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When two or more waves meet while traveling along the same medium, they interfere with each other. This can occur in several ways, but two primary types are known as constructive and destructive interference. In constructive interference, the amplitudes of the waves add together, creating a larger wave. Conversely, in destructive interference, the waves cancel each other out, resulting in a reduced or completely nullified wave.

Standing waves are a special scenario of wave interference. They occur when two waves of the same frequency and amplitude travel in opposite directions and interfere. Instead of appearing as separate traveling waves, a standing wave appears to stand still, with certain points called nodes (points of no displacement) and antinodes (points of maximum displacement) staying fixed along the medium.

In the given problem, the standing wave is formed by the addition of two waves traveling in opposite directions. By using trigonometric identities, the interference pattern creates a sinusoidal standing wave where nodes and antinodes can be clearly identified.

Wave velocity is the speed at which a wave travels through a medium. It's an essential aspect to understand the movement of waves, particularly in problems related to standing waves. The velocity can be calculated using the formula:

• v = f\(\lambda\)

in which \(v\) represents wave velocity, \(f\) is the frequency, and \(\lambda\) (lambda) is the wavelength of the wave.

In the context of our exercise, with the given wave equation, the angular wave number \(k\) can be extracted, allowing the calculation of the wavelength \(\lambda\) using \(\lambda = \frac{2\pi}{k}\). This gives us a wavelength of 0.5 meters. Given the periodic nature of the system, we can determine the wave speed as 100 m/s.

Wave velocity is crucial for solving how far each wave travels over a certain time interval. In half of a wave period (0.0025 seconds in this scenario), the waves in the exercise move 0.25 meters along the string. Always remember that knowing the wave speed helps in predicting how quickly wave energies are transferred through a medium.

In standing waves, antinodes and nodes are fundamental concepts that help describe wave behavior. Nodes are points along the wave that experience no displacement. Mathematically, they occur where the wave interference results in a consistent zero amplitude, effectively meaning the medium at this point does not move at all.

Antinodes, on the other hand, are the points of maximum displacement within a standing wave. These are located midway between nodes, where the wave interference has maximum constructive interference, resulting in the highest amplitude.

In our exercise, we identify antinodes through the equation \(4\pi x = \left(n + \frac{1}{2}\right)\pi\). Solving for \(x\) gives us the positions where all antinodes are located. These are found at intervals of \(\left(n + \frac{1}{2}\right)\frac{1}{4}\) meters, confirming that antinodes repeat regularly along the length of the string.

Understanding the locations of antinodes and nodes is vital for correctly analyzing wave patterns, especially in practical applications such as sound or light wave phenomena, where such patterns can disrupt or enhance signals based on their position relative to barriers or interfaces.