Sound WavesHard
Question
The (x, y) coordinates of the corner of a square plate are (0, 0), (L, 0), (L, L) and (0, L). The edge of the plate are clamped and transverse standing waves are st 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(πx / 2L) cos(πy / 2L)
B.a sin(πx / L) sin (πy / L)
C.a sin(πx / L) sin (2πy / L)
D.a cos(2πx / L) sin (πy / L)
Solution
Since, the edge are clamed, displacement of the edges u(x, y) = 0 for
Line, OA i.e., y = 0 0 ≤ x ≤ L
AB i.e., x = 0 0 ≤ y ≤ L
BC i.e., y = L 0 ≤ x ≤ L
OC i.e., x = 0 0 ≤ y ≤ L
The above conditions are satisfied only in alternative
(b) and (c).
Note that u(x, y) = 0 for all values eg, 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 option (b)
and (c), u(x, y) = 0 for x = 0, y = 0, x = L and y = L.
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