91Ó°ÊÓ

When we speak of the amplitude of a wave, we're referring to the maximum height of the wave's crest above its central axis, or equally, the maximum depth of its trough below the axis. The amplitude is critical because it's directly related to the energy transported by the wave; greater amplitude means more energy. In the given problem, we're looking at a wave equation of the form
\(y = A \sin(kx - \omega t)\),where \(A\) represents the amplitude. For the exercise's wave equation,
\(y = (4.0 \, \mathrm{cm}) \sin(2.0x - 3.0t)\),the amplitude is clearly given as \(4.0 \, \mathrm{cm}\), which is the coefficient before the sine function. This value indicates the energy strength of the wave and can affect how it interacts with objects and mediums in its path.

The wavelength of a wave is the distance over which the wave's shape repeats, and it is inversely related to the wave number \(k\). To find the wavelength \(\lambda\), we look at the wave equation and identify the coefficient in front of the variable \(x\), which represents \(k\). In the exercise's equation,\(y = (4.0 \, \mathrm{cm}) \sin(2.0x - 3.0t)\),the wave number is \(2.0 \, \mathrm{cm}^{-1}\), meaning one wavelength stretches over \(\frac{1}{2.0} \, \mathrm{cm}\), or \(0.5 \, \mathrm{cm}\). This mathematical relationship between wave number and wavelength helps determine how 'tightly' the wave oscillates as it travels.

Frequency is the number of waves that pass a point in a given period of time and it is measured in hertz (Hz). High frequency means more waves are passing through a point each second. In the context of our wave equation,
\(y = (4.0 \, \mathrm{cm}) \sin(2.0x - 3.0t)\),the frequency can be deduced from the angular frequency \(\omega\), which is the coefficient of the time variable \(t\). In this scenario, \(\omega\) is \(3.0 \, \mathrm{s}^{-1}\), and we use the formula\(f = \frac{\omega}{2\pi}\)to find the frequency \(f\). Calculating \(f\), we get approximately \(0.48 \, \mathrm{Hz}\), revealing that almost half a wave cycle occurs every second.

The period of a wave is the time it takes for a single cycle to pass a point. It is the inverse of frequency, reflecting the temporal length of the wave cycle. If a wave has a high frequency, it has a short period; conversely, a low frequency results in a longer period. From the frequency calculated in the previous segment, we establish the wave's period \(T\) using the relationship
\(T = \frac{1}{f}\).With \(f\) being approximately \(0.48 \, \mathrm{Hz}\), the period \(T\) is therefore around
\(2.08 \, \mathrm{s}\),which describes the duration of time for one full cycle of this wave to occur.

The direction in which a wave travels can be understood by examining the phase of the sine equation in its wave function. A phase \(kx - \omega t\) indicates a wave moving in the positive x-direction, whereas \(kx + \omega t\) would show motion in the negative x-direction. Given our wave function
\(y = (4.0 \, \mathrm{cm}) \sin(2.0x - 3.0t)\),we see that the negative sign in front of \(3.0t\) signifies the wave is propagating in the positive x-direction. This directional component is crucial for understanding how waves interact with their environments and the orientation of their energy transfer.