EP02

EP02: String Vibration and the Wave Equation

PDE, Fourier级数

▶ — Physics/AcousticsHarmonic Analysis

后续拓展

EP09 Combination Tones and Nonlinear Acoustics EP15 Opera Acoustics and the Singer's Formant

小红书 Watch on 小红书

Overview

When you pluck a guitar string, why does it produce a rich spectrum of overtones rather than a single pure tone? The answer lies in a second-order partial differential equation derived by d’Alembert in 1747 and the separation-of-variables technique that Fourier refined in the early 19th century.

This episode derives the 1D wave equation from Newton’s second law, establishes d’Alembert’s general solution as a superposition of traveling waves, and then reframes everything as Fourier series — infinite sums of standing-wave modes whose frequencies form the integer harmonic series $f_1, 2f_1, 3f_1, \ldots$

中文: “从波动方程到傅里叶级数，我们看到了数学如何揭示音乐的物理本质。达朗贝尔的行波解告诉我们波如何传播，傅里叶的级数解告诉我们泛音如何叠加。”

Prerequisites

Multivariable calculus: partial derivatives, Taylor expansion
Ordinary differential equations: constant-coefficient second-order ODEs
Basic real analysis: convergence of series
Cochlear nonlinearity (EP09) — this wave equation is the linear model from which EP09 departs
Microphone physics (EP28) — the same ODE structure reappears as a damped oscillator

Definitions

Definition 2.1 (The Wave Equation)

Let $u(x,t)$ denote the transverse displacement of a string at position $x \in [0,L]$ and time $t \geq 0$ . The 1D wave equation is

\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}, \qquad c = \sqrt{\frac{T}{\mu}}

where $T > 0$ is the string tension (force) and $\mu > 0$ is the linear mass density (mass per unit length). The constant $c$ has units of velocity and is the propagation speed of waves on the string.

Definition 2.2 (Dirichlet Boundary Conditions)

For a string with fixed endpoints, the boundary conditions are

u(0,t) = 0 \quad \text{and} \quad u(L,t) = 0 \qquad \text{for all } t \geq 0

Combined with initial conditions $u(x,0) = \phi(x)$ (initial shape) and $u_t(x,0) = \psi(x)$ (initial velocity), this forms a well-posed initial-boundary value problem.

Definition 2.3 (Normal Modes)

The $n$ -th normal mode (or standing wave) is the particular solution

u_n(x,t) = \sin\!\left(\frac{n\pi x}{L}\right)\!\left(A_n \cos\omega_n t + B_n \sin\omega_n t\right), \qquad n = 1, 2, 3, \ldots

where

\omega_n = \frac{n\pi c}{L}, \qquad f_n = \frac{\omega_n}{2\pi} = \frac{nc}{2L}

The frequency $f_1 = c/(2L)$ is the fundamental frequency (first harmonic). The frequency $f_n = n f_1$ is the $n$ -th harmonic or $(n-1)$ -th overtone.

Definition 2.4 (Fourier Series Solution)

The general solution to the wave equation with Dirichlet boundary conditions is the Fourier series

u(x,t) = \sum_{n=1}^{\infty} \sin\!\left(\frac{n\pi x}{L}\right)\!\left(A_n \cos\omega_n t + B_n \sin\omega_n t\right)

The coefficients are determined by initial conditions via

A_n = \frac{2}{L}\int_0^L \phi(x)\sin\!\left(\frac{n\pi x}{L}\right) dx, \qquad B_n = \frac{2}{n\pi c}\int_0^L \psi(x)\sin\!\left(\frac{n\pi x}{L}\right) dx

Definition 2.5 (Modal Energy)

The total mechanical energy of the string is

E = \frac{1}{2}\int_0^L \left[\mu\, u_t^2 + T\, u_x^2\right] dx

The energy of the $n$ -th mode is

E_n = \frac{L}{4}\left[\mu\, \omega_n^2(A_n^2 + B_n^2) + T\!\left(\frac{n\pi}{L}\right)^2(A_n^2 + B_n^2)\right] = \frac{L\mu\omega_n^2}{4}(A_n^2 + B_n^2)

(using $T(n\pi/L)^2 = \mu\omega_n^2$ ).

Main Theorems

Theorem 2.1 (Wave Equation from Newton's Second Law)

Under the small-amplitude approximation (displacements much smaller than string length), Newton’s second law applied to an infinitesimal string element of length $dx$ yields the wave equation

u_{tt} = c^2\, u_{xx}, \qquad c = \sqrt{T/\mu}

Proof.

Consider the string element occupying $[x, x+dx]$ with transverse displacement $u(x,t)$ . The element has mass $dm = \mu\, dx$ .

The tension acts tangentially at both ends. At the right end, the string makes angle $\theta(x+dx,t) \approx u_x(x+dx,t)$ with the horizontal (small angle approximation: $\sin\theta \approx \tan\theta \approx \theta$ ). At the left end, angle is $\theta(x,t) \approx u_x(x,t)$ .

The net transverse force on the element is:

dF = T\sin\theta(x+dx,t) - T\sin\theta(x,t) \approx T\left[u_x(x+dx,t) - u_x(x,t)\right] = T\, u_{xx}(x,t)\, dx

Newton’s second law: $dF = dm \cdot u_{tt}$ , so

T\, u_{xx}\, dx = \mu\, dx\cdot u_{tt}

Dividing by $\mu\, dx$ :

u_{tt} = \frac{T}{\mu}\, u_{xx} = c^2\, u_{xx} \qquad \square

Visual: Draw a curved string element with tangent lines at both endpoints. The net upward force from tension is proportional to the curvature $u_{xx}$ — positive curvature (concave up) creates upward restoring force, driving oscillation.

Theorem 2.2 (D'Alembert's General Solution)

Every solution of $u_{tt} = c^2 u_{xx}$ on $\mathbb{R} \times \mathbb{R}$ can be written as

u(x,t) = f(x - ct) + g(x + ct)

for arbitrary twice-differentiable functions $f$ and $g$ . The term $f(x-ct)$ represents a rightward-traveling wave; $g(x+ct)$ represents a leftward-traveling wave. Given initial conditions $u(x,0) = \phi(x)$ and $u_t(x,0) = \psi(x)$ , the unique solution is

u(x,t) = \frac{\phi(x-ct) + \phi(x+ct)}{2} + \frac{1}{2c}\int_{x-ct}^{x+ct}\psi(s)\,ds

Proof.

Step 1 — Change of variables. Let $\xi = x - ct$ , $\eta = x + ct$ . By the chain rule:

u_{xx} = u_{\xi\xi} + 2u_{\xi\eta} + u_{\eta\eta}, \qquad u_{tt} = c^2(u_{\xi\xi} - 2u_{\xi\eta} + u_{\eta\eta})

The wave equation $u_{tt} = c^2 u_{xx}$ becomes

c^2(u_{\xi\xi} - 2u_{\xi\eta} + u_{\eta\eta}) = c^2(u_{\xi\xi} + 2u_{\xi\eta} + u_{\eta\eta})

Simplifying: $-4c^2 u_{\xi\eta} = 0$ , so $u_{\xi\eta} = 0$ .

Step 2 — Integration. Integrate $u_{\xi\eta} = 0$ first in $\eta$ : $u_\xi = h(\xi)$ for some function $h$ . Then integrate in $\xi$ : $u = F(\xi) + G(\eta)$ where $F' = h$ . Setting $f = F$ , $g = G$ gives $u(x,t) = f(x-ct) + g(x+ct)$ .

Step 3 — Initial conditions. From $u(x,0) = f(x) + g(x) = \phi(x)$ and $u_t(x,0) = -cf'(x) + cg'(x) = \psi(x)$ :

f'(x) - g'(x) = -\frac{\psi(x)}{c}

Integrating: $f(x) - g(x) = -\frac{1}{c}\int_0^x \psi(s)\,ds + K$ . Solving the system of two equations for $f$ and $g$ , then substituting back yields the stated formula. $\square$

Theorem 2.3 (Separation of Variables: Normal Modes)

The only solutions of $u_{tt} = c^2 u_{xx}$ of the separable form $u(x,t) = X(x)T(t)$ satisfying Dirichlet boundary conditions $X(0) = X(L) = 0$ are (up to scalar multiples) the normal modes:

X_n(x) = \sin\!\left(\frac{n\pi x}{L}\right), \quad T_n(t) = A_n\cos\omega_n t + B_n\sin\omega_n t, \quad \omega_n = \frac{n\pi c}{L}

for $n = 1, 2, 3, \ldots$

Proof.

Substituting $u = X(x)T(t)$ into $u_{tt} = c^2 u_{xx}$ :

X(x)T''(t) = c^2 X''(x)T(t)

Dividing both sides by $c^2 X(x)T(t)$ (assuming both nonzero):

\frac{T''(t)}{c^2 T(t)} = \frac{X''(x)}{X(x)} = -\lambda

where $-\lambda$ is a separation constant (it must be constant since the left side depends only on $t$ and the right only on $x$ ).

Spatial ODE: $X'' = -\lambda X$ with $X(0) = X(L) = 0$ .

If $\lambda \leq 0$ : the general solution is exponential or linear; neither can satisfy both boundary conditions with $X \not\equiv 0$ .
If $\lambda > 0$ : write $\lambda = (n\pi/L)^2$ . Then $X(x) = A\sin(\sqrt{\lambda}\,x) + B\cos(\sqrt{\lambda}\,x)$ . Condition $X(0) = 0$ gives $B = 0$ . Condition $X(L) = 0$ gives $\sin(\sqrt{\lambda}\,L) = 0$ , so $\sqrt{\lambda} = n\pi/L$ for $n = 1, 2, 3, \ldots$

Temporal ODE: $T'' = -\lambda c^2 T = -\omega_n^2 T$ (with $\omega_n = n\pi c/L$ ), which has general solution $T_n(t) = A_n\cos\omega_n t + B_n\sin\omega_n t$ . $\square$

Visual: The first five normal modes: mode 1 has one arch (no interior nodes), mode 2 has one interior node at $x=L/2$ , mode $n$ has $n-1$ interior nodes at $x = L/n, 2L/n, \ldots, (n-1)L/n$ . Each mode vibrates at frequency $f_n = n f_1$ .

Theorem 2.4 (Energy Conservation and Modal Orthogonality)

The total energy decomposes as a sum over modes with no cross terms:

E = \sum_{n=1}^{\infty} E_n

where $E_n = \frac{L\mu\omega_n^2}{4}(A_n^2 + B_n^2)$ , and each $E_n$ is independently conserved in time.

Proof.

Compute the kinetic energy $K = \frac{\mu}{2}\int_0^L u_t^2\,dx$ .

Differentiate the Fourier series:

u_t = \sum_{n=1}^\infty \sin\!\left(\frac{n\pi x}{L}\right)\!(-A_n\omega_n\sin\omega_n t + B_n\omega_n\cos\omega_n t)

u_t^2 = \sum_{n=1}^\infty\sum_{m=1}^\infty \sin\!\left(\frac{n\pi x}{L}\right)\!\sin\!\left(\frac{m\pi x}{L}\right) \cdot (\text{time factors})

Integrate over $[0,L]$ using the orthogonality of sine functions:

\int_0^L \sin\!\left(\frac{n\pi x}{L}\right)\!\sin\!\left(\frac{m\pi x}{L}\right) dx = \frac{L}{2}\,\delta_{nm}

All cross terms ( $n \neq m$ ) vanish. The same orthogonality applies to the potential energy $\frac{T}{2}\int_0^L u_x^2\,dx$ . Adding kinetic and potential contributions for mode $n$ and using $T(n\pi/L)^2 = \mu\omega_n^2$ :

E_n = \frac{\mu}{2}\cdot\frac{L}{2}\,\omega_n^2(A_n^2 + B_n^2) + \frac{T}{2}\cdot\frac{L}{2}\cdot\left(\frac{n\pi}{L}\right)^2\!(A_n^2 + B_n^2) = \frac{L\mu\omega_n^2}{4}(A_n^2 + B_n^2)

Each $E_n$ is constant in time (the $A_n^2 + B_n^2$ factor does not depend on $t$ ). $\square$

Prop 2.1 (Harmonic Series)

The frequencies of the normal modes of a fixed string form the harmonic series:

f_1 : f_2 : f_3 : f_4 : \cdots = 1 : 2 : 3 : 4 : \cdots

where $f_1 = c/(2L) = \frac{1}{2L}\sqrt{T/\mu}$ is the fundamental frequency.

Proof.

By Definition 2.3, $f_n = \omega_n/(2\pi) = nc/(2L) = n f_1$ . The ratio $f_n / f_1 = n$ . $\square$

Physical interpretation: Doubling the string length halves $f_1$ (one octave lower). Doubling the tension multiplies $f_1$ by $\sqrt{2}$ (a musical tritone up). Quadrupling tension doubles $f_1$ (one octave up).

Worked Example: Plucked String

Setup: A string of length $L = 1$ is plucked at the midpoint to height $h$ and released from rest. Initial conditions:

\phi(x) = \begin{cases}2hx & 0 \leq x \leq 1/2 \\ 2h(1-x) & 1/2 \leq x \leq 1\end{cases}, \qquad \psi(x) = 0

Fourier coefficients: Since $\psi = 0$ , all $B_n = 0$ . For $A_n$ :

A_n = 2\int_0^1 \phi(x)\sin(n\pi x)\,dx = \frac{8h}{n^2\pi^2}\sin\!\left(\frac{n\pi}{2}\right)

This equals $8h/(n^2\pi^2)$ for $n \equiv 1 \pmod 4$ , $-8h/(n^2\pi^2)$ for $n \equiv 3 \pmod 4$ , and $0$ for even $n$ .

Conclusion: Plucking at the midpoint eliminates all even harmonics ( $A_2 = A_4 = \cdots = 0$ ). The resulting timbre contains only odd harmonics $f_1, 3f_1, 5f_1, \ldots$ — this is the mathematical explanation for why plucking position shapes timbre.

中文: “傅里叶级数的威力在于：任何初始形状，无论多复杂，都可以分解为这些正弦驻波的叠加。系数Aₙ由初始条件决定，而这些系数正是傅里叶系数。”

Musical Connection

音乐联系

The Overtone Series is the Harmonic Series

Theorem 2.3 shows that a vibrating string produces frequencies in the exact ratio $1:2:3:4:5:6:\ldots$ This is called the overtone series or harmonic series in music. The first several intervals it generates are:

Harmonic ratio	Interval
$2:1$	Octave
$3:2$	Perfect fifth
$4:3$	Perfect fourth
$5:4$	Major third
$6:5$	Minor third
$7:6$	“Harmonic seventh” (slightly flat of B $\flat$ )

The ratios $3:2$ (perfect fifth) and $5:4$ (major third) that define the consonant intervals in Western music emerge directly from the integer structure of the harmonic series. This is why Pythagorean tuning (built from pure $3:2$ fifths) and just intonation (built from pure $5:4$ thirds) have mathematical foundations in the wave equation.

Timbre as spectral envelope: Different instruments playing the same pitch $f_1$ differ only in the amplitudes $A_n$ of each harmonic. A clarinet emphasizes odd harmonics (similar to our midpoint-pluck example, because of its cylindrical bore closed at one end). A violin’s spectral envelope depends on bow speed and contact point. The piano’s hammer position at roughly $1/8$ of the string length from one end suppresses harmonics $8, 16, 24, \ldots$ , avoiding the harsh high partials that would come from the exact hammer resonance.

中文: “泛音的相对强度决定了音色。这就是为什么钢琴和吉他演奏同一个音符，听起来却完全不同——它们的泛音结构不同。”

Limits and Open Questions

Small-amplitude approximation: The derivation in Theorem 2.1 requires $|u_x| \ll 1$ . For large-amplitude vibrations (e.g., hard piano hammering), nonlinear terms enter. The nonlinear string equation introduces overtone inharmonicity — the higher modes deviate from exact integer multiples, which piano technicians compensate for by stretching the octave during tuning.
Real strings have stiffness: Physical strings are not perfectly flexible. The corrected model adds a bending stiffness term:
$u_{tt} = c^2 u_{xx} - \frac{EI}{\mu}u_{xxxx}$ where $EI$ is the bending stiffness. This yields frequencies $f_n \approx nf_1\sqrt{1 + Bn^2}$ for a small constant $B$ (inharmonicity coefficient). The effect is musically significant for piano bass strings and is discussed in EP33 (physical modeling) .
Two and three dimensions: Drumhead vibrations satisfy the 2D wave equation $u_{tt} = c^2(u_{xx} + u_{yy})$ . The normal modes involve Bessel functions rather than sines, and the frequency ratios are no longer integer multiples — hence drums do not produce a clear pitch the way strings do.
Damping and decay: Real strings have internal friction. Adding a damping term $-2\beta u_t$ to the right side shifts the normal mode frequencies to $\omega_n^{\text{damp}} = \sqrt{\omega_n^2 - \beta^2}$ and causes exponential amplitude decay $e^{-\beta t}$ , which gives each string note its characteristic sustain and decay envelope.

Conjecture (Optimal Plucking Position)

For a plucking position at

x = L/k

(where

k

is an integer

\geq 2

), the Fourier coefficients satisfy

A_{mk} = 0

for all positive integers

m

. Therefore plucking at

1/k

eliminates all harmonics that are multiples of

k

. Among all plucking positions,

x = L/2

minimizes the total power in even harmonics and

x = L/3

minimizes power in harmonics

3, 6, 9, \ldots

Is there a plucking position (not at a rational multiple of

L

) that produces the smoothest (most rapidly decaying

A_n

) spectral envelope? The answer relates to the Diophantine approximation properties of the plucking position.

Academic References

Strauss, W. A. (2008). Partial Differential Equations: An Introduction (2nd ed.). Wiley. Ch. 1–2.
Haberman, R. (2013). Applied Partial Differential Equations (5th ed.). Pearson. Ch. 4 (Wave equation), Ch. 3 (Fourier series).
Fletcher, N. H., & Rossing, T. D. (1998). The Physics of Musical Instruments (2nd ed.). Springer. Ch. 2 (Vibrating strings).
Rayleigh, J. W. S. (1877). The Theory of Sound (2nd ed., 1894). Macmillan. Vol. 1, Ch. 6.
d’Alembert, J. (1747). Recherches sur la courbe que forme une corde tendue mise en vibration. Histoire de l’Académie Royale des Sciences et Belles-Lettres de Berlin, 3, 214–219.
Morse, P. M. (1948). Vibration and Sound (2nd ed.). McGraw-Hill. Ch. 3.
Terhardt, E. (1974). Pitch, consonance, and harmony. Journal of the Acoustical Society of America, 55(5), 1061–1069.