Final answer:
The expressions for wave functions of a sinusoidal wave traveling in the negative x-direction are based on the wave function sin(kx + wt + p), and their sum satisfies the linear wave equation, which confirms the combined function is a valid wave solution with velocity v.
Step-by-step explanation:
The question involves finding expressions for wave functions for a sinusoidal wave traveling along a string in the negative x-direction and verifying these expressions with respect to the linear wave equation. A general wave function for a sinusoidal wave traveling in the negative x-direction is given by y (x, t) = A sin(kx + wt + p), where A is the amplitude, k is the wave number (related to the wavelength), w (omega) is the angular frequency, t is time, p (phi) is the phase constant, and x is the position along the string. The linear wave equation which these functions must satisfy is ñt2 y = v^2 ñx2 y, where v is the wave velocity.
When adding two wave functions such as y1 (x, t) = A sin (kx - wt) and y2 (x, t) = A sin (kx + wt + p), the resultant wave is a superposition of these two waves. By applying trigonometric identities, we can show that this sum is still a solution to the linear wave equation. The superposed wave has a wave velocity of v, and its wave function will be a mathematical combination of the two original wave functions.