A 120 cm length of string is stretched between fixed...

A $120 \mathrm{~cm}$ length of string is stretched between fixed supports. What are the (a) longest, (b) second longest, and (c) third longest wavelength for waves traveling on the string if standing waves are to be set up? (d) Sketch those standing waves.

Key Concepts

Wavelength Relationship for Fixed Strings

The wavelength (?) of the standing waves on a string fixed at both ends is determined by the length of the string (L) and the harmonic number (n), based on the formula ? = 2L/n. This formula directly relates the physical constraint of the string's length to the possible wavelengths that can form standing wave patterns on it.

Standing Waves

Standing waves are patterns of vibration that remain stationary on a medium, created by the interference of two waves traveling in opposite directions. In the context of a string fixed at both ends, standing waves manifest when the waves reflect back and forth, leading to fixed points (nodes) where there is no motion, and points (antinodes) where the amplitude is maximized.

Boundary Conditions for Fixed Ends

For a string with fixed endpoints, the boundary conditions require that the displacement of the wave is zero at both ends. This restriction means that the standing wave patterns must have nodes at the two fixed supports, fundamentally influencing the possible wavelengths and frequencies of the resulting vibrations.

Harmonics (Normal Modes)

Harmonics, or normal modes, refer to the specific patterns of standing waves that can exist on a string with fixed ends. The simplest mode (the fundamental) has the longest wavelength and lowest frequency, while higher harmonics correspond to higher frequencies and shorter wavelengths. Each harmonic is characterized by an integer (n), where n=1 represents the fundamental mode and higher integers represent subsequent modes.

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