when a mechanical wave is reflected by a fixed support it experiences a phase change of pi .... without getting into maths can i qualitatively prove this ? i cannot analyse how the forces will be acting .
where c is the wavespeed and [itex] y_1(.), y_2(.) [/itex] can be anything and have to be determined by initial conditions. [itex] y_1(x-ct) [/itex] is a wave moving in the +x direction and [itex] y_2(x+ct) [/itex] is a wave moving in the -x direction and the two waves just add up (superimpose). now let's say that your string is anchored at x=0. that means that
[tex] y(0,t) = y_1(-ct) + y_2(ct) = 0 [/tex]
for all time t. now about the only way for that to happen is if
[tex] y_1(-ct) = -y_2(ct) [/tex]
that means, at x=0, that the wave that is moving in one direction has to be the exact negative of the wave moving in the other direction for them to add to zero and they have to add to zero because of the "fixed support". reversing the polarity is the same as a phase change of [itex] \pi [/itex] radians.
The easiest way to see the pi pahse change is that the incident and reflected wave mus cancel at the fixed wall. This means the reflected wave must be the negative of the incident wave.