Adding mass at a node leaves that mode unchanged

Evidence

The exact center of a uniform string tied at both ends is a special place: all the even modes—the second, fourth, sixth, and so on partials—have a node there; the string does not move at that point. The shape of any specific mode and its frequency do not change if any of its nodes is tied down: it is already tying itself down, in effect. It does not matter if we place a weight, the extra heavy bead, at the node. The bead would not move, and the string would not “know” that the bead is there. This is not true of the odd modes, which all have an antinode at the center of the string. The heavy bead makes the odd modes oscillate more slowly—roughly speaking, the formula 2πf = (K/m)1/2 applies, where K is the force constant resisting deformation of the string, and m is the mass. By adding a heavy bead, we are increasing the mass of those modes that move the bead, without increasing the resistance to bending. The frequency would be expected to drop. — Eric J. Heller, Why You Hear What You Hear- An…, 857

Categorization

• XREF: A tiny putty spot measurably slowing a violin string’s odd harmonics connects to why stringed instruments are so sensitive to even tiny geometric imperfections.
• CANDY: A 2 mm putty spot on a violin string dramatically changes its sound — a fun, graspable everyday fact.