Cos × Cos = Beats ⚡ Waves Shortcut
❓ Concept Question What happens when two waves of slightly different frequencies superpose? Why do we hear beats ? 🖼 Concept Image ✍️ Short Concept When two waves with nearly equal frequencies interfere, the resultant wave shows periodic variation in amplitude . This phenomenon is called Beats . 🔷 Step 1 — Superposition of Two Waves 💯 When two harmonic waves move in same direction: x = x 1 + x 2 x = x_1 + x_2 If frequencies are close: x 1 = a cos ( ω 1 t ) x_1 = a \cos(\omega_1 t) x 2 = a cos ( ω 2 t ) x_2 = a \cos(\omega_2 t) Resultant motion becomes product form. 🔷 Step 2 — Product Form = Beat Signal After applying trigonometric identity: x = 2 a cos ( ω 1 − ω 2 2 t ) cos ( ω 1 + ω 2 2 t ) x = 2a \cos\left(\frac{\omega_1 - \omega_2}{2} t\right) \cos\left(\frac{\omega_1 + \omega_2}{2} t\right) Two parts appear: 👉 Slow oscillation 👉 Fast oscillation 🔷 Step 3 — Envelope Creates Beats Slow cosine term forms the envelope . Fast cosine term is...