Acoustic Resonance
Acoustic resonance occurs when an object or air column vibrates at its natural frequency in response to an external sound source at that same frequency, resulting in a dramatic amplification of the sound. The phenomenon arises when standing waves are set up within the resonating object, with nodes and antinodes at fixed positions. Acoustic resonance is exploited in all musical instruments — strings, pipes, and percussion — as well as in architectural acoustics, industrial machinery fault detection, and medical imaging.
Key Formula
f_n = n × v / (2L)
LaTeX: f_n = \frac{nv}{2L}
| Symbol | Meaning | Unit |
|---|---|---|
| f_n | Frequency of the nth harmonic | Hz |
| n | Harmonic number (1, 2, 3…) | dimensionless |
| v | Speed of sound in the medium | m/s |
| L | Length of the resonating air column or string | m |
Worked Example
Problem
An organ pipe that is open at both ends has a length of 0.85 m. If the speed of sound is 340 m/s, what are the frequencies of the first three harmonics?
Solution
Step 1: Use f_n = nv / (2L). Step 2: f₁ = 1 × 340 / (2 × 0.85) = 340 / 1.7 = 200 Hz. Step 3: f₂ = 2 × 340 / 1.7 = 400 Hz. Step 4: f₃ = 3 × 340 / 1.7 = 600 Hz.
Answer
First harmonic = 200 Hz, Second = 400 Hz, Third = 600 Hz
Resonance in Different Pipe Configurations (v = 340 m/s, L = 0.5 m)
| Pipe Type | Boundary Conditions | Fundamental Frequency | Harmonics Present | Example Instrument |
|---|---|---|---|---|
| Open-open | Antinode at both ends | 340 Hz | All (1st, 2nd, 3rd…) | Flute |
| Open-closed | Antinode + node | 170 Hz | Odd only (1st, 3rd, 5th…) | Clarinet |
| Closed-closed | Node at both ends | 340 Hz | All | Organ pipe (stopped) |
| String (fixed-fixed) | Node at both ends | Depends on tension | All | Guitar, violin |
Interactive Tools
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Related Terms
Sound Beats
Sound beats are periodic variations in amplitude — heard as a rhythmic pulsing or "wah-wah" sound — that occur when two sound waves of slightly different frequencies interfere. The beat frequency equals the absolute difference between the two source frequencies, and the sound alternately gets louder (constructive interference) and quieter (destructive interference). Musicians use beats to tune instruments: when no beats are heard, the two sources are in tune; as beats slow to zero, the frequencies converge.
Sound Intensity
Sound intensity is the power carried by a sound wave per unit area perpendicular to the direction of propagation, measured in watts per square metre (W/m²). It quantifies how much acoustic energy passes through a given surface each second and decreases with the square of the distance from a point source — the inverse square law. Sound intensity is the physical basis for the decibel scale and is central to audiology, architectural acoustics, and occupational noise exposure standards.
Constructive Interference
Constructive interference occurs when two or more waves overlap in such a way that their displacements add together, producing a resultant wave with greater amplitude than either individual wave. This phenomenon arises when the waves are in phase — that is, their crests and troughs align — leading to a net increase in energy at that point. It is fundamental to technologies such as noise-cancelling headphones (in reverse), optical coatings, and phased-array antennas.
From Latin "resonare" (to resound, echo), from "re-" (again) + "sonare" (to sound). The systematic study of acoustic resonance is attributed to Marin Mersenne (1636) and later formalised by Helmholtz in the 19th century.