When a wave propagating on a string gets reflected from a fixed rigid support,what is the change in its phase?

The $(x, y)$ coordinates of the corners of a square plate are $(0, 0), (L, 0), (L, L)$ and $(0, L).$ The edges of the plate are clamped and transverse standing waves are set up in it. If $u(x, y)$ denotes the displacement of the plate at the point $(x, y)$ at some instant of time,the possible expression$(s)$ for $u$ is(are) ($a =$ positive constant).

If you set up the seventh harmonic on a string fixed at both ends,how many nodes and antinodes are set up in it?

$A$ wave $y = a \cos (kx + \omega t)$ superimposes on another wave to form a stationary wave having a node at $x = 0$. What is the equation of the other wave?

$A$ wave represented by the equation $y_1 = a \cos(kx - \omega t)$ is superimposed with another wave to form a stationary wave such that the point $x = 0$ is a node. The equation for the other wave is

$A$ standing wave is formed by the superposition of two waves travelling in opposite directions. The transverse displacement is given by $y(x, t) = 0.5 \sin(\frac{5\pi}{4}x) \cos(200\pi t)$. What is the speed of the travelling wave moving in the positive $x$ direction in $m/s$? ($x$ and $t$ are in meter and second,respectively.)