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pde waves: string, quantum packet, and e&m 485Figure 18.3 The vertical displacement as a function of position x and time t. A string initiallyplaced in a standing wave on a string with friction. Notice how the standing wave moves upand down with time. (Courtesy of J. Wiren.)Modify the program to incorporate this initial condition and see if a normalmode results.6. Observe the motion of the wave for initial conditions corresponding to thesum of two adjacent normal modes. Does beating occur?7. When a string is plucked near its end, a pulse reflects off the ends and bouncesback and forth. Change the initial conditions of the model program to onecorresponding to a string plucked exactly in its middle and see if a travelingor a standing wave results.8. ⊙ Figure 18.4 shows the wave packets that result as a function of time forinitial conditions corresponding to the double pluck indicated on the left inthe figure. Verify that initial conditions of the formy(x, t =0)0.005⎧0, 0.0 ≤ x ≤ 0.1,10x − 1, 0.1 ≤ x ≤ 0.2,−10x +3, 0.2 ≤ x ≤ 0.3,⎪⎨= 0, 0.3 ≤ x ≤ 0.7,10x − 7, 0.7 ≤ x ≤ 0.8,−10x +9, 0.8 ≤ x ≤ 0.9,⎪⎩0, 0.9 ≤ x ≤ 1.0lead to this type of a repeating pattern. In particular, observe whether thepulses move or just oscillate up and down.−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 485