If [tex]\Psi[/tex] = f(x, y) and x = g(t), y = h(t), then (d[tex]\Psi[/tex] / dt) = (p[tex]\Psi[/tex] / px)(dx / dt) + (p[tex]\Psi[/tex] / py)(dy / dt), where "p" symbolizes the partial derivative here and throughout the rest of this posting.
Derive the phase velocity equation +/- v = -(p[tex]\Psi[/tex] / pt) / (p[tex]\Psi[/tex] / px)
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[tex]\Psi[/tex] / 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 [tex]\Psi[/tex]? I know that [tex]\Psi[/tex] describes a surface wave, and that the absolute value of the expression (p[tex]\Psi[/tex] / pt) equals [tex]\omega[/tex], the angular frequency of the wave, and that the absolute value of (p[tex]\Psi[/tex] / px) = k, the propagation number.