A sinusoidal wave traveling on a string is moving in the positive \(x\) -direction. The wave has a wavelength of \(4 \mathrm{~m}, \mathrm{a}\) frequency of \(50.0 \mathrm{~Hz},\) and an amplitude of \(3.00 \mathrm{~cm} .\) What is the wave function for this wave?

A sinusoidal wave is a smooth, periodic oscillation that is mathematically described by the sine or cosine function. Imagine it like the ebb and flow of ocean waves, where each wave crest and trough follows a smooth and regular pattern. These waves can exist in various mediums, such as strings, air (as sound), or even the surface of water.

The defining properties of any sinusoidal wave include amplitude, wavelength, frequency, and phase. The amplitude (\( A \) in the equation) is the height of the wave crests, which represents the maximum displacement of the wave from its equilibrium position. In the context of the exercise, the amplitude of the wave is given as 3.00 cm.

Sinusoidal waves are fundamental to our understanding of a variety of physical phenomena, especially in the fields of acoustics, optics, and electromagnetism. They're also essential in our daily lives, from the music we enjoy to the wireless communications we rely on.

The wavelength and frequency of a wave are inversely related to each other. Wavelength, often denoted by \( \lambda \) for sinusoidal waves, is the distance between two consecutive points that are in phase, such as two successive crests or troughs. It informs us about the spatial period of the wave—the greater the wavelength, the more stretched out the wave is.

Frequency (denoted by \( f \)), on the other hand, is the number of wave crests that pass a given point in one second and is measured in hertz (Hz). The frequency tells us about how quickly the wave oscillates. As per the exercise information, our wave has a wavelength of 4 meters and a frequency of 50.0 Hz. These two values are connected by the simple equation \( c = \lambda f \), where \( c \) is the speed of the wave.

Understanding this relationship helps to predict how a wave will behave when it enters a different medium or interacts with various objects, which is valuable in designing wave-related technology such as musical instruments, radio transmitters, and even sonographic equipment.

The angular frequency (represented by \( \omega \) in our equations) is a measure of how fast something rotates or revolves, relating to waves as to how quickly their cycles are completed. It's expressed in radians per second (rad/s) and is calculated as \( \omega = 2\pi f \). This quantity can be thought of as the 'speed' of the wave's oscillation, from a circular motion perspective.

In the step-by-step solution provided, the angular frequency was found by multiplying the standard frequency by \( 2\pi \) yielding \( 100\pi \) rad/s. This value signifies that the wave completes \( 50 \) oscillations (or cycles) each second, and at each oscillation, the point on the wave traces out a path that covers \( 2\pi \) radians.

Angular frequency is particularly useful in electrical engineering for understanding alternating currents and in mechanical systems that involve rotations or vibrations, such as engines or oscillating circuits.