Preview
IB DP Physics — Unit C: Wave Behaviour | SL & HL | C.4 Standing Waves and Resonance
📋 What You Will Learn
- Observe how two travelling waves superpose to form a standing wave, using real-world examples from music and engineering
- Define nodes, antinodes, harmonics, and all key notation used in standing wave analysis
- Derive the harmonic frequency equations for strings and open/closed pipes from first principles
- Apply the harmonic series to calculate resonant frequencies and wavelengths for strings and pipes
- Analyse displacement diagrams and frequency spectra for different harmonics
- Evaluate common misconceptions about standing waves and connect this topic to sound, resonance, and wave superposition across the IB Physics course
🌍 Real-World Context When a guitarist plucks a string, the string does not simply vibrate back and forth uniformly — it forms a beautiful, stable pattern of loops and still points that persist as long as the string vibrates. The same phenomenon occurs inside a flute, an organ pipe, or even the air column inside a bottle when you blow across its opening. These stable vibration patterns are not random: they occur only at very specific frequencies, and it is precisely this selectivity that allows musical instruments to produce pure, defined pitches rather than noise. Understanding why only certain frequencies are "allowed" — and what determines them — is the central question of this topic.
🔍 Guiding Question How do boundary conditions on strings and in pipes determine which frequencies of vibration are possible, and what is the physical origin of the discrete harmonic series observed in musical instruments?
Physical Setup: How Standing Waves Form
Imagine sending a continuous wave pulse along a stretched string that is fixed at both ends. When the wave reaches the fixed end, it reflects back along the string. Now there are two waves travelling in opposite directions on the same string simultaneously — the original wave moving forward and its reflection moving backward. These two waves continuously pass through each other, and at every point on the string, the displacements from both waves add together according to the principle of superposition.
At most frequencies, this superposition produces a complicated, ever-changing pattern. But at certain special frequencies, something remarkable happens: the pattern of displacement on the string appears to stand still. The string oscillates up and down, but the shape of the pattern does not travel along the string — it stays fixed in space. This is a standing wave, and it arises from the perfect constructive and destructive interference of the two counter-propagating waves.
Notice in the diagram that at the points labelled N (nodes), the two waves always cancel each other perfectly — these points never move. At the points labelled A (antinodes), the two waves always reinforce each other — these points oscillate with maximum amplitude. The distance between two adjacent nodes is exactly half a wavelength, \( \lambda/2 \). This spatial relationship is the key to understanding why only certain frequencies fit inside a string or pipe of fixed length.
Core Concepts and Definitions
When a wave reflects from a fixed boundary (such as the end of a string clamped to a wall), the reflected wave is inverted — it undergoes a phase change of \( \pi \) radians (180°). This means that at a fixed boundary, the displacement is always zero. This is not a coincidence: it is a boundary condition imposed by the physics of the fixed end. The standing wave pattern must