Where a Piano Hammer Should Strike

A piano hammer gives a string a sharp kick of velocity at one point. The string then vibrates as a sum of standing waves, and any harmonic with a node at the strike point is never excited. Strike at 1/7 of the length and the 7th harmonic all but disappears.

How it works

The vibrating string: standing waves and harmonics

In one sentence

The wave equation $u_{tt} = c^2u_{xx}$ with fixed ends $u(0,t) = u(L,t) = 0$ , initial displacement $u(x,0) = 0$ and initial velocity $u_t(x,0) = g(x)$ has the solution $u = \sum B_n\sin\frac{n\pi ct}{L}\sin\frac{n\pi x}{L}$ with $B_n = \frac{2}{n\pi c}\int_0^L g(x)\sin\frac{n\pi x}{L}\,dx$ . Mode $n$ vibrates at frequency $f_n = nc/(2L)$ .

General formula

u(x, t) = \sum_{{\htmlData{sym=4}{n}}=1}^{\infty} {\htmlData{sym=0}{B_n}}\sin\frac{{\htmlData{sym=4}{n}}\pi {\htmlData{sym=1}{c}}t}{\htmlData{sym=2}{{L}}}\sin\frac{{\htmlData{sym=4}{n}}\pi x}{\htmlData{sym=2}{{L}}}, \qquad {\htmlData{sym=0}{B_n}} = \frac{2}{{\htmlData{sym=4}{n}}\pi {\htmlData{sym=1}{c}}}\int_0^{\htmlData{sym=2}{L}} {\htmlData{sym=3}{g(x)}}\sin\frac{{\htmlData{sym=4}{n}}\pi x}{\htmlData{sym=2}{{L}}}\,dx

Each harmonic is a standing wave with fixed nodes. Kicking the string at one of a mode's nodes can't set that mode moving.

What the symbols mean

$B_n$
Harmonic amplitude
How strongly the strike excites harmonic n.
$c$
Wave speed
How fast waves run along the string, set by its tension and mass.
$L$
String length
The vibrating length between the fixed ends.
$g(x)$
Initial velocity
The kick the hammer gives each point of the string.
$n$
Harmonic number
1 is the fundamental; n vibrates n times as fast.

The hammer starts the string with no displacement but a burst of velocity $g(x)$ around the strike point.
Separation of variables gives standing waves $\sin\frac{n\pi x}{L}$ , each oscillating at frequency $f_n = \frac{nc}{2L}$ : whole-number multiples of the fundamental.
Matching the initial velocity gives each amplitude $B_n$ by orthogonality. A harmonic with a node at the strike point, where $\sin\frac{n\pi p}{L} = 0$ , gets nothing.
A wider, softer hammer also tames the high harmonics, because it can't deliver a kick sharper than its own width.

Learn the math

The Wave EquationPDE · Lesson 5, Foundations The Vibrating StringPDE · Lesson 10, Separation of Variables