A guitar string of length L is plucked in such a way that the total wave produced is the sum of the fundamental and the second harmonic. That is, the standing wave is given by y?? ,? ?t?) = ?y?1 (?x,? ?t?)? ?)y?2(?x,? t where y?1(? ?? ?) ? ? sin ???1?t? sin ?k? 1 ?x y?2(? ?? ?) ? ? sin? ? t sin ?? ?x with ???1 = ??k?1 and ???2 = ??k?2 (a) At what values of ?x are the nodes of ?y?1? (b) At what values of ?x are the nodes of y ?? ? (c) Graph the total wave at ?t = 0, (d) Does the sum of the two standing waves ?y1 ? and ?y2? produce a standing wave? Explain.
Solution 79P Introduction In this question we have to first find the nodes for the standing waves. Then we have to draw the graph of the wave for the given times and then we have to discuss if two standing wave makes a standing wave or not. Step 1 The notes are the point for which y(x,t) is zero for all t. Hence for the fundamental mode we can write that y(x) = C sin(k1x) = 0 Now for fundamental mode, we know that = 2L, hence the wavenumber is given by 1 k = 2= 2 1 1 2L Using this values in the above equation we have y(x) = C sin(2L) = 0 x L n x = nL = 0,L Hence the position of the nodes for y1are 0 and L.