Example 1
Waves contribute +3.0mm and −1.0mm at one point. Find combined displacement.
Add signed displacements: y=+3.0+(−1.0)=+2.0mm.
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Year 11 · SACE Stage 1
Superposition, standing waves and resonance
Calculate combined displacement, beats and simple harmonics.
Method and worked examples
Superposition adds simultaneous signed displacements at the same position. Equal in-phase disturbances reinforce; equal opposite disturbances cancel at that instant. Similar frequencies produce beats at ∣f1−f2∣. Resonance gives a large forced response near a system’s natural frequency, with damping limiting its amplitude. A standing wave forms from opposing waves of the same frequency and has fixed nodes and antinodes.
For a string fixed at both ends or an ideal pipe open at both ends, λn=n2L and fn=(2L)nv, with n=1,2,3… . For a pipe closed at one end, λn=(2n−1)4L, giving odd harmonics. Ignore end corrections when stated. A string’s fixed end is a displacement node; an ideal open air-column end is a displacement antinode.
Example 1
Example 2
An ideal 0.85m pipe is closed at one end. Use sound speed 340m/s and ignore end correction. Find fundamental frequency.
The fundamental has λ=4L=3.4m. f=λv=3.4340=100 Hz. The next allowed mode is 300 Hz.
Watch out: Nodes are fixed zero-displacement positions of the standing pattern. Travelling components create that pattern by interference.