# StraussPDEch2s1p6.pdf - Strauss PDEs 2e Section 2.1...

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Strauss PDEs 2e: Section 2.1 - Exercise 6 Page 1 of 1 Exercise 6 In Exercise 5, find the greatest displacement, max x u ( x, t ), as a function of t . Solution Note that Exercise 5 is called, “The hammer blow.” The string is at rest with zero displacement up until t = 0 when the hammer strikes. Thus, we will only be looking for positive values of t in this exercise. As can be seen from the graphs of the solution in Exercise 5, the greatest displacement of the string, max x u ( x, t ), occurs at x = 0 for all t > 0. The solution of the wave equation was written there as u ( x, t ) = 1 2 c ˆ x + ct x - ct ψ ( s ) ds, where ψ ( x ) was the given initial condition, ψ ( x ) = ( 1 | x | < a 0 | x | ≥ a . If we plug in x = 0 to this, we get u (0 , t ) = 1 2 c ˆ ct - ct ψ ( s ) ds. This integral has different values depending on what
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