The `(x, y)` co-ordinates of the corners of a square plate are `(0, 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)`

Options

A. `a cos ((pi x)/(2 L)) cos ((pi y)/(2 L))`
B. `a sin((pi x)/(L)) sin ((pi y)/(L))`
C. `a sin((pi x)/(L)) sin ((2 pi y)/(L))`
D. `a cos((2 pi x)/(L)) sin ((pi y)/(L))`

Answer

B, C

Explanation

Since, the edges are clamed, 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 ley leL),("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 i nalternatives (b) and (c)
Now that `u(x,y) =0` , for all four value e.g. in alternative `(d),u(x,y)=0` for `y =0,y=L` but is not zero for 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 options (b) and (c), `u(x,y) =0` for `x =0,y=0,x=L` and `y =L`