👂Acoustics Unit 6 Review

Every physical object has certain frequencies at which it "wants" to vibrate. These are its natural frequencies, and the specific patterns of motion that occur at each one are called modes of vibration. Together, they explain everything from how a guitar string produces a note to why a poorly designed bridge can shake itself apart.

Natural Frequencies and Vibration Modes

A natural frequency is a frequency at which a system tends to oscillate when it's disturbed and then left alone. Which frequencies qualify depends on the object's physical properties (mass, stiffness, shape) and its boundary conditions (how the ends or edges are held in place). A guitar string fixed at both ends, for example, has a different set of natural frequencies than the same string with one end free.

Each natural frequency comes with a vibration mode, which is the spatial pattern of motion the system takes on at that frequency. On a vibrating string, modes look like standing waves with different numbers of loops. On a drum head, they form two-dimensional patterns (some circular, some radial). The first mode is the simplest pattern; higher modes get progressively more complex.

Resonance occurs when you drive a system at one of its natural frequencies. The amplitude of vibration builds up dramatically because energy is being added in sync with the system's preferred motion. Classic examples include a singer shattering a wine glass by matching its natural frequency, or the Tacoma Narrows Bridge collapse in 1940, where wind excited a structural resonance.

Natural frequencies and vibration modes, Normal Modes of a Standing Sound Wave – University Physics Volume 1

Fundamental Frequency vs. Harmonics

The fundamental frequency (also called the first harmonic, $f_1$ ) is the lowest natural frequency of a system. It corresponds to the simplest mode shape and determines the perceived pitch of a vibrating object.

Harmonics are the natural frequencies above the fundamental. For ideal systems like strings and open pipes, harmonics are integer multiples of the fundamental:

$f_n = n \cdot f_1$

where $n = 1, 2, 3, \ldots$ and $f_n$ is the frequency of the $n$ th harmonic.

So if a string's fundamental is 100 Hz, its second harmonic is 200 Hz, its third is 300 Hz, and so on. These higher harmonics are what give an instrument its timbre (tonal color). A flute and a violin can play the same pitch, but they sound different because they produce different relative strengths of harmonics.

One important distinction: closed pipes (closed at one end, open at the other) only support odd harmonics ( $n = 1, 3, 5, \ldots$ ). This is why a clarinet, which behaves roughly like a closed pipe, has a distinctly different tone quality from a flute, which behaves like an open pipe supporting all harmonics.

Natural frequencies and vibration modes, Standing Waves in Strings

Mode Shapes in Acoustic Systems

Strings vibrate as sinusoidal standing waves. Each mode has nodes (points of zero displacement) and antinodes (points of maximum displacement). The fundamental has nodes only at the two fixed ends and one antinode in the middle. The second harmonic adds a node at the center, creating two half-loops. The $n$ th mode has $n$ antinodes and $n + 1$ nodes (including the endpoints).

Pipes work with pressure standing waves rather than displacement waves:

Open pipes (open at both ends): Pressure nodes form at each open end, with pressure antinodes inside. All harmonics are present. A flute is a good approximation.
Closed pipes (closed at one end): A pressure antinode forms at the closed end and a pressure node at the open end. Only odd harmonics are supported. A clarinet approximates this behavior.

Cavities (rooms, car interiors, instrument bodies) support three-dimensional standing wave patterns. The mode shapes depend on the cavity's geometry. A rectangular room, for instance, has modes along its length, width, and height, plus combinations of all three. These room modes are why certain bass frequencies can sound boomy in specific spots while nearly disappearing in others.

Factors Influencing Vibration Modes

Boundary conditions set the rules for where nodes and antinodes can exist:

Strings: A fixed end forces a node; a free end creates an antinode.
Pipes: An open end acts as a pressure node; a closed end acts as a pressure antinode.
Cavities: Rigid walls reflect sound efficiently and enforce strong standing waves; flexible walls absorb energy and weaken resonances.

Material properties determine how fast waves travel through the system:

Density affects the inertia of the vibrating medium. A heavier string vibrates more slowly, producing a lower pitch. Steel strings are denser than nylon, which is one reason they sound different even at the same tension and length.
Elasticity (stiffness) provides the restoring force. A stiffer material snaps back faster, raising the natural frequencies.
Tension in a string directly affects frequency. Tightening a guitar string raises its pitch because the wave speed increases with tension.

Geometry shapes which modes are possible and at what frequencies:

Length is inversely proportional to frequency. Doubling a string's length halves its fundamental frequency (drops the pitch by one octave).
Cross-sectional area changes the mass per unit length, which also affects frequency.
Cavity shape determines the spatial pattern of modes. A rectangular room has a different set of resonances than a cylindrical or spherical one.

Temperature shifts natural frequencies by changing wave speed:

In air, the speed of sound increases with temperature (roughly 0.6 m/s per °C). Wind instruments go sharp in warm environments and flat in cold ones.
In solids, thermal expansion can change dimensions and material stiffness, subtly altering tuning. This is why orchestras tune up after the instruments have warmed to performance temperature.

👂Acoustics Unit 6 Review