3. Solve the two dimensional wave equation utt = 4(uxx +...

3. Solve the two dimensional wave equation u_{tt} = 4(u_{xx} + u_{yy}) on the square 0 < x < pi, 0 < y < pi, with boundary conditions u(0,y,t) = u(pi,y,t) = 0 u_y(x,0,t) = u_y(x,pi,t) = 0, and initial conditions u(x,y,0) = 4 sin(2x) - 3 sin(3x) cos(2y) + 7 sin(8x) cos(10y) u_t(x,y,0) = sin(x) - 3 sin(5x) cos(y).

Transcript

00:01 So we have given here wave equation u the double differentiation with respect to t is equal to u the double differentiation with respect to x plus u double differentiation with respect to y.

00:16 So our equation will be del square u upon del t square is equal to del square u upon del x square plus del square u upon del y square.

00:42 So we have a function u x comma y comma t and that is equal to we have given x of x y y of y into t of t.

01:00 So from here we can write that is 1 upon x d square x upon dx square plus 1 upon y d square y upon dy square and is equal to we have here that is 1 upon t d square t upon dt square and that is is equal to minus of w square.

01:39 Now our equation will be that is 1 upon x d square x upon x square 1 upon y minus of 1 upon y d square y upon dy square minus of omega square is equal to minus of p square.

02:05 Now we can write from here 1 upon x d square x upon dx square is equal to minus of p square.

02:19 Now so from here we can write 1 upon x into d square x upon dx square plus p square is equal to 0.

02:33 And let's say that is our equation number first and implies that from we can write 1 upon y d square and that is capital y upon dy square and that is small y is equal to u square minus of p square.

02:57 So from here we can write that is 1 upon y d square y upon dy square is equal to p square minus omega square.

03:13 So implies that we can write from here that is d square y upon dy square is equal to p square that is small p minus omega square multiplied by y.

03:30 So from here we can write that is d square y upon dy square plus omega square minus p square y is equal to 0.

03:47 Now let's say let omega square minus p square is equal to q...

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