Are these travelling waves?

[shanepitts]

Homework Statement

Consider the following expressions:
(a) y(z,t)=A{sin[4π(t+z)]}^2
(b) y(z,t)=A(z-t)
(c) y(z,t)=A/[bz^(2)-t]

Which of them describe traveling waves? Prove it. Moreover, for the expressions that represent waves find the magnitude and direction of wave velocity.

Homework Equations

y(z,t)=A(kx-wt)

v=w/k

The Attempt at a Solution

 

[Simon Bridge]

Did you want to know something specific or are you just inviting comment? It is important to say.

I think you have worked harder than you needed to.
Definition: A shape of form $y=f(z)$ traveling in the +z direction with speed $v$ has form $y(z,t)=f(z-vt)$
... this will be a traveling wave if it also satisfies the wave equation. (Do all such functions satisfy the wave equation?)

... a definition like that allows positive values of v to mean that the waveform propagates in the positive z direction - making it easier to keep track of minus signs.

For (a): $y(z,t)=A\sin^2 4\pi(t+z)$ ... this is a traveling wave with form $f(z)=A\sin^2 4\pi z$
This means that $z-vt = t+z \implies v=1\text{ (unit)}$ ... i.e. the wave propagates in the +z direction.
See how that somes easily?

There is also no need to go into wave numbers and angular frequencies.
You don't need the $\pm$ sign in your definitions unless you insist that the constants $\omega$ and $k$ can only take positive values.

Fortunately you don't have to prove that (c) is not a traveling wave.