What is the formal definition of a normal mode for a string? And how does this relate to the definition from e.g. wiki that seem to be applied to discrete systmes of particles only? Also on a string what makes: $$y=A\cos(kx)\sin(\omega t)$$ a normal mode, and $$y=A\sin(\omega t+kx)$$ not? (I know why the firt is a statioary wave and the second is not, but that is not whay I am asking here, I am spefically concerned with the definition of normal modes).
Normal modes are the separable solutions to the string's (linear) partial differential equation
$$y(t) = X(x)T(t)$$
that arise from applying the solution method of separation of variables.
These solutions form an orthogonal (normal) basis for any solution.
Due to the form, a function of space only multiplied by a function of time only, the shape of the mode does not change with time, only the amplitude.
- Each moving part oscillates with the same frequency
- Every moving part crosses the equilibrium at the same time.
A normal mode of a system is a pattern of motion (to borrow Wikipedia's term) where every point of the system oscillates with the same frequency and are in phase with each other (with the caveat that some points of the system may have a negative amplitude, which is equivalent to having a positive amplitude but being 180 degrees out of phase).
Your second equation does not meet the criteria for a normal mode because different points will be out of phase with one another.