Standing Waves and Harmonics
On a finite string clamped at both ends — u(0, t) = u(L, t) = 0 —
separation of variables
gives the wave equation's normal modes. The spatial part is again
\sin\frac{n\pi x}{L}; the time part, with no damping, oscillates:
u_n(x, t) = \sin\frac{n\pi x}{L}\,\Big(a_n\cos\omega_n t + b_n\sin\omega_n t\Big), \qquad \omega_n = \frac{n\pi c}{L}.
Each mode is a standing wave: a fixed sine shape that breathes up and down at
its own frequency \omega_n, with nodes (still points)
where the sine vanishes.
The harmonic series you can hear
The frequencies are evenly spaced: \omega_n = n\,\omega_1. The lowest,
\omega_1, is the fundamental — the pitch you hear — and
the higher modes are its harmonics (overtones) at 2\times,
3\times, the frequency. This integer ladder is exactly what makes a
plucked string sound musical rather than noisy.
A real pluck is a superposition of modes. Releasing the string from an initial shape
f(x) at rest gives
u(x, t) = \sum_{n=1}^{\infty} b_n\,\cos(\omega_n t)\,\sin\frac{n\pi x}{L},
with the b_n the
Fourier sine coefficients
of the pluck — so the shape of the pluck sets the mix of overtones, which is the string's
timbre.
- Modes \sin(n\pi x/L)\cos(\omega_n t) with \omega_n = n\pi c/L.
- Mode n has n - 1 interior nodes (still points).
- Frequencies form the harmonic series \omega_n = n\omega_1: a fundamental plus integer overtones.
- A pluck is a Fourier sine sum of modes; the coefficients set the timbre.
Pick a harmonic, watch it ring
On [0, \pi] with c = 1, mode
n is \cos(nt)\sin(nx). Choose
n to switch harmonics — count the nodes — and advance
t to watch it vibrate. Higher harmonics have more nodes and oscillate
faster, just as your ear reports.