Animated Solution for Physics - Waves: The correct figure that shows schematically, the wave pattern produced by superposition of two waves of frequencies 9 Hz and 11 Hz, is

Imagine you are standing in a room where two tuning forks are struck simultaneously. One vibrates at a frequency of $11\text{ Hz}$, and the other at a slightly lower frequency of $9\text{ Hz}$. As these two sound waves travel through the air, they don't just pass each other by; they interact. This interaction is governed by the Principle of Superposition, which states that the resultant displacement of the medium is the algebraic sum of the individual displacements.

Because their frequencies are so close, the waves will periodically fall perfectly into step (constructive interference), creating a loud sound, and then gradually fall out of step (destructive interference), canceling each other out to create silence. This rhythmic pulsing of sound volume is what we call beats.

To understand the visual representation of this phenomenon, we first need to determine how often these beats occur. The number of loud-soft cycles per second is known as the beat frequency, and it is elegantly simple to calculate. It is just the absolute difference between the two original frequencies:

Plugging in our given values:

This tells us that we will hear exactly two beats every second. But to match this with a graph, we need the time duration of a single beat. The time period of one beat is simply the reciprocal of the beat frequency:

This is a crucial piece of the puzzle! The envelope of our resultant wave—the dashed line that bounds the amplitude—must pinch down to zero every $0.5$ seconds.

While the envelope pulses every $0.5$ seconds, what is happening inside that envelope? The actual sound wave oscillating within the bounds of the envelope has a frequency that is the average of the two original frequencies. We call this the internal or carrier frequency:

Let's calculate it for our waves:

This means the internal wave completes $10$ full cycles every second.

Now, let's bring it all together to find the correct graph. We know that one beat lasts for $0.5$ seconds, and the internal wave oscillates at a rate of $10$ cycles per second. How many internal cycles fit perfectly inside one single beat?

Each beat must contain exactly $5$ complete cycles.

When we examine the given options, we must look for two distinct features: 1. The nodes (where the amplitude is zero) must occur every $0.5$ seconds (at $t=0.5,1.0,1.5,2.0$). 2. Between any two nodes, there must be exactly $5$ peaks of the internal wave.

Option (a) is the only graph that flawlessly depicts a beat envelope pinching to zero every $0.5$ seconds while containing exactly $5$ rapid internal oscillations within each beat. The math perfectly dictates the geometry of the wave!