Standing wave equation phase shift and boundary conditions

[elemis]

Both of the highlighted equations deal with a standing wave. However, they are slightly different in the sense that the latter has a phase shift in it.

Why ?

Also, how does one go from the latter equation for a standing wave to :

2A*sinkx*sinwt. For a string with both ends fixed

And

2A*sinkx*coswt. For one end fixed and the other free

And

2A*coskx*coswt. For both free.

 

[Simon Bridge]

eq. 4.4.2 deals with the specific situation where the amplitude of the wave is zero everywhere at t=0, and the point x=0 is always a node.

However, t=0 is just when we happen to press the button on the stopwatch and x=0 is just where we happen to put the ruler at the time. These are arbitrary: the wave in front of you could be at any stage in it's motion at t=0 for eg. and the ruler may not start at a node.

The phase term is needed to account for the possibility that at t=0, and x=0, the wave is not in such a tidy state.

The other possibilities come from the phase terms ... you change from a sine to a cosine by changing phase by $\pi/2$.