For a string fixed at both ends or a pipe open at both ends, use and . For a pipe closed at one end, only odd harmonics occur: and , where .
For a string fixed at both ends or a pipe open at both ends, use and . For a pipe closed at one end, only odd harmonics occur: and , where .
This is SL and HL content from C.4 Standing waves and resonance. A standing wave forms when two identical waves travelling in opposite directions superpose. The permitted wavelengths are determined by the boundary conditions at the ends.
| System | Boundary condition and permitted pattern | Wavelength and frequency |
|---|---|---|
| String fixed at both ends | Displacement node at each end; half-wavelengths fit into | , so and |
| Pipe open at both ends | Displacement antinode at each end; half-wavelengths fit into | and |
| Pipe closed at one end | Displacement node at the closed end and antinode at the open end | and , for odd only |
The wave equation gives frequency once the permitted wavelength is known. For example, a open pipe with sound speed has third-harmonic frequency
For a closed pipe of the same length, the third harmonic instead gives . Here, “third harmonic” means , which is the second permitted mode after the first harmonic.
For a stretched string, calculate the wave speed using , where is tension and is mass per unit length. Increasing tension therefore increases every harmonic frequency, while increasing linear density decreases it.
A common misconception is that a closed pipe has every harmonic. It supports only the first, third, fifth, and subsequent odd harmonics. In IB terminology, write first harmonic, not “fundamental” or “overtone.”
Identify the end conditions before selecting a formula. In a Calculate question, show the permitted-mode relationship, substitute , , and , then give wavelength in metres or frequency in hertz.