Overview
This lecture explains standing waves on a string fixed at both ends, how they form, and how to calculate their possible wavelengths and harmonics.
Waves in Boundless vs. Bounded Media
- In an unbounded medium, waves can have any wavelength or frequency.
- In a bounded medium with boundaries, waves reflect and can overlap with themselves.
Standing Waves and Nodes
- Reflected waves on a bounded medium can form standing waves.
- Only specific wavelengths and frequencies (resonances) form standing waves.
- A node is a point on the string with no motion, typically at the fixed ends.
- Anti-nodes are points of maximum displacement.
String Fixed at Both Ends: Example
- A string fixed at both ends (e.g., guitar, piano) enforces nodes at each end.
- Plucking the string creates waves that reflect back and overlap.
- Most wavelengths produce messy patterns, but standing waves emerge at special wavelengths.
Harmonics and Wavelengths
- The fundamental (first harmonic) is the simplest standing wave with only two nodes at the ends.
- The second harmonic has an additional node in the middle.
- The third harmonic has two nodes in the middle, and so on.
- For each harmonic, the standing wave fits a specific wavelength into the string's length.
Calculating Allowed Wavelengths
- For a string of length ( L ), the possible wavelengths are found by fitting nodes at both ends.
- The pattern for the nth harmonic: ( \lambda_n = \frac{2L}{n} ), where ( n = 1, 2, 3, ... ).
- The fundamental wavelength is ( 2L ), the second is ( L ), the third is ( 2L/3 ), etc.
Key Terms & Definitions
- Standing wave — a wave that oscillates up and down in place without traveling along the medium.
- Node — a point on the medium with zero motion (no displacement).
- Anti-node — a point on the medium with maximum displacement.
- Harmonic — a standing wave pattern, labeled by the integer number ( n ) (first, second, etc.).
- Fundamental — the lowest frequency (first harmonic) standing wave.
Action Items / Next Steps
- Practice drawing the first few harmonics for a string with fixed ends.
- Memorize the formula ( \lambda_n = \frac{2L}{n} ) for node-node standing waves.
- Review examples with strings of various lengths and calculate possible wavelengths.