Wave Motion 2 Question 32
32. 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 $)$
(1998, 2M)
(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) $a \cos (2 \pi x / L) \sin (\pi y / L)$
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Answer:
Correct Answer: 32. $(b, c)$
Solution:
- Since, the edges are clamped, displacement of the edges $u(x, y)=0$ for
Line,
$$ \begin{aligned} O A \text { i.e. } y & =0 & & 0 \leq x \leq L \\ A B \text { i.e. } x & =L & & 0 \leq y \leq L \\ B C \text { i.e. } y & =L & & 0 \leq x \leq L \\ O C \text { i.e. } x & =0 & & 0 \leq y \leq L \end{aligned} $$
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 $y=L$.
