Question

QUESTION 2 The vertical vibration at position i and at time t of a stretched homogeneous and infinitely long string is determined by the function u = u(x,y). Suppose that when the string is straight, it has a linear density of 2 and the tension at any given point of the string is 8. a. Derive the partial differential equation satisfied by u. b. If the initial position is 1 – r and the initial velocity is e 1 write down the initial-value problem which models the above situation. c. Use the appropriate formula to solve the initial value problem found in (b).

Answer 1

u T is For infinitely long string, ihe vertical Vibration of string at borition x and at time t if t if u cxit) Then satiopienne following PDE : Uzt - Cunn=0 KGIR, too where of T/e tension of string, e in the deurity (linear) of string. given inat To8 po2, .C'> TVP=4,7).C=2 Hence. Required PDE Uzt -44XX=0 t20. given U (2.0) |-X, Ut(x,0) = becomed Then intial Daluod problem Utt 4 Uxx. HE IR, 620 U(KO)I- KER UE (R16) - e12, nGR. a ☺ XGIR, (lo XER, ēW/2, XER.

x+ 2t © using @'Alembert formula, u(0.6) 4(x+2+) + f (x-2) AS, C=2) 242 J 4 (6) ds + 2 X-2t where (a)? } I-X 4 (2) 042 x+2t G X-2t = (-x) + 4yX(-9) Cop) -Ql+2)/2 Q-20/2 ? (1-2) – ņē > (-x) - 112 [et. 6)/2 е -1/2 t 2 (1-x) + e -t e 2 -X12 (1-x) + ē Sinht et et : Sioht 2 ucxt) - (-X) + ē92 Sinht 2 0