91Ó°ÊÓ

A plane progressive wave is a type of wave that travels continuously in a specific direction through a medium. It's characterized by its ability to represent the wave's motion in a mathematical equation. This wave can transport energy without transporting the medium's particles permanently. In simple terms, the medium's particles move temporarily back and forth as the wave passes without shifting permanently.An equation like the one given, \( y = (5 \times 10^{-5} \text{ m}) \sin(100 \pi t) \), represents a plane progressive wave. Here, the equation describes the displacement of the wave at a specific point, indicating how the wave varies with time while staying within the plane it travels. At \( x = 0 \), this equation shows how the wave's displacement changes solely with time, suggesting the wave's advancement through space towards the positive \(x\)-axis. The presence of sine or cosine functions in these equations reflects the wave's oscillatory nature.

Frequency is a fundamental property of waves, representing the number of oscillations or cycles a wave completes per unit of time, typically measured in Hertz (Hz). In the given problem, the frequency of the wave is \(50 \text{ Hz}\).This means the wave completes 50 cycles every second. Frequency is crucial in understanding waves as it influences other wave characteristics like wavelength and energy. The higher the frequency, the more cycles occur in a given timeframe, and vice versa. Frequency is often used to determine other properties of waves:

• Wave speed: It is related to frequency and wavelength, defined in the equation \( v = f\lambda \), where \( v \) is wave speed and \( \lambda \) is wavelength.
• Energy: Higher frequency means more energy is transmitted by the wave.

Understanding frequency helps in tackling problems involving wave analysis, as it is a pivotal factor in predicting wave behavior.

Angular frequency (c9) is a measure of how fast the wave oscillates in radians per second. It's closely linked to the standard frequency but expressed in a different unit. Angular frequency offers an intuitive understanding of waves in terms of circular motion, as waves often behave like circular motions through sinusoidal functions.The given wave has an angular frequency denoted by \(100\pi\). This can be derived from the frequency using the formula \(\omega = 2\pi f\). For our problem with \(f = 50 \text{ Hz}\), the angular frequency becomes \(\omega = 2\pi \times 50 = 100\pi\). This means that in one second, the wave completes \(2\pi \times 50\) radians worth of its cycle. Angular frequency provides a deeper insight into the oscillatory nature of waves compared to standard frequency, especially when dealing with sine or cosine wave equations.

Wave speed defines how fast a wave propagates through a medium, calculated as the distance it covers per unit of time. For the described wave, the wave speed is given as \(300 \text{ m/s}\).Wave speed is an integral aspect of wave mechanics, determining how quickly wave patterns move from one point to another within the medium. The relationship between wave speed, frequency, and wavelength is given in the equation \(v = f\lambda\). Key notes about wave speed include:

• It remains constant for a given medium and type of wave, assuming unchanged conditions like temperature and pressure.
• The speed helps calculate wavelength when frequency is known: \(\lambda = \frac{v}{f}\).

In our case, wave speed plays a critical role in determining other wave parameters, such as the wave number, \(k\), using \(v = \frac{\omega}{k}\). Understanding wave speed allows one to predict where the wave's position will be at any given time, making it essential for solving wave-related problems.