Final answer:
For a sinusoidal wave traveling in the negative x-direction with given amplitude, frequency, and wavelength, the wave function y(x, t) takes the form of y(x, t) = A sin(kx + ωt + φ), considering the initial conditions for y at specific values of t and x.
Step-by-step explanation:
The expression for y as a function of x and t for a sinusoidal wave traveling along a rope in the negative x direction, with the given characteristics, should take the general form of a sinusoidal wave, which is y(x, t) = A sin(kx - ωt + φ). Since we know that y(0, t) = 0 at t = 0 and y(x, 0) = 0 at x = 10.0 cm, we need to incorporate these initial conditions into our wave equation.
To compute the wave speed v, we use the formula v = λf, where λ is the wavelength and f is the frequency. The wave number k is calculated as 2π/λ, and the angular frequency ω is 2πf. However, because the wave travels in the negative x-direction, we have to ensure that the sign in front of the kx term is positive, representing a change in phase as the wave travels. Based on the given amplitude A and considering the phase shifts, the wave function in part (b) assuming y(x, 0) = 0 at x = 10.0 cm will be y(x, t) = A sin(kx + ωt + φ) with the appropriate value of φ to satisfy the initial condition.