If an explosion loud enough to be heard around the world occurred and the world was a perfectly smooth sphere, the sound waves would not cancel each other out completely at the exact opposite area on the globe. While sound waves can undergo interference and cancelation in certain conditions, the cancellation effect would not be significant in this scenario for several reasons:
Sound attenuation: Sound waves naturally experience attenuation as they propagate through a medium, such as air. The further the sound travels, the more it attenuates, which reduces its intensity. By the time the sound waves reach the opposite side of the globe, their intensity would have significantly decreased, making the cancellation effect less pronounced.
Multiple reflections and diffraction: Sound waves would encounter various obstacles, such as mountains, buildings, and other geographical features, as they propagate around the world. These obstacles would cause the sound waves to reflect, diffract, and scatter, altering their direction and spreading them out. This scattering effect would prevent the sound waves from reaching the exact opposite side of the globe with sufficient coherence and alignment for complete cancellation.
Non-uniformity of the Earth's surface: The Earth's surface is not perfectly smooth and features variations in elevation, topography, and atmospheric conditions. These irregularities would introduce additional complexities into the propagation of sound waves, preventing precise cancellation at the opposite side of the globe.
Absorption and dispersion: Sound waves also experience absorption and dispersion as they interact with the atmosphere. Different frequencies of sound may be absorbed to varying degrees, affecting the overall cancellation effect. Additionally, dispersion can cause the sound waves to spread out and lose coherence, further limiting cancellation.
While the concept of sound cancellation exists and can be observed in controlled environments, the complex and non-uniform nature of the Earth's surface, as well as the natural attenuation, scattering, and absorption of sound waves, would prevent a complete cancellation of sound waves reaching the exact opposite area on the globe.