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)
IIT 1998, Diffcult
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(d) Since the edges are clamped, displacement of the edges $u(x,y) = 0$ for line -
$OA$ i.e. $y = 0$, $0 \le x \le L$
$AB$ i.e.$x = L$, $0 \le y \le L$
$BC$ i.e.$y = L$, $0 \le x \le L$
$OC$ i.e.$x = 0$, $0 \le y \le L$
The above conditions are satisfied only in alternatives $(b)$ and $(c).$
Note that $u(x,y) = 0$, for all four values e.g. in alternative $(d)$, $u(x,y) = 0$ for $y = 0,y = L$ but it is not zero for $x = 0$ or $x = L$. Similarly in option $(a).$ $u(x,y) = 0$ at $x = L,y = L$ but it is not zero for $x = 0$ or $y = 0$,
$OA$ i.e. $y = 0$, $0 \le x \le L$
$AB$ i.e.$x = L$, $0 \le y \le L$
$BC$ i.e.$y = L$, $0 \le x \le L$
$OC$ i.e.$x = 0$, $0 \le y \le L$
The above conditions are satisfied only in alternatives $(b)$ and $(c).$
Note that $u(x,y) = 0$, for all four values e.g. in alternative $(d)$, $u(x,y) = 0$ for $y = 0,y = L$ but it is not zero for $x = 0$ or $x = L$. Similarly in option $(a).$ $u(x,y) = 0$ at $x = L,y = L$ but it is not zero for $x = 0$ or $y = 0$,
while in options $(b)$ and $(c),$ $u(x,y) = 0$ for $x = 0,y = 0,x = L$ and $y = L$
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