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 = posotive constant)

Options

A. (a) `a cos (pix//2L)cos (piy//2L)`
B. (b) `a sin (pix//L)sin (piy//L)`
C. ( c ) `a sin (pix//L)sin (2piy//L)`
D. (d) `a cos (2pix//2L)sin(piy//L)`

Answer

B, C

Explanation

(b,c) Due to the clamping of the square plate at the edges, its displacement along the `x` and `y` axies will individually be zero at the edges. Only the choices `(b) and ( c )` predict these displacements correctly. This is because `sin 0 = 0`
Option (a): `u(x, y) = 0` at `x = L, y = L`
u(x, y) != 0` at `x = 0, y = 0`
Option (b): u(x, y) = 0` at `x = 0, y = 0 [because sin 0 = 0]`
u(x, y) = 0` at `x = L, y = L [because sin pi = 0]`
Option ( c ): u(x, y) = 0` at `x = 0, y = 0 [because sin 0 = 0]`
u(x, y) = 0` at `x = L, y = L [because sin pi = 0, sin pi = 0]`
Option (d) : u(x, y) = 0` at `Y = 0, y = L [because sin = 0, sin pi = 0]`
u(x, y) != 0` at `x = 0, x = L [because cos0 = 1, cos 2pi = 1]`