Three sound waves of equal amplitudes have frequencies (v - 1), v, (v + 1). They superpose to give b

Q: Why do we take the maximum difference and not the sum of differences?

Beats are perceived as the periodic variation in loudness. The fastest variation in the envelope is governed by the most extreme frequency components. The intermediate frequencies just modulate the shape of the beat, but the primary beat rate is determined by the maximum frequency spread.

Q: Will all the beats sound equally loud?

No! If you analyze the mathematical superposition, you'll find that one beat is very loud (when all three waves are in phase) and the other is softer (when two are in phase and one is out of phase).

Q: What if the amplitudes were different?

If amplitudes are different, the beats will still occur at the same rate, but the minimum intensity will not be zero. The sound will not completely fade out between beats.

Q: Is the beat frequency always an integer?

Not necessarily. If the frequencies were $\nu$, $\nu+1.5$, the beat frequency would be 1.5 Hz, meaning you'd hear 3 beats every 2 seconds.

Q: Can we hear beats if the frequencies are very far apart, like 500 Hz and 1000 Hz?

No, the human ear can only distinguish beats if the frequency difference is less than about 10-15 Hz. Beyond that, we hear two distinct tones or a rough, continuous sound.

Question

Three sound waves of equal amplitudes have frequencies $(
u - 1)$, $
u$, $(
u + 1)$. They superpose to give beat. The number of beats produced per second will be

Options:

(A) 4

(B) 3

(C) 2

(D) 1

Sigma Percentile · Accepted Answer

### The Phenomenon of Beats

Imagine you are sitting in a concert hall, and two musicians are tuning their instruments. If they play notes that are very close in frequency but not exactly the same, you won't hear two distinct notes. Instead, you will hear a single note that periodically grows louder and softer. This rhythmic pulsing of sound is what physicists call **beats**.

The mathematics behind this is beautifully simple. When two waves of frequencies $f_1$ and $f_2$ superpose, they interfere constructively and destructively over time. The rate at which this loudness oscillates—the beat frequency—is exactly equal to the absolute difference between the two frequencies:

$$f_{beat} = |f_1 - f_2|$$

### Analyzing the Given Frequencies

In our problem, we are not dealing with two, but **three** sound waves. They all have the same amplitude $A$, but their frequencies are slightly staggered:

1. $f_1 = 
u - 1$
2. $f_2 = 
u$
3. $f_3 = 
u + 1$

When these three waves superpose, the situation becomes a bit more complex. To find the number of beats produced per second, we need to look at the frequency differences between all possible pairs of these waves.

Let's calculate the differences:
- Between the first and second wave: $|
u - (
u - 1)| = 1	ext{ Hz}$
- Between the second and third wave: $|(
u + 1) - 
u| = 1	ext{ Hz}$
- Between the first and third wave: $|(
u + 1) - (
u - 1)| = 2	ext{ Hz}$

### The Rule of Maximum Difference

When multiple waves superpose, the overall beat frequency that our ears perceive is dictated by the **maximum frequency difference** present in the mixture. The extreme frequencies create the fastest envelope of modulation, while the intermediate frequencies simply shape the internal structure of that envelope.

Since the maximum difference in our set is between $f_1$ and $f_3$, which is $2	ext{ Hz}$, the system will produce **2 beats per second**.

### The Mathematical Proof

If you are wondering *why* the maximum difference dictates the beat frequency, let's dive into the rigorous mathematics of superposition. The total displacement $y$ of the air particles is the sum of the individual displacements:

$$y = A\sin(2\pi(
u - 1)t) + A\sin(2\pi
u t) + A\sin(2\pi(
u + 1)t)$$

Let's group the first and third terms and apply the trigonometric identity $\sin C + \sin D = 2\sin\left(\frac{C+D}{2}ight)\cos\left(\frac{C-D}{2}ight)$:

$$y = A[\sin(2\pi(
u - 1)t) + \sin(2\pi(
u + 1)t)] + A\sin(2\pi
u t)$$

$$y = A\left[2\sin(2\pi
u t)\cos(2\pi t)ight] + A\sin(2\pi
u t)$$

Now, we can factor out the common term $A\sin(2\pi
u t)$:

$$y = A\sin(2\pi
u t) [1 + 2\cos(2\pi t)]$$

This equation is a masterpiece. It tells us that the resultant wave oscillates at the central frequency $
u$, but its amplitude is modulated by an envelope function: $A_{env} = A|1 + 2\cos(2\pi t)|$.

The loudness, or **intensity** $I$, of the sound is proportional to the square of this amplitude envelope:

$$I \propto A^2(1 + 2\cos(2\pi t))^2$$

To find the number of beats per second, we need to find how many times this intensity reaches a maximum in a one-second interval (from $t=0$ to $t=1$).

- At $t = 0$, $\cos(0) = 1$, so $I \propto (1 + 2)^2 = 9$.
- At $t = 0.5$, $\cos(\pi) = -1$, so $I \propto (1 - 2)^2 = 1$.
- At $t = 1$, $\cos(2\pi) = 1$, so $I \propto (1 + 2)^2 = 9$.

In exactly one second, the intensity hits a massive peak (value 9), then a smaller peak (value 1), and then returns to a massive peak. Because there are **two distinct maxima** (one major and one minor) in one second, the listener will perceive **2 beats per second**.

### Conclusion

Whether you use the intuitive shortcut of finding the maximum frequency difference or the rigorous mathematical derivation of the intensity envelope, the conclusion is identical. The superposition of these three waves creates a complex beat pattern that pulses twice every second.

Animated Solution for Physics - Waves: Three sound waves of equal amplitudes have frequencies $(ν - 1)$ , $ν$ , $(ν + 1)$ . They superpose to give beat. The number of beats produced per second will be

Select Answer:

Visualized Solution

$f_{1}, f_{2}, f_{3}$

$f_{b e a t} = ∣ f_{a} - f_{b} ∣$

$Δ f_{p ai r s}$

$Δ f_{ma x}$

$I_{ma x}$

The Sigma Insight: Interference of Waves

The Phenomenon of Beats

Analyzing the Given Frequencies

The Rule of Maximum Difference

The Mathematical Proof

Conclusion

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