15.2 Standing Waves on Strings

For a string with both ends fixed, each endpoint is a displacement node. Only standing-wave shapes that satisfy node-node conditions at exactly the string endpoints are allowed. This quantization means not every wavelength can exist on the string; only a discrete family fits the boundaries.

The first harmonic has one loop (two nodes and one antinode). The second has two loops, and so on. Neighboring loops oscillate in antiphase, while points within the same loop oscillate in phase. This phase structure helps when questions ask about relative velocity directions of points on different parts of the string.

Because mode geometry is fixed by L and n, frequency changes come through wave speed v. For strings, v depends on tension and linear density: higher tension raises speed and frequencies, while larger linear density lowers them. Conceptually, stronger restoring force speeds signal transmission, while larger inertia slows it.

This equation is one reason harmonic demonstrations are sensitive to setup. A small adjustment in tension can move a driven system out of one resonant mode and into another. In laboratory tuning tasks, frequency and tension are not independent if the same string and mode number are enforced.

Given a mode sketch, first count loops to identify n. Then compute wavelength from lambda_n = 2L/n. Next use frequency-speed relation to solve for unknown quantity. For mixed conceptual-numerical questions, keep a sentence in your solution explaining why this mode is allowed by node-node boundary conditions.

If a question changes tension while length and driving frequency stay fixed, the excited harmonic number may change. The system searches for the nearest allowed mode that matches boundary constraints. This is why static formulas and dynamic tuning intuition should be used together.