How can simple waves build a complex signal?
Where will a third harmonic create a spectral peak?
- Select signal and spectrum.
- Set the fundamental to 2 Hz.
- Change the third-harmonic amplitude; compare the time trace and DFT.
What to measure
Sample rate f_s, N, bin spacing Δf=f_s/N, Nyquist limit f_s/2, and peak frequencies.
The idea underneath
The DFT projects samples onto sinusoidal basis functions. The spectrum is calculated from the synthetic signal, not decoration. A one-second observation resolves bins 1 Hz apart. Noninteger cycles cause spectral leakage. Frequencies beyond half the sample rate can alias into lower ones; practical electronics use antialias filters.
s(t)=sin(2πf₀t)+A₃sin(6πf₀t); X_k=Σₙ sₙ exp(−i2πkn/N)f₀ in Hz; A₃ is third-harmonic amplitude; n=0…N−1.
Build the argument
- Write a travelling disturbance as y = A sin(kx − ωt + φ), with k = 2π/λ and ω = 2πf.
- Keeping phase constant gives v = ω/k = fλ. Add displacements, not amplitudes, when phases differ.
- Two waves travelling in opposite directions form nodes when their sum is zero for all times.
Worked example 1
For f₀=2 Hz and A₃=0.3, peaks occur at 2 and 6 Hz with amplitudes 1 and 0.3.
Worked example 2
For f_s=256 Hz and N=256, Δf=1 Hz and Nyquist is 128 Hz. A 150 Hz component aliases to 106 Hz.
Applications
The same language describes musical instruments, communications, seismic waves and interference-based measurement. Frequency content reveals structure hidden in a time trace.
Common misconception
The moving pattern is not the motion of a piece of the medium. Interference redistributes energy. A DFT has finite resolution and a Nyquist limit.
A further investigation
At fixed f_s, how does doubling observation time change Δf?
Check your understanding
For an ideally bandlimited signal below 10 Hz, sample rate must exceed…
- 10 Hz
- 20 Hz
- 5 Hz
Explanation
Ideal sampling requires f_s>2f_max, with practical filtering and margin. Doubling observation time halves Δf.
Model limits
Analytic linear educational model, not a numerical fluid solver. No damping, dispersion or nonlinear coupling.