91Ó°ÊÓ

The wave amplitude is a measure of the height of the wave. It's like how tall a wave is from its resting position. In mathematical terms, it's the maximum displacement of the wave from its equilibrium position. When looking at the wave function given in the problem,

• \(y(x, t) = (0.0200~m) \sin\left[ (6.35~m^{-1}) x + (2.63~s^{-1}) t \right]\)

You can see that the wave amplitude is represented by the constant in front of the sine function. Here, the amplitude is

• \(A = 0.0200~m\)

This tells us that the wave reaches up to 0.0200 meters above and below its central rest position. Essentially, if you imagine the wave on a graph, it would move 0.0200 meters up and 0.0200 meters down from the middle line.

The wave period is how long it takes for one complete cycle of the wave to pass a given point. It measures time, typically in seconds, for a repeating pattern to completely repeat itself.
In the wave function

• \(y(x, t) = (0.0200~m) \sin\left[ (6.35~m^{-1}) x + (2.63~s^{-1}) t \right]\)

You can determine the period from the angular frequency \(\omega\). Here, \(\omega = 2.63~s^{-1}\). To find the period \(T\), use the formula

• \(\omega = \frac{2\pi}{T}\)

Solving for \(T\), we find

• \(T = \frac{2\pi}{2.63~s^{-1}} \approx 2.39~s\)

This means each wave cycle takes approximately 2.39 seconds to complete.

The wave wavelength is the distance between two corresponding points on consecutive cycles of the wave, such as peak to peak or trough to trough. It's essentially the length of one complete wave cycle.
Given the wave function

• \(y(x, t) = (0.0200~m) \sin\left[ (6.35~m^{-1}) x + (2.63~s^{-1}) t \right]\)

The wave number \(k\) provides us with this information. Here, \(k = 6.35~m^{-1}\). To find the wavelength \(\lambda\), use the relationship

• \(k = \frac{2\pi}{\lambda}\)

Solving for \(\lambda\), we find

• \(\lambda = \frac{2\pi}{6.35~m^{-1}} \approx 0.990~m\)

Thus, the wavelength is approximately 0.990 meters, indicating how long one wave cycle is in physical space.

Wave speed tells us how fast the wave is moving, which in turn shows how quickly energy is transferred by the wave. It's calculated by multiplying the frequency and wavelength.
In our example, having calculated

• Wavelength \(\lambda = 0.990~m\)
• Period \(T \approx 2.39~s\)

The wave speed \(v\) can be found using the formula

• \(v = \frac{\lambda}{T}\)

Substitute the values to find

• \(v = \frac{0.990~m}{2.39~s} \approx 0.414~m/s\)

This means the wave travels at a speed of approximately 0.414 meters per second.

Wave direction indicates the path along which the wave energy moves. It is crucial as it impacts how energy is distributed in the medium through which the wave is moving.
The sign in front of the term combinations \(kx\) and \(\omega t\) in a wave function often hints at direction. For the given wave equation

• \(y(x, t) = (0.0200~m) \sin\left[ (6.35~m^{-1}) x + (2.63~s^{-1}) t \right]\)

The positive sign between \(kx\) and \(\omega t\) points to the wave moving in the negative x-direction. Typically, the familiar form of a rightward wave is \(y(x,t) = A \sin(kx - \omega t)\).
Since our function instead resembles \(y(x,t) = A \sin(kx + \omega t)\), this indicates movement in the opposite direction, confirming the wave is traveling towards the negative x-direction.