:

• Strings need fixed endpoints (or a combination of fixed and free) to reflect waves back and create standing wave patterns. The wave speed on the string depends on tension and linear mass density.
• Open pipes have both ends open to the atmosphere. Pressure nodes (points of minimum pressure variation) form at each open end.
• Closed pipes have one end closed and one end open. A pressure node forms at the open end, and a pressure antinode (point of maximum pressure variation) forms at the closed end.
• Cavities are enclosed volumes of air with rigid walls that reflect sound waves internally, trapping energy at specific frequencies determined by the cavity's dimensions.

Resonance Calculations and Analysis

Resonant Frequency Formulas

Each type of resonator has its own formula for predicting which frequencies will resonate. The key variable in all of them is the relationship between wave speed and the system's physical dimensions.

Strings:

The fundamental (lowest resonant) frequency of a string fixed at both ends is:

$f_1 = \frac{1}{2L}\sqrt{\frac{T}{\mu}}$

where $L$ is the string length, $T$ is the tension in newtons, and $\mu$ is the linear mass density (mass per unit length, in kg/m). All higher harmonics are integer multiples of the fundamental:

$f_n = n f_1 \quad (n = 1, 2, 3, \ldots)$

So a string with a fundamental of 110 Hz also resonates at 220 Hz, 330 Hz, 440 Hz, and so on. Strings support all harmonics.

Open pipes:

An open pipe resonates at:

$f_1 = \frac{v}{2L}$

where $v$ is the speed of sound in the medium inside the pipe. Like strings, open pipes support all harmonics:

$f_n = n f_1 \quad (n = 1, 2, 3, \ldots)$

Closed pipes:

A pipe closed at one end resonates at:

$f_1 = \frac{v}{4L}$

The closed end forces an antinode there, which means the pipe is effectively "half" of an open pipe for its fundamental. Closed pipes support only odd harmonics:

$f_n = (2n - 1) f_1 \quad (n = 1, 2, 3, \ldots)$

This gives frequencies at $f_1, 3f_1, 5f_1, \ldots$ and is why closed-pipe instruments (like a clarinet, which behaves approximately as a closed pipe) have a distinctly different timbre from open-pipe instruments like a flute.

Rectangular cavities:

A rigid-walled rectangular cavity with dimensions $L_x$, $L_y$, and $L_z$ has resonant frequencies given by:

$f_{lmn} = \frac{v}{2}\sqrt{\left(\frac{l}{L_x}\right)^2 + \left(\frac{m}{L_y}\right)^2 + \left(\frac{n}{L_z}\right)^2}$

Here $l$, $m$, and $n$ are mode numbers (non-negative integers, not all zero). Each combination of $(l, m, n)$ represents a different standing wave pattern inside the room. This is why small rooms tend to have problematic resonances at low frequencies: their dimensions correspond to wavelengths in the audible range.

Effects of System Changes on Resonance

Changing the physical properties of a resonator shifts its resonant frequencies in predictable ways. Understanding these relationships is how instrument builders and acousticians tune systems.

Strings:

• Increasing length $L$ lowers all resonant frequencies (this is why a bass guitar has longer strings than a standard guitar).
• Increasing tension $T$ raises frequencies (tuning a guitar string tighter raises its pitch).
• Increasing linear mass density $\mu$ lowers frequencies (thicker, heavier strings produce lower notes).

Pipes (open and closed):

• Increasing length $L$ lowers frequencies. A trombone extends its slide to lower pitch.
• Temperature affects the speed of sound ($v \approx 331 + 0.6T_C$ m/s in air, where $T_C$ is temperature in °C). Warmer air means higher $v$, which raises all resonant frequencies. This is why wind instruments go sharp in warm environments.

Rectangular cavities:

• Increasing any dimension lowers the resonant frequencies associated with that axis.
• Changing the medium inside (e.g., filling with a different gas) changes $v$, shifting all modal frequencies.

General principles across all systems:

• Adding mass to a vibrating system typically lowers its resonant frequencies (weighted piano keys vibrate more slowly).
• Increasing stiffness typically raises resonant frequencies (tightening a drum head raises its pitch).
• Altering boundary conditions can change the resonant modes entirely. Switching an organ pipe from closed to open changes which harmonics are present and shifts the fundamental frequency.