Cos × Cos = Beats ⚡ Waves Shortcut

 

❓ Concept Question

What happens when two waves of slightly different frequencies superpose?

Why do we hear beats?

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🖼 Concept Image

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✍️ Short Concept

When two waves with nearly equal frequencies interfere, the resultant wave shows periodic variation in amplitude.

This phenomenon is called Beats.

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🔷 Step 1 — Superposition of Two Waves 💯

When two harmonic waves move in same direction:

$$x = x_1 + x_2$$

If frequencies are close:

$$x_1 = a \cos(\omega_1 t)$$
$$x_2 = a \cos(\omega_2 t)$$

Resultant motion becomes product form.

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🔷 Step 2 — Product Form = Beat Signal

After applying trigonometric identity:

$$x = 2a \cos\left(\frac{\omega_1 - \omega_2}{2} t\right) \cos\left(\frac{\omega_1 + \omega_2}{2} t\right)$$

Two parts appear:

👉 Slow oscillation
👉 Fast oscillation

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🔷 Step 3 — Envelope Creates Beats

Slow cosine term forms the envelope.

Fast cosine term is the carrier wave.

So:

Beats come from the slow varying amplitude.

Fast oscillation simply carries the wave.

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🔷 Step 4 — Beat Angular Frequency

Beat frequency depends on difference of frequencies.

$$\omega_{beat} = |\omega_1 - \omega_2|$$

👉 Smaller difference
👉 Slower beats

Which means beats become clearly audible.

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🔷 Step 5 — Beat Time Period

Time period of beats:

$$T_{beat} = \frac{2\pi}{\omega_{beat}}$$

⚠️ Always use radians when angular frequency is given.

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✅ Final Takeaway

When two waves with nearly equal frequencies interfere:

$$\boxed{\text{Beat frequency} = |f_1 - f_2|}$$

Amplitude periodically increases and decreases.

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⭐ Golden JEE Insight

Whenever expression looks like:

$$\cos(\omega_1 t) \cos(\omega_2 t)$$

Immediately think:

👉 Beats formation

Slow term = envelope
Fast term = carrier wave.