Help!!! To model standing waves on a rectangular plate, we can use our usual app

Question

Help!!!

To model standing waves on a rectangular plate, we can use our usual approach of separating the problem into two parts, one along each perpendicular axis. For the following 2 sets of conditions, find the expressions for waves which are combinations of the lowest three frequency modes in each direction. For both cases, the length in the x direction is 1 meter and the length in the y direction is 0.75 meters. Case 1: The edges at x=0 and y=0 are fixed (stationary). The other 2 edges are free.

Z(x,y,t) = A sin(kx)sin(jy)

Case2: All edges are fixed.

Explanation / Answer

case 1: so we know that it will be some combination of sin and cos

so Z = ( A sin kx + B cos kx) ( C sin jy + D cos jy)

firs condition x = 0, Z = 0

so Z = 0 = B*(C sin jy + D cosjy) but has to be true for all y so B = 0

similary for y =0, so D = 0

so Z = A sin kx * sin jy

case 2: has the x = 0 and y = 0 condition

but now has x = 1 and y = 0.75 Z = 0

x=1 condition

A sin k sin j y = 0

so k = n pi

y = 3/4 condition

so sin 3 j/4 = 0

so 3 j/4 = m pi

so j = 4 m pi/3

so Z = A sin n pi x sin 4 m pi x/3 where n and m are integers

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