Destructive Wave Interference

When two waves of the same frequency and amplitude overlap 180° out of phase (peaks lining up with troughs), they cancel each other completely. This is destructive interference — and it's the principle behind noise-cancelling headphones, anti-glare coatings on optics, and the dark fringes in a double-slit experiment.

The Math

If $y_1(t) = A\sin(\omega t)$ and $y_2(t) = A\sin(\omega t + \pi) = -A\sin(\omega t)$, then:

$y_{\text{sum}} = y_1 + y_2 = A\sin(\omega t) - A\sin(\omega t) = 0$

The two waves vanish identically. At every instant and every location, they cancel.

Step-by-Step Solution

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

Find: The combined wave $y_{\text{sum}}(t)$.

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Step 1 — Simplify $y_2$ using a trig identity.

The identity $\sin(\theta + \pi) = -\sin\theta$ gives us:

$y_2(t) = \sin(t + \pi) = -\sin t$

So $y_2$ is the negative of $y_1$ — at every $t$.

Step 2 — Add the two waves.

$y_{\text{sum}}(t) = \sin t + (-\sin t) = 0$

The sum is identically zero for all $t$.

Step 3 — Verify at a few specific points.

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

Everywhere, exactly zero.

Step 4 — What if the phase isn't exactly $\pi$?

For a general phase difference $\varphi$, the sum's amplitude is $2A\cos(\varphi/2)$ (a famous identity). The cancellation is complete only at $\varphi = \pi$ (and $3\pi$, $5\pi$, ...). At intermediate phases the amplitude shrinks but doesn't vanish.

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Answer: Two waves of equal amplitude and frequency that are 180° out of phase produce a sum that is identically zero — total cancellation:

$\boxed{\;y_{\text{sum}}(t) = 0\;}$

Where did the energy go? It didn't disappear — it went to other regions of space where the same two source waves are interfering constructively. The total energy is conserved; it's just redistributed.

Try It

• Adjust the phase widget — start at $\varphi = \pi$ (default, full cancellation), then sweep toward 0 (constructive) or toward $2\pi$ (constructive again).
• The sum's amplitude follows a $|\cos(\varphi/2)|$ pattern — maximum at $\varphi = 0, 2\pi$, zero at $\varphi = \pi$.
• The HUD shows the live sum amplitude.