91Ó°ÊÓ

Wave speed is a fundamental property of waves, describing how quickly they travel through a medium. It's represented by the symbol \( v \) and is expressed in meters per second (m/s). In our exercise, the transverse waves on the string have a given speed of 8.00 m/s.

Wave speed is related to two primary characteristics of a wave:

• Wavelength (\( \lambda \)): The distance over which the wave's shape repeats. For our wave, \( \lambda = 0.320 \, \text{m} \).
• Frequency (\( f \)): The number of cycles per second, measured in Hertz (Hz).

These properties are linked through the formula for wave speed:\[v = f \times \lambda\\]Using this relation, you can calculate any of the properties if the other two are known. Understanding wave speed allows us to predict how a wave will propagate through a medium over time.

To calculate the frequency of a wave, you can use the formula:\[f = \frac{v}{\lambda}\\]This equation demonstrates that frequency is the wave speed divided by the wavelength. In our specific exercise, substituting the known values gives:\[f = \frac{8.00}{0.320} = 25.0 \, \text{Hz}\\]Frequency tells us how many times a wave oscillates each second. It is a key variable that not only defines a wave's behavior, but also influences other characteristics like the wave's period.

• Period (\( T \)): The time it takes for one full cycle of the wave, calculated as the reciprocal of the frequency: \( T = \frac{1}{f} = 0.040 \, \text{s} \). The period is inversely proportional to frequency; hence, a high frequency means a short period and vice versa.

The wave function describes the shape and behavior of a wave at any given point in space and time. It provides a complete mathematical model for what the wave does. For a wave traveling in the -x direction, the general form is:\[y(x,t) = A \sin(kx + \omega t + \phi)\\]Here's what each term signifies:

• \( A \): Amplitude, the maximum displacement from the equilibrium position. Here \( A = 0.0700 \, \text{m} \).
• \( k \): Wave number, calculated by \( k = \frac{2\pi}{\lambda} \approx 19.63 \, \text{m}^{-1} \). It measures how many wave cycles occur in a unit distance.
• \( \omega \): Angular frequency, given by \( \omega = 2\pi f \approx 157.08 \, \text{rad/s} \).
• \( \phi \): Phase constant, which determines the wave's initial phase. In this exercise, it's set to \( \phi = \frac{\pi}{2} \) since the particle is at its highest point at \( t=0 \).

The specific wave function for our exercise becomes:\[y(x,t) = 0.0700 \sin(19.63x + 157.08t + \frac{\pi}{2})\\]This function can be used to compute the wave's position at any location and time, giving deep insight into its dynamic behavior.

Transverse displacement refers to the vertical shift of a point on the string from its equilibrium position as the wave passes. Calculating this involves substituting the desired values for position \( x \) and time \( t \) into the wave function derived earlier.

For instance, in this specific problem, you need to find the displacement at \( x = 0.360 \, \text{m} \) and \( t = 0.150 \, \text{s} \):\[y(0.360, 0.150) = 0.0700 \sin(19.63 \times 0.360 + 157.08 \times 0.150 + \frac{\pi}{2})\\]By performing the calculations, we find that the displacement at these coordinates is approximately 0.048 m.

This tells us how much a point on the string has moved from its resting position at that specific place and time. Understanding transverse displacement helps in visualizing wave motion and examining how different points on a medium move in response to a wave's energy. This is crucial in fields like engineering, meteorology, and even music, where wave behavior is fundamental.