Question

A standing wave has maximum amplitude 7 and nodes at 0, /2, a, 31/2, 2T, as shown in the figure. Each point that is not a node moves up and down with period 4T. Find a function of the form y(x, t) A sin(ax) cos(Bt) that models this wave. y(x, t) AA y Need Help? Read It Talk to a Tutor

Answer 1

Solution ; Given that, The The Standing wave Amplitude is = 7 given nodes are =0,5/2, 4, 37/2 , 27 goes up and down with In addition the waves Period "HT". Let we sketch the node bedow.

From the data , we have the Amplitude of the wave is =) A = 7, Here ,ß= 40 because the "cos ßt term is constant when the value of "x' is fixed. This means that it is a scaling factor for how the wave moves up and down . 5) To find 'đ , consider the behaviour of the wave when 't is fixed. This is because the sinax term describe how each individual point behaves at particular time to - As the nodes are at o 7,5, 34 a t the periodic dength must be t', since a period is

the value for one complete cycle. Here to solve - & Period = a0 a T = 20 here, =) a=2 substitute 2=9, and ß=47 and A= 7 the standard form ie; y oct) = Asind accosßt into y= A sin a cosßt, ->) 7 sin accoshat. the function of wave Hence is given by g(x, t) = 7 sin ax cos(htt)