Solution:** After 4 seconds, the radius of the expanding wavefront is $ r = 3 \times 4 = 12 $ cm. The area covered by the circular wave is given by the formula for the area of a circle:

["Understanding the Expansion of a Wavefront: Calculating the Area Coverage in Real Time", "In physics and wave propagation, understanding how a wave expands over time is crucial for both theoretical study and practical applications. One common scenario involves circular wavefronts expanding outward from a central point. Let’s explore a key moment in this process: when the radius of the expanding wavefront reaches 48 centimeters—specifically after 4 seconds—how quickly the area covered by the circular wave grows can be clearly determined using fundamental geometric principles.", "The Radius After 4 Seconds", "According to the problem, after 4 seconds, the radius of the expanding circular wavefront is:", "[\nr = 3 \ imes 4 = 12 \ ext{ cm}\n]", "This means that in 4 seconds, the wavefront has expanded to a radius of 12 centimeters, illustrating how rapidly energy, sound, or energy waves propagate outward from a source.", "The Area Covered by the Wavefront", "The wavefront expands in a circular pattern, so the area ( A ) covered by the circular wave at any time is calculated using the standard formula for the area of a circle:", "[\nA = \pi r^2\n]", "Substituting ( r = 12 ) cm:", "[\nA = \pi (12)^2 = \pi \ imes 144 = 144\pi \ ext{ cm}^2\n]", "Using the approximation ( \pi \approx 3.1416 ), the numerical value is approximately:", "[\nA \approx 144 \ imes 3.1416 = 452.39 \ ext{ cm}^2\n]", "Key Insight: Sudden Increase in Coverage", "Though the radius increases slowly (in this case by 3 cm per second), the area grows quadratically. This means that even after just 4 seconds—when the radius reaches 12 cm—the wavefront already covers nearly 452 cm². This highlights a fundamental characteristic of circular wave expansion: small changes in radius result in rapidly increasing area coverage.", "Practical Implications", "Understanding this relationship helps in fields such as seismology, underwater acoustics, and wireless signal propagation. Engineers and scientists rely on precise calculations of expanding wavefront areas to model how signals, vibrations, or energy pulses propagate through various media, ensuring effective system design and early detection of phenomena like earthquakes or radio interference.", "Conclusion", "The expression ( r = 3 \ imes 4 = 12 ) cm captures a key moment in wave expansion, where the radius grows linearly with time and area coverage expands quadratically. The area at 4 seconds is elegantly computed using the circle area formula, emphasizing how geometry underpins real-world wave dynamics. This simple yet powerful calculation enables clearer insight into the speed and scale of wave propagation.", "---", "Keywords: wavefront expansion, circular wave area formula, r = 3 × 4 = 12 cm, circular wave calculation, radius and area relationship, wave propagation, physics of waves"]