Question

The \[\left( {x,y} \right)\] coordinates of the corners of a square plate are \[\left( {0,0} \right)\], \[\left( {L,0} \right)\], \[\left( {L,L} \right)\] and \[\left( {0,L} \right)\]. The edge of the plate is clamped and transverse standing waves are set up in it. If \[u\left( {x,y} \right)\] deNote:s the displacement of the plate at \[\left( {x,y} \right)\] some instant in time, the possible expression(s) for u is(are)
(A) \[a\cos \left( {\dfrac{{\pi x}}{{2L}}} \right)\cos \left( {\dfrac{{\pi y}}{{2L}}} \right)\]
(B) \[a\sin \left( {\dfrac{{\pi L}}{L}} \right)\sin \left( {\dfrac{{\pi y}}{L}} \right)\]
(C) \[a\cos \left( {\dfrac{{\pi x}}{L}} \right)\sin \left( {\dfrac{{\pi y}}{L}} \right)\]
(D) None of these

Answer 1

Hint: The displacement of the edges \[u\left( {x,y} \right) = 0\]. We will find all the points of edges and put all the points in each option individually and check whether \[u\left( {x,y} \right) = 0\]. The option which satisfied all the edges of the square plate will be considered as the correct answer.

Complete step by step answer:
It is given in the question that the \[\left( {x,y} \right)\] coordinates of the corners of a square plate are \[\left( {0,0} \right)\], \[\left( {L,0} \right)\], \[\left( {L,L} \right)\] and \[\left( {0,L} \right)\]. The edge of the plate is clamped and transverse standing waves are set up in it. \[u\left( {x,y} \right)\] deNote:s the displacement of the plate at \[\left( {x,y} \right)\] some instant of time then we have to find the possible expression for u –

We know that A standing wave occurs when an incident wave meets a reflected wave on a string.
As it is mentioned in the question that the edges of the plate are clamped and transverse standing waves are set up in it. Then the displacement of the edges \[u\left( {x,y} \right) = 0\] for If we see the line AB, we have \[Y = 0;0 \leqslant X \leqslant L\]
Similarly, if we see the line DC, we have \[Y = L;0 \leqslant X \leqslant L\]
Inline BC, we have \[X = L;0 \leqslant Y \leqslant L\]
Similarly, in line AD, we have \[X = 0;0 \leqslant Y \leqslant L\].
Now, we will check all the options individually by putting the values of coordinates of \[\left( {x,y} \right)\] that if the value of \[u = 0\].
In option A we have \[a\cos \left( {\dfrac{{\pi x}}{{2L}}} \right)\cos \left( {\dfrac{{\pi y}}{{2L}}} \right)\]. Now we will check it along the line AB, if we put the value of y as 0, we get-
\[a\cos \left( {\dfrac{{\pi x}}{{2L}}} \right)\cos \left( {\dfrac{{\pi 0}}{{2L}}} \right) = a\cos \left( {\dfrac{{\pi x}}{{2L}}} \right)\cos 0\].
We know that the value of \[\cos 0 = 1\], it means the value of option a U does not equal to 0.
So, option A is incorrect.
In option B we have \[a\sin \left( {\dfrac{{\pi x}}{{2L}}} \right)\sin \left( {\dfrac{{\pi y}}{L}} \right)\]. Now we will check it along the line AB, if we put the value of Y as O, we get
\[a\sin \left( {\dfrac{{\pi x}}{{2L}}} \right)\sin \left( {\dfrac{{\pi 0}}{L}} \right) = a\sin \left( {\dfrac{{\pi x}}{{2L}}} \right)\sin 0 = 0\].
We will check option B along the line BC and put the value of X as L we get-
\[a\sin \left( \pi \right)\sin \left( {\dfrac{{\pi y}}{L}} \right) = 0\]
We will check option B along the line CD and put Y equals to L we get-
\[a\sin \left( {\dfrac{{\pi x}}{L}} \right)\sin \left( {\dfrac{{\pi L}}{L}} \right) = a\sin \left( {\dfrac{{\pi x}}{L}} \right)\sin \left( \pi \right) = 0\]
We will check option B along with the line DA and put x equals to 0 we get-
\[a\sin \left( {\dfrac{{\pi 0}}{L}} \right)\sin \left( {\dfrac{{\pi y}}{L}} \right) = a\sin \left( {\dfrac{{\pi y}}{L}} \right)\sin \left( 0 \right) = 0\]
So, option B is satisfied on all the four edges thus, option B is correct.
Now we will check option C \[a\cos \left( {\dfrac{{\pi x}}{L}} \right)\sin \left( {\dfrac{{\pi y}}{L}} \right)\]along with BC. we will put L in place of x we get-
\[a\cos \left( {\dfrac{{\pi L}}{L}} \right)\sin \left( {\dfrac{{\pi y}}{L}} \right) = a\cos \left( \pi \right)\sin \left( {\dfrac{{\pi y}}{L}} \right)\]
We know that \[\cos \left( \pi \right)\] does not equal 0 it means option C does not satisfy the required condition of \[u\left( {x,y} \right) = 0\]. So, option C is incorrect.

Only option B satisfied all the points therefore, option C is correct.

Additional information:
A standing wave has some points that remain flat due to destructive interference. These are called antinodes. whereas the points on a standing wave that have reached maximum oscillation do so from constructive interference and are called nodes. Every point in the medium containing a standing wave oscillates up and down and the amplitude of the oscillations depends on the location of the point.

Note:
The solution is quite complex, so the probability of making mistakes is quite high. One can easily make a mistake in substituting the values of coordinates in options and this will change the whole answer. Thus, it is strongly recommended to substitute value carefully and cross-check the value again after doing the calculation. This may consume some time but always remember that getting the correct answer on the first attempt will surely give you an edge over doing silly mistakes in doing fast calculations.