Deriving the phase velocity of a wavefunction

[buffordboy23]

Homework Statement

If $$\Psi$$ = f(x, y) and x = g(t), y = h(t), then (d$$\Psi$$ / dt) = (p$$\Psi$$ / px)(dx / dt) + (p$$\Psi$$ / py)(dy / dt), where "p" symbolizes the partial derivative here and throughout the rest of this posting.

Derive the phase velocity equation +/- v = -(p$$\Psi$$ / pt) / (p$$\Psi$$ / px)

Homework Equations

The Attempt at a Solution

This answer is given in the back of the book, but I am trying to understand both mathematically and physically about what is happening during this derivation.

The solution: Let y = t, then through substitution the equation is derived.

Question #1: Is let y = at, where a = constant, a better a solution since the result leads to the same derived equation?

Question #2: What happens physically if we let y = a(t^b), where a and b are both constants? Substitution reveals that the grouping of terms with the variable y become b(p$$\Psi$$ / pt), which leads to the same derived equation but with the right-hand side multiplied by the constant b. This result shows that the phase velocity in the x-spatial dimension relies on the phase velocity in the y-spatial dimension, but can this be? Is this alternate derivation due to superposition of the x- and y-spatial dimension component waveforms?

Question #3: To derive the required equation, I understand that the equation must be transformed into terms of x and t. In deriving the equation using the solution from question one, what are we saying both mathematically and physically about $$\Psi$$? I know that $$\Psi$$ describes a surface wave, and that the absolute value of the expression (p$$\Psi$$ / pt) equals $$\omega$$, the angular frequency of the wave, and that the absolute value of (p$$\Psi$$ / px) = k, the propagation number.

 

[BigJDubs]

Question #4: What is the physical meaning of the minus sign in the derived equation? I think it's the direction of phase propagation, but I'm not sure.