Fourier transform of an assumed solution to a propagating wave

[erba]

We have a wave ψ(x,z,t). At t = 0 we can assume the wave to have the solution (and shape)

ψ = Q*exp[-i(kx)]
where k = wavenumber, i = complex number

The property for a Fourier transform of a time shift (t-τ) is

FT[f(t-τ)] = f(ω)*exp[-i(ωτ)]

Now, assume ψ(x,z,t) is shifted in time. Thus, ψ(x,z,t) = P(x,z,t-τ). And I want to express the shifted version of ψ (i.e. P) in the frequency domain. Me myself think this is done as:

FT[P(x,z,t-τ)] = Q(x,z,ω)*exp[-i(kx)]*exp[-i(ωτ)] = Q(x,z,ω)*exp[-i(kx - ωτ)]

Though, my teacher claims that, when we assume the solution to be Q(x,z,ω)*exp[-i(kx - ωτ)], we have not necessarily transformed the it into the Fourier domain yet. He says that this solution can be given to a wave in the spatial-time domain as well.
I claim, and think that we have transformed it! But, I guess that my teacher is right and I am wrong.

So please, could you guide me through my confusion here?

PS. Hope the explanation of my question is good enough. Otherwise, ask me :)

 

[perplexabot]

Now, assume ψ(x,z,t) is shifted in time. Thus, ψ(x,z,t) = P(x,z,t-τ). And I want to express the shifted version of ψ (i.e. P) in the frequency domain. Me myself think this is done as:

FT[P(x,z,t-τ)] = Q(x,z,ω)*exp[-i(kx)]*exp[-i(ωτ)] = Q(x,z,ω)*exp[-i(kx - ωτ)]

Hey! Correct me if I am wrong but you are trying to get the Fourier transform of ψ(x,z,t-T)? but you solved for ψ(x,z,t-T), where t=0, or ψ(x,z,T). Or is that what you are trying to do?