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now, the book attempts to explain the solution in the back of the book, but i don't quite get it. it says both a) and b) are waves since they are twice differentiable functions of (z-vt) and (x+vt) respectively. Therefore, for a)Which of the following expressions correspond to traveling waves? For each of those, what is the speed of the wave? The quantities a, b, and c are positive constants.

a) psi(z,t) = (az - bt)^2

b) psi(x,t) = (ax + bt + c)^2

c) psi(x,t) = 1/(ax^2 + b)

**psi = a^2(z - bt/a)^2**and the velocity is b/a in the positive z-direction. For b)

**psi = a^2(x + bt/a + c/a)^2**and the velocity is b/a in the negative x-direction.

i don't understand where the bolded equations come from. I understand how they got the velocity and direction from those equations, but no idea how they were derived.