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For a wave equation η(u,v) = f

I'm a little confused with the velocity initial condition; shouldn't the time derivative of η

_{1}(u) + f_{2}(v) where u = x - ct and v = x + ct, consider an initial displacement η = η_{0}(x) and an initial velocity ∂_{t}η = [itex]\dot{η_{0}}(x)[/itex].I'm a little confused with the velocity initial condition; shouldn't the time derivative of η

_{0}(x) be 0?
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