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The two-dimensional wave equation string is describing the vibrations of an infinite 2²n 20² n Ət² дх2 where n = n(x, t), -∞ < x < ∞ and c> 0 is a constant. Show that the general solution can be written as n(x, t) = F(x- ct) + G(x + ct) with F and G arbitrary functions. Explain the physical interpretation of F and G in the overall shape of the string. = If the string's initial position is given by n(x, t velocity by (x, t = 0) = 0 determine F, G and hence n(x, t). 0) 1+4x² = and its initial

The two-dimensional wave equation string is describing the vibrations of an infinite 2²n 20² n Ət² дх2 where n = n(x, t), -∞ < x < ∞ and c> 0 is a constant. Show that the general solution can be written as n(x, t) = F(x- ct) + G(x + ct) with F and G arbitrary functions. Explain the physical interpretation of F and G in the overall shape of the string. = If the string's initial position is given by n(x, t velocity by (x, t = 0) = 0 determine F, G and hence n(x, t). 0) 1+4x² = and its initial

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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The two-dimensional wave equation describing the vibrations of an infinite string is 2²n_ 20²m Ət² əx² where n = n(x, t), -∞< x < ∞ and c> 0 is a constant. (i) Show that the general solution can be written as n(x, t) = F(x - ct) + G(x + ct) with F and G arbitrary functions. Explain the physical interpretation of F and G in the overall shape of the string. (ii) If the string's initial position is given by n(x, t = 0) = 1 1+4x² velocity by (x, t = 0) = 0 determine F, G and hence n(x, t). Ət (iii) Now consider a semi-infinite string 0 < x <∞ which in addition to the wave equation satisfies the boundary condition an -(x = 0, t) = 0 and its initial əx for all t > 0. If at t = = 0 the string is at rest and has displacement n(x, t 0) = x(5x)e-², find F, G and hence n(x, t), stating clearly the solution for x < ct and x > ct.
Transcribed Image Text:The two-dimensional wave equation describing the vibrations of an infinite string is 2²n_ 20²m Ət² əx² where n = n(x, t), -∞< x < ∞ and c> 0 is a constant. (i) Show that the general solution can be written as n(x, t) = F(x - ct) + G(x + ct) with F and G arbitrary functions. Explain the physical interpretation of F and G in the overall shape of the string. (ii) If the string's initial position is given by n(x, t = 0) = 1 1+4x² velocity by (x, t = 0) = 0 determine F, G and hence n(x, t). Ət (iii) Now consider a semi-infinite string 0 < x <∞ which in addition to the wave equation satisfies the boundary condition an -(x = 0, t) = 0 and its initial əx for all t > 0. If at t = = 0 the string is at rest and has displacement n(x, t 0) = x(5x)e-², find F, G and hence n(x, t), stating clearly the solution for x < ct and x > ct.
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