a ) We have ( for small amplitude osillayione). ( 1) 1 2 Is = linear mass density T = string tension and , with a trial solution y( x, t ) = s( x) glt ) (1) -> 9"( 4 ) of " ( x ) 9 ( t ) of ( x ) = constant 2 ( 2 ) lindep, of yit ) arbitrary at this ( 2 ) -> g(t) : a cos ( wit) + bo sinkwt) point f (x ) = a cos (lex) +d sin(kx) So the solutions are of the form ens ( wt) cos (lex ) (3) eds lust ) Sin (ks) ( 4 ) Sinlut) cos ( kx ) (5 ) sin(wt ) sin ( tex ) show that ((x,t ) = sin ( wt) sin( tex ) solves ( 1) as long as a Cie, * = + (w / N ) ) 6) We also shamed ( sas p. (2 ) of examples of wave eand , notes ) that forms ( 3 )-5 ( ( ) implay possible solutions of the form : cos ( lex - wit ) ( 7)sin (kx - wt ) (8 ) cos ( kx + wit ) ( 9 ) sin (lex + wt ) ( 10) show that y(x, t) = cos( texawit ) solves (1) ( again, with 12 w/ k? ) ( ) look at the form yex, t ) cos ( lex- wt) = cos( KEx- WE] ) cos ( kix- NE] ) at t= 0, y( x,0)= cos (lex ) = cos ( Do ( x ) ) : a short time At later , we have y( x at) = castlebounty . . . . = cos ( Pat ( XAt ) ) Note : We could plot y(x ,0 ) and y( xz , at ) and we would see that y (x, At ) is the same as y(x,0) but shifted to greater By how much is it shifted? show that Pat ( *it ) = d ( x ) when XAt = X + N At which means the whole wave moves in the + X direction at N (= Nphase = velocity of constant place Points ) ( here or = w/k, or k = w/ N with N = (I ) / from ( 1 ) re .d ) The forms ( 7 ) - ( 10 ) , and sums (or integrals ) of such forms for different K's and w's would imply that any ( differentiable ) functions of the forms 5 ( x mart) and g(xtart) showled salve (1. Show that y(xt) = 5(x-at) solves (1). ( hint : rese the chain rule ) ( ) A solution having one frequency ( w ) only is called a normal mode ( it would give a pure tone a violin string ) . ex: say we have a violin string length 1 = 0.3 3m and u = 1x10 'Kg ( this is an E string ). We hold the string down at X=0 and X mm ( /1) (12 ) so ylot) =0 and y( h, t) = 0 for all t. ") with yext) =asinckx) sin(wt) the form as possible solutions, what values of k satisfy the boundary conditions (1 1) and ( 12 ) ? " ) what is the smallest value of k in ? ) ? What is the corresponding value of w ? If we want the value 27. 5 = 271 ( 660 / s ) what T is here needed