Final answer:
The amplitude of the wave is 0.005 m, the wavelength 4 m, the frequency 1 Hz, the wave speed 4 m/s, and the direction of travel is in the positive x-direction. The maximum perpendicular speed of the cord's particles is 0.0314 m/s.
Step-by-step explanation:
Given the wave equation y (x, t) = (0.50 cm)sin (1.57 m⁻¹x - 6.28 s⁻¹t), we can determine various characteristics of the wave:
- The amplitude is the coefficient of the sine function which is 0.50 cm (or 0.005 m).
- The wavelength (λ) can be found from the wavenumber k, which is given as 1.57 m⁻¹. Since k = 2π/λ, we find that λ = 2π/1.57 m⁻¹, which simplifies to approximately 4 m.
- The frequency (f) of the wave is determined from the angular frequency ω (6.28 s⁻¹), where f = ω/(2π). This results in a frequency of 1 Hz.
- The wave velocity (v) can be calculated using the formula v = λ * f, which gives us 4 m/s. The direction of wave travel can be inferred from the wave equation, where the negative sign in front of the time variable indicates that the wave is moving in the positive x-direction.
- The maximum speed of the particles in the string perpendicular to the direction of wave motion can be found by differentiating the wave function with respect to time and evaluating it at its maximum value. This gives us a maximum speed of 0.0314 m/s.