Visual Harmonic Wave Superposition, Beats & Phase Interference Lab
By the Principle of Linear Superposition, two overlapping waves $y_1 = A_1\sin(k_1 x)$ and $y_2 = A_2\sin(k_2 x + \phi)$ sum point-by-point to $y_{\text{net}}(x) = y_1(x) + y_2(x)$, producing constructive amplification at $\phi = 0^\circ$ and destructive cancellation at $\phi = 180^\circ$.
y_net(x) = A₁·sin(f₁·x) + A₂·sin(f₂·x + φ)Inputs & Parameters (5)
20-Row Comparison Table
| # | Wave 1 Amplitude (A₁) (u) | Peak Constructive Envelope | Beat Freq |f₁−f₂| | Node Min |
|---|---|---|---|---|
| 01 | 0 u | 2 u | 0.5 Hz | 2 u |
| 02 | 0.25 u | 2.25 u | 0.5 Hz | 1.75 u |
| 03 | 0.5 u | 2.5 u | 0.5 Hz | 1.5 u |
| 04 | 0.75 u | 2.75 u | 0.5 Hz | 1.25 u |
| 05 | 1 u | 3 u | 0.5 Hz | 1 u |
| 06 | 1.25 u | 3.25 u | 0.5 Hz | 0.75 u |
| 07 | 1.5 u | 3.5 u | 0.5 Hz | 0.5 u |
| 08 | 1.75 u | 3.75 u | 0.5 Hz | 0.25 u |
| 09 | 2 u (Current) | 4 u | 0.5 Hz | 0 u |
| 10 | 2.25 u | 4.25 u | 0.5 Hz | 0.25 u |
| 11 | 2.75 u | 4.75 u | 0.5 Hz | 0.75 u |
| 12 | 3 u | 5 u | 0.5 Hz | 1 u |
| 13 | 3.25 u | 5.25 u | 0.5 Hz | 1.25 u |
| 14 | 3.5 u | 5.5 u | 0.5 Hz | 1.5 u |
| 15 | 3.75 u | 5.75 u | 0.5 Hz | 1.75 u |
| 16 | 4 u | 6 u | 0.5 Hz | 2 u |
| 17 | 4.25 u | 6.25 u | 0.5 Hz | 2.25 u |
| 18 | 4.5 u | 6.5 u | 0.5 Hz | 2.5 u |
| 19 | 4.75 u | 6.75 u | 0.5 Hz | 2.75 u |
| 20 | 5 u | 7 u | 0.5 Hz | 3 u |
Key Analytical Takeaways
- 01Set $f_1 = f_2 = 3\text{ Hz}$ and slide Phase $\phi$ from $0^\circ$ (doubled amplitude $A_1 + A_2$) to $180^\circ$ (complete flatline cancellation $|A_1 - A_2| = 0$).
- 02Set slightly detuned frequencies (e.g., $f_1 = 4.0\text{ Hz}, f_2 = 4.5\text{ Hz}$) to observe periodic beat envelopes oscillating at $f_{\text{beat}} = |f_1 - f_2|$.
- 03Set $f_2 = 3f_1$ with $A_2 = A_1/3$ to see the first two odd Fourier terms synthesize a square wave.
Frequently Asked Questions
How do noise-canceling headphones use destructive wave interference?+
Microphones sample ambient noise and generate an anti-phase waveform shifted by exactly φ = 180° (π radians) with matching amplitude, summing to zero acoustic pressure at your eardrum.
What causes acoustic beats when tuning two guitar strings?+
Two waves with slightly different frequencies f₁ and f₂ alternate between being in-phase and out-of-phase at a beat frequency equal to |f₁ − f₂|.
How is acoustic beat frequency related to two close component frequencies?+
Superposing two pure tones f₁ and f₂ of similar amplitude produces periodic loudness pulsations (beats) at the absolute difference frequency f_beat = |f₁ − f₂|.