Resonance in strings and pipes Knowledge-Based Study

Study notes

1. Resonance occurs when a vibrating system is driven at one of its natural frequencies, resulting in a dramatic increase in oscillation amplitude. In mechanical media such as stretched strings and air columns, resonance manifests as standing waves.

   Formation of Standing Waves

   A standing wave is produced by the superposition of two identical waves travelling in opposite directions with equal frequency and amplitude. Instead of propagating through the medium, the resulting disturbance creates a stationary profile defined by two landmark features:

   • Nodes: Locations in the medium that experience zero oscillation amplitude due to continuous destructive interference.
   • Antinodes: Locations that oscillate with maximum amplitude due to constructive interference.

   The distance between two consecutive nodes, or between two consecutive antinodes, is equal to half a wavelength: $\frac{\lambda}{2}$. The distance between any node and the nearest antinode is equal to one quarter of a wavelength: $\frac{\lambda}{4}$.

   A uniform string clamped securely at both ends must have a displacement node at each boundary because the physical supports prevent motion.

   Harmonic Series for Strings

   For a string of length $L$, standing waves can only be sustained if an integer number of half-wavelength loops fit between the fixed ends:

   $L = n \frac{\lambda_n}{2}$, where $n = 1, 2, 3, \dots$

   Solving for the allowed resonant wavelengths gives:

   $\lambda_n = \frac{2L}{n}$

   Using the wave speed relationship $v = f \lambda$, the resonant frequencies are:

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

   • Fundamental Frequency (First Harmonic, $n = 1$): $f_1 = \frac{v}{2L}$. The string vibrates in a single loop with two boundary nodes and one central antinode.
   • Second Harmonic (First Overtone, $n = 2$): $f_2 = 2 f_1$. The string forms two loops separated by a central node.
   • Third Harmonic (Second Overtone, $n = 3$): $f_3 = 3 f_1$. The string vibrates in three equal loops.

   The wave propagation speed along the string depends on the tension $T$ and linear mass density $\mu$ through $v = \sqrt{\frac{T}{\mu}}$. Increasing tension raises the wave speed and shifts all resonant frequencies higher.

   In air-filled columns, sound travels as longitudinal pressure and displacement waves. The acoustic boundary conditions depend directly on the pipe ends:

   • Open End: Air particles move freely across the boundary, creating a displacement antinode (a pressure node where acoustic pressure equals atmospheric pressure).
   • Closed End: The rigid barrier prevents longitudinal motion, forcing a displacement node (a pressure antinode).

   Open-Open Pipes (Open at Both Ends)

   An open-open pipe has displacement antinodes at both ends. Its resonant condition is identical in form to that of a clamped string:

   $L = n \frac{\lambda_n}{2}$ and $f_n = \frac{n v}{2L} = n f_1$, where $n = 1, 2, 3, \dots$

   Here $v$ is the speed of sound in air. An open-open tube supports all integer harmonics.

   Open-Closed Pipes (Closed at One End)

   An open-closed pipe has a displacement node at the closed end and a displacement antinode at the open end. The length $L$ must span an odd number of quarter-wavelengths:

   $L = m \frac{\lambda_m}{4}$, where $m = 1, 3, 5, \dots$

   The corresponding resonant frequencies are:

   $f_m = \frac{m v}{4L} = m f_1$, where $m = 1, 3, 5, \dots$

   An open-closed pipe supports only odd harmonics (1st, 3rd, 5th, etc.). Even harmonics cannot form because they violate the required boundary conditions.

   Contrasting strings and different pipe geometries highlights essential physical principles governing musical instruments:

   • Pipe Geometry and Pitch: For two pipes of identical length $L$, the fundamental frequency of an open-closed pipe is half that of an open-open pipe ($f_{1,\text{closed}} = \frac{v}{4L}$ compared to $f_{1,\text{open}} = \frac{v}{2L}$). Closing one end drops the fundamental pitch by one full octave.
   • Harmonic Content and Timbre: Strings and open-open pipes generate full harmonic spectra (all integer multiples of $f_1$), creating bright tonal qualities. Open-closed columns lack even harmonics, producing a characteristic hollow timbre.
   • Temperature Effects: The speed of sound in air increases with thermodynamic temperature according to $v \propto \sqrt{T}$. As an instrument warms during play, the sound speed increases, causing its resonant frequencies to sharpen slightly.