When two waves of almost equal frequencies $n_1$ and $n_2$ reach at a point simultaneously, what is the time interval between successive maxima?

When two waves of almost equal frequencies n₁ and n₂ reach at a point simultaneously, what is the time interva

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Concept understanding — Beats Frequency Analysis

Beats Frequency Analysis

The Intuition: When Two Tones "Wobble"

Imagine you're tuning a guitar. You pluck the string you're tuning, and at the same time, you play a reference note from a tuning fork. If the two notes are exactly the same pitch, you hear a single steady tone. But if they are slightly different — say one is 440 Hz and the other is 442 Hz — you don't hear two separate notes. Instead, you hear a single tone that wobbles in loudness: it gets louder, then softer, then louder again, in a slow, rhythmic pulse.

That pulse is called a beat. The phenomenon is beats.

Why does this happen? Because the two sound waves are constantly going in and out of sync. When their crests align, they add up to a louder sound (constructive interference). When a crest meets a trough, they cancel partially (destructive interference). The result is a wave whose amplitude rises and falls at a rate equal to the difference between the two original frequencies.

Note

You don't hear the individual frequencies when they are very close. Your ear perceives the average frequency (around 441 Hz in the example), but the loudness fluctuates at the beat frequency.

The Precise Statement

Let two sound waves of slightly different frequencies $f_1$ and $f_2$ (with $f_1 > f_2$ ) travel through the same medium. Their displacements at a point can be written as:

y_1 = A \sin(2\pi f_1 t)

y_2 = A \sin(2\pi f_2 t)

By the principle of superposition, the resultant displacement is:

y = y_1 + y_2 = A \left[ \sin(2\pi f_1 t) + \sin(2\pi f_2 t) \right]

Using the trigonometric identity $\sin P + \sin Q = 2 \sin\left(\frac{P+Q}{2}\right) \cos\left(\frac{P-Q}{2}\right)$ , we get:

y = 2A \cos\left( 2\pi \frac{f_1 - f_2}{2} t \right) \sin\left( 2\pi \frac{f_1 + f_2}{2} t \right)

This is the key equation. It describes a wave with two parts:

The carrier wave: $\sin\left( 2\pi \frac{f_1 + f_2}{2} t \right)$ — this oscillates at the average frequency $f_{\text{avg}} = \frac{f_1 + f_2}{2}$ . This is the pitch you actually hear.
The envelope: $2A \cos\left( 2\pi \frac{f_1 - f_2}{2} t \right)$ — this modulates the amplitude of the carrier. The envelope oscillates at half the difference frequency.

f_{\text{beat}} = |f_1 - f_2|

Why $|f_1 - f_2|$ and not half of it? Because the cosine term goes through a full cycle (from maximum to minimum and back to maximum) when its argument changes by $2\pi$ . That happens when $\frac{f_1 - f_2}{2} t = 1$ , i.e., $t = \frac{2}{f_1 - f_2}$ . So the time period of the envelope is $T_{\text{envelope}} = \frac{2}{|f_1 - f_2|}$ . But the loudness (intensity) goes through two maxima per envelope cycle — one at each positive peak of the cosine and one at each negative peak (since squaring the amplitude gives intensity). So the period of the beat (the time between successive loudness maxima) is half of that: $T_{\text{beat}} = \frac{1}{|f_1 - f_2|}$ . Hence, the beat frequency is $f_{\text{beat}} = \frac{1}{T_{\text{beat}}} = |f_1 - f_2|$ .

Watch out

A common mistake is to think the beat frequency is $\frac{|f_1 - f_2|}{2}$ . That is the frequency of the envelope oscillation, not the beat. The ear detects two loudness peaks per envelope cycle, so the beat frequency is double the envelope frequency.

Key Conditions for Beats

Small difference: The two frequencies must be close (typically less than about 10–15 Hz apart). If the difference is too large, the ear perceives two separate tones instead of beats.
Comparable amplitudes: The amplitudes should be roughly equal for maximum contrast in loudness. If one is much louder, beats are still present but less noticeable.
Same medium: The waves must overlap in the same region of space.

Why This Matters for Exams …

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When two waves of nearly equal frequency interfere, they produce beats—periodic variations in amplitude. The time between successive maxima (loud sounds) is the reciprocal of the beat frequency: $\frac{1}{|n_1 - n_2|}$ .

Why beats occur

When two waves of slightly different frequencies superpose, they drift in and out of phase with each other. Sometimes their crests align and reinforce (constructive interference, maximum amplitude); half a beat cycle later, a crest meets a trough and they cancel (destructive interference, minimum amplitude). This periodic rise and fall in intensity is what we hear as beats.

The mathematics reveals that the combined wave oscillates at the average frequency $\frac{n_1 + n_2}{2}$ , but its amplitude itself oscillates slowly at the beat frequency $|n_1 - n_2|$ . Each complete cycle of the amplitude envelope—from one maximum through a minimum and back to the next maximum—takes one beat period.

Step-by-step derivation

Write the two waves. Assume both have the same amplitude $A$ and meet at a point. Their displacements are:

y_1 = A \sin(2\pi n_1 t), \quad y_2 = A \sin(2\pi n_2 t).

Superpose them. The resultant displacement is:

y = y_1 + y_2 = A[\sin(2\pi n_1 t) + \sin(2\pi n_2 t)].

Apply the sum-to-product identity. Recall that $\sin C + \sin D = 2 \sin\left(\frac{C+D}{2}\right)\cos\left(\frac{C-D}{2}\right)$ . Here:

y = 2A \cos\left(2\pi \frac{n_1 - n_2}{2} t\right) \sin\left(2\pi \frac{n_1 + n_2}{2} t\right).

Identify the modulation. The sine term oscillates rapidly at the average frequency $\frac{n_1 + n_2}{2}$ (the carrier). The cosine term oscillates slowly at frequency $\frac{|n_1 - n_2|}{2}$ and acts as a time-varying amplitude envelope.
Recognize that intensity depends on amplitude squared. The amplitude envelope is:

A_{\text{env}}(t) = 2A \left|\cos\left(\pi (n_1 - n_2) t\right)\right|.

The intensity (proportional to $A_{\text{env}}^2$ ) reaches a maximum whenever $\cos^2(\pi (n_1 - n_2) t) = 1$ , which happens twice per cycle of the cosine—once at each peak of $|\cos|$ . …

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Q.When two waves of almost equal frequencies n1n_1n1​ and n2n_2n2​ reach at a point simultaneously, what is the time interval between successive maxima?

Why beats occur

Step-by-step derivation

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Q.When two waves of almost equal frequencies $n_1$ and $n_2$ reach at a point simultaneously, what is the time interval between successive maxima?

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