The Vibration of Strings theory

The dynamics of a stretched flexible string represent the prototypical 1D wave system in continuum mechanics. Consider an infinitesimal string segment of unstretched length dx and linear mass density μ under mechanical tension T. When displaced transversely by a small deflection y(x, t), the net vertical restoring force acting on the element arises from the difference in the vertical components of tension at its ends: dF_y = T sin(θ + dθ) − T sin(θ) ≈ T (∂²y/∂x²) dx, under the small-angle approximation where tan(θ) ≈ ∂y/∂x. Applying Newton's second law to the segment mass dm = μ dx yields: μ dx (∂²y/∂t²) = T (∂²y/∂x²) dx. Dividing by μ dx produces d'Alembert's classic 1D wave equation: ∂²y/∂t² = (T / μ) (∂²y/∂x²). Comparing this with the standard wave equation ∂²y/∂t² = v² (∂²y/∂x²) identifies the transverse wave propagation speed: v = √(T / μ). This fundamental relation demonstrates that the speed of transverse mechanical waves depends exclusively on the restoring tension T (the elastic stiffness factor) and the linear density μ (the inertial factor). For a steel wire (diameter d = 0.50 mm, μ = 1.541 g/m) under a tension of T = 29.42 N (load mass M = 3.0 kg), transverse waves propagate at v = √(29.42 / 0.001541) = 138.16 m/s.

In 1636, French polymath Marin Mersenne published 'Harmonie Universelle', establishing the empirical relationships that govern vibrating strings. By subjecting strings of various lengths, tensions, and materials to rigorous pitch comparison, Mersenne deduced three fundamental proportionalities that form the cornerstone of musical acoustics: (1) First Law (Length): At constant tension and linear density, fundamental frequency is inversely proportional to vibrating length: f ∝ 1/L. Halving the length raises the pitch by an exact octave (ratio 2:1). (2) Second Law (Tension): At constant length and linear density, fundamental frequency is directly proportional to the square root of tension: f ∝ √T. To raise the pitch by an octave, the tension must be increased by a factor of four. (3) Third Law (Linear Density): At constant length and tension, fundamental frequency is inversely proportional to the square root of linear mass density: f ∝ 1/√μ. For strings of identical material (density ρ), linear density scales with cross-sectional area: μ = ρ (π d² / 4), so that frequency is inversely proportional to wire diameter: f ∝ 1/d. Combining all three laws yields the unified formula f = (1 / 2L) √(T / μ), which Brook Taylor formally derived from dynamic principles in 1713.

When a stretched string is terminated by rigid bridges at x = 0 and x = L, boundary conditions enforce zero displacement at both extremities: y(0, t) = 0 and y(L, t) = 0. Expressing the transverse wave motion as a Fourier superposition of standing normal modes, the spatial shape function must satisfy Y(x) = A sin(k x). Enforcing Y(L) = A sin(k L) = 0 restricts the allowable wavenumbers to discrete eigenvalues: k_n L = n π for integer n = 1, 2, 3, ... Substituting k_n = 2π / λ_n yields the wavelength quantization condition: L = n (λ_n / 2) ⟹ λ_n = 2L / n. The corresponding harmonic frequencies are f_n = v / λ_n = n [v / (2L)] = n f₁, with n = 1 (fundamental), n = 2 (second harmonic / first overtone, octave), n = 3 (third harmonic, twelfth), n = 4 (fourth harmonic, double octave), etc. Because a string fixed at both ends satisfies symmetric boundary conditions identical to an open organ pipe, it generates a complete harmonic spectrum containing all integer multiples (1f, 2f, 3f, 4f, ...). Plucking or bowing the string at different fractional positions excites specific subsets of these normal modes, shaping the harmonic spectrum and timbre.

Direct plucking of a thin string produces remarkably little sound because the small wire surface area has extremely poor acoustic impedance matching with the unconfined air (Z_air = 415 Pa s/m), radiating virtually no acoustic power. In a sonometer, the string rests upon hardwood bridges mounted directly on a hollow spruce soundboard. When the string vibrates, its transverse dynamic forces drive the bridges, which flex the large wooden soundboard. Because the soundboard has a large surface area, it radiates sound into the air with hundreds of times greater acoustic efficiency. Conversely, when an external tuning fork is struck and its stem is pressed against the soundboard, the process reverses: vibrational acoustic energy transmits through the soundboard into the bridges and excites transverse waves in the wire. Under non-resonant conditions, this forced excitation produces microscopic wire amplitudes. However, when the natural fundamental frequency of the wire segment matches the driving frequency of the fork (f_string = f_fork), resonant constructive interference builds up standing wave energy cycle after cycle, amplifying the midpoint displacement until a lightweight paper rider straddling the antinode is agitated violently and thrown clear of the wire. This sympathetic resonance provides an objective, visual indicator of resonance requiring zero electrical instrumentation.

When two acoustic harmonic sources of slightly different frequencies f₁ and f₂ vibrate simultaneously, their superposition produces the phenomenon of acoustic beats. Let the acoustic pressure waves arriving at the listener's ear be p₁(t) = A cos(2π f₁ t) and p₂(t) = A cos(2π f₂ t). Applying the trigonometric sum-to-product identity yields: p_tot(t) = 2 A cos[2π (f₁ − f₂) / 2 · t] cos[2π (f₁ + f₂) / 2 · t]. This represents a high-frequency carrier wave vibrating at the average frequency f_avg = (f₁ + f₂) / 2 whose amplitude is slowly modulated by an envelope cos[2π (f_beat / 2) t]. Because acoustic loudness (intensity) is proportional to the square of pressure amplitude, loudness peaks occur twice per envelope cycle, yielding a beat frequency equal to the absolute difference: f_beat = |f₁ − f₂|. When tuning a sonometer wire to a tuning fork, the experimenter listens for these rapid pulsations in volume. As the bridge is adjusted toward perfect unison, the beat frequency decreases: 8 beats/s → 4 beats/s → 2 beats/s → 1 beat every several seconds. When the beats vanish entirely, the wire is tuned to exact mathematical unison with the fork.

To determine the transverse wave speed and linear mass density without relying on a single fallible measurement, experimentalists record the fundamental resonant segment length L across a calibrated set of tuning forks of known frequencies f. From the fundamental standing wave relation L = λ₁ / 2 = v / (2f), rearranging yields: L = (v / 2) (1 / f). Designating x = 1 / f (in seconds) as the independent variable and y = L (in metres) as the dependent variable, the data fit the canonical linear regression equation y = m x + b. Here, the slope m represents half of the transverse wave velocity: m = v / 2, which allows the wave speed to be determined directly as v = 2m. Crucially, ordinary least-squares regression averages out random bridge setting errors across all frequency stations. Once v is obtained, the linear mass density μ of the wire is calculated from the known mechanical tension T = M g via μ = T / v² = M g / (4m²). Furthermore, the vertical intercept b evaluates whether any zero-offset or bridge knife-edge indentation error is present (theoretically b = 0). Comparing experimental linear density μ_exp with the gravimetric value μ = m_wire / L_wire or micrometer value μ = ρ (π d² / 4) provides rigorous confirmation of continuum mechanics.

Mersenne's laws govern the design and scaling of all stringed musical instruments, most notably the concert grand piano. A standard piano spans over seven octaves from A0 (27.5 Hz) to C8 (4186 Hz)—a frequency ratio of more than 150:1. If piano designers relied solely on Mersenne's first law (f ∝ 1/L) with uniform steel wire, an 88-key piano whose treble strings are 5 cm long would require bass strings exceeding 7.5 metres (25 feet!) in length. To maintain a practical instrument length (~2 to 3 metres), piano makers combine all three of Mersenne's laws: bass strings are made with thicker steel cores and wrapped with dense copper wire winding. The copper winding increases the linear mass density μ by a factor of over 100 without increasing the bending stiffness of the steel core, lowering the fundamental pitch into the deep bass register. However, real piano strings possess finite bending stiffness (elastic modulus E), which adds a small restoring moment to tension. This bending stiffness makes the string behave slightly like a rigid bar, shifting overtones slightly sharp according to Fletcher's inharmonicity formula: f_n ≈ n f₁ √(1 + B n²), where B = (π³ E d⁴) / (64 T L²) is the inharmonicity coefficient. Piano tuners must 'stretch' piano octave tuning to compensate for this physical inharmonicity.