question_answer

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; Orissa PMT 2004]

A)            \[a\cos \frac{\pi x}{2L}\cos \frac{\pi y}{2L}\]             

B)            \[a\sin \frac{\pi x}{L}\sin \frac{\pi y}{L}\]

C)            \[a\sin \frac{\pi x}{L}\sin \frac{2\pi y}{L}\]                  

D)            \[a\cos \frac{2\pi x}{L}\cos \frac{\pi y}{L}\]

Correct Answer: B

Solution :

  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\[y=0\], while in options (b) and (c), \[u(x,y)=0\]for \[x=0,y=0,x=L\] and \[y=L\]