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Transverse waves are a fundamental type of wave where the particles of the medium move perpendicular to the direction of the wave's travel. Imagine a rope being flicked up and down; as a result, the wave moves horizontally along the rope, while the particles of the rope move vertically. This distinct perpendicular motion characterizes transverse waves and differentiates them from other wave types, such as longitudinal waves where particle movement is parallel to wave direction.

In our problem, we consider a transverse wave traveling along a string. The variables in our wave equation, like amplitude and frequency, describe the wave’s characteristics, and understanding these helps us predict the wave’s motion. For instance, the amplitude (1.50 mm) suggests how far particles in the medium move from their rest position, showing the wave's energy level.

The wave equation describes the displacement of a wave as it moves through a medium. For a transverse wave, the equation often takes the form: \[ y(x, t) = A \sin(\omega t - kx) \] where

• \( y(x, t) \) is the wave's displacement at position \( x \) and time \( t \).
• \( A \) is the amplitude, representing the maximum displacement.
• \( \omega \) is the angular frequency, determining how many oscillations occur per second.
• \( k \) is the wave number, related to the wavelength and direction of the wave.

The terms in the sine function determine the wave’s progression over time and space. The negative sign in \(\omega t - kx\) indicates the wave travels in the positive x-direction. This clear framework allows for predicting how the wave evolves, revealing information like speed and direction by substituting known values.

The wavelength \( \lambda \) of a wave is the distance over which the wave's shape repeats. It holds the key to understanding how many wave crests (or troughs) fit within a unit of distance. In our scenario, the relationship between the wave number \( k \) and the wavelength is given by: \[ k = \frac{2\pi}{\lambda} \] From this equation, we can rearrange to find the wavelength: \[ \lambda = \frac{2\pi}{k} \] Plugging the values, we determine that \( \lambda \approx 0.150\, \text{m} \).

A smaller wavelength indicates a denser packing of wave peaks and troughs, which often means higher frequency, and vice versa. The wavelength is crucial in applications like communication, where signals are transmitted in specific wave bands.

Angular frequency \( \omega \) is a measure of how quickly a wave oscillates and is central to understanding wave behavior. It is expressed in radians per second. The relationship between angular frequency and regular frequency \( f \) is straightforward: \[ \omega = 2\pi f \] Using this relationship, we calculate \( f \) from a given \( \omega \). In our case, \( \omega = 157\, \text{s}^{-1} \), which translates to a frequency \( f \approx 25.0 \) Hz, meaning 25 oscillations per second occur.

This concept is essential in fields such as acoustics and optics, where understanding how fast wave crests pass a certain point informs us about the energy and nature of the wave. Angular frequency ensures a concise description of this oscillatory motion, compactly expressing its periodic nature.