Final answer:
For the x-variable, the function is expected to be an oscillating solution A0 sin(kx) + B0 cos(kx), and for the y-variable, it depends on whether the system exhibits growth/decay (exponentials) or oscillations (sine and cosine) in that direction.
This correct answer is none of the above.
Step-by-step explanation:
Given the boundary conditions of the problem and some constant k, the general function in the x-variable is expected to be A0 sin(kx) + B0 cos(kx), while the function in the y-variable is expected to be A1 exp(ky) + B1 exp(-ky) or A1 sin(ky) + B1 cos(ky), depending on the nature of the boundary conditions.
The function options suggest different behaviors: exponential functions imply a growing or decaying solution, while sinusoids imply oscillating solutions.
Taking into account the information provided, the expected general function for the wave equation would involve oscillating solutions in the form of sine and cosine for the x-variable, as shown with Yk (x) = Bk sin(kx). If the wave is confined within boundaries, this form must respect the boundary conditions such as Yk (x) = 0 at the endpoints, leading to specific values of k.
As for the y-variable, the form would depend on the specific problem being considered. If the problem expects an oscillating nature in both directions, the y-function should also involve sine and cosine.
However, if it implies growth or decay in the y-direction, then exponential components are expected. Therefore, the correct general function could be either option B or C, depending on the behavior of the system in the y-direction.
This correct answer is none of the above.