Constructive Wave Interference

When two waves of the same frequency overlap in phase (peaks line up with peaks, troughs with troughs), they add together to make a wave with double the amplitude. This is constructive interference — and it's why noise-cancelling headphones do the opposite (destructive interference) to remove sound.

The Math

If $y_1(t) = A\sin(\omega t)$ and $y_2(t) = A\sin(\omega t)$ (same wave, in phase), then:

$y_{\text{sum}} = y_1 + y_2 = A\sin(\omega t) + A\sin(\omega t) = 2A\sin(\omega t)$

The amplitude doubles, but the frequency stays the same. The combined wave has the same shape as either input — just twice as tall.

More generally, if two waves have a phase difference $\varphi$:

$A\sin(\omega t) + A\sin(\omega t + \varphi) = 2A\cos\!\left(\dfrac{\varphi}{2}\right)\sin\!\left(\omega t + \dfrac{\varphi}{2}\right)$

The amplitude becomes $2A\cos(\varphi/2)$ — maximum at $\varphi = 0$ (perfect constructive), zero at $\varphi = \pi$ (perfect destructive).

Step-by-Step Solution

Given: Two waves $y_1(t) = \sin t$ and $y_2(t) = \sin t$ (same frequency, in phase, amplitude 1).

Find: The combined wave amplitude.

---

Step 1 — Add the waves directly.

$y_{\text{sum}}(t) = \sin t + \sin t = 2\sin t$

The new amplitude is 2.

Step 2 — Verify at a few specific points.

• At $t = 0$: $y_1 = 0$, $y_2 = 0$, sum $= 0$ ✓
• At $t = \pi/2$: $y_1 = 1$, $y_2 = 1$, sum $= 2$ ✓
• At $t = \pi$: $y_1 = 0$, $y_2 = 0$, sum $= 0$ ✓
• At $t = 3\pi/2$: $y_1 = -1$, $y_2 = -1$, sum $= -2$ ✓

Step 3 — Compute the energy ratio.

The energy of a wave is proportional to its amplitude squared. So the combined wave has energy $\propto 2^{2} = 4$, but each input wave has energy $\propto 1^{2} = 1$. Together, the two waves carry 4 units of energy — not 2! Constructive interference doesn't violate conservation of energy globally, because somewhere else (where there's destructive interference), energy goes to zero. The total averages out.

---

Answer: Two identical sine waves added in phase produce a wave with double the amplitude and the same frequency:

$\boxed{\;y_{\text{sum}}(t) = 2\sin t\;}$

The peaks and troughs of the inputs reinforce each other perfectly. Energy in the constructive region is four times the energy of either single wave — but this is balanced by destructive regions elsewhere.

Try It

• Adjust the amplitude widget — both component waves grow together, and so does their sum (which is always twice as tall).
• Adjust the frequency — both waves and the sum oscillate faster or slower in lockstep.
• Compare with the destructive interference visualization — two waves of equal amplitude can also cancel to zero if they're 180° out of phase.