A sum of sinusoids is periodic only if their frequencies share a common divisor

Evidence

This principle—that the sum of the solutions is the same as the solution of the sum of the sine waves— is called superposition and is a powerful aspect of the Fourier approach. — David Howard, Jamie Angus, coustics and Psychoacoustics, 439 (#expands, via linked claim)

Categorization

• XREF: connects to: Only a sinusoid has a single frequency; every other shape contains multiple frequencies; Superposition lets complex waveforms be solved as sums of sine wave solutions — David Howard, Jamie Angus, coustics and Psychoacoustics, 439; Any waveform can be approximated by summing orthogonal sine waves

Related

• Superposition lets complex waveforms be solved as sums of sine wave solutions (similarity 0.89)
• Only a sinusoid has a single frequency; every other shape contains multiple frequencies (similarity 0.89)
• Any waveform can be approximated by summing orthogonal sine waves (similarity 0.88)
• Sine waves are fully described by amplitude, frequency, and phase (similarity 0.86)
• Any complex waveform can be decomposed into simple sine waves (similarity 0.84)
• All complex sounds, including noise, are superpositions of pure sinusoids (similarity 0.82)
• Only a pure sinusoid comprises exactly one frequency. (similarity 0.81)
• Any waveform can be exactly modeled by combining sine waves (similarity 0.81)
• Linear superposition lets separate vibrations coexist independently while interfering in shared space (similarity 0.81)
• Any complex waveform can be built from independent orthogonal sine waves (similarity 0.80)
• The period of a sinusoid is the inverse of its frequency (similarity 0.80)
• A sum of sinusoids takes the period of the slowest common component (similarity 0.80)
• Superposed sinusoids with incommensurate frequencies yield nonperiodic motion (similarity 0.79)
• Fourier analysis recovers constituent sinusoids from any summed waveform (similarity 0.79)
• A hypothetical compound tuning fork illustrates how complex vibrations decompose into sine waves (similarity 0.78)

Broader Connections

• Superposition lets complex waveforms be solved as sums of sine wave solutions #expands (0.72) — David Howard, Jamie Angus, coustics and Psychoacoustics, 439
• Any waveform can be approximated by summing orthogonal sine waves #expands (0.70)