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Introduction to the Physics of Waves by Tim Freegarde

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12

Boundary conditions

12.1 The imposition of external constraints

We have established in Chapter 11 that the motion of an unobstructed wave is generally constrained by two conditions, which, for our guitar-string example, are the continuity of the wave displacement ψ and of its first derivative dψ/dx. For a string to follow a wave motion in which ψ were discontinuous would involve the finite extension of the arbitrarily small element containing the discontinuity, and hence require that that element be stretched beyond its linear range, causing it to snap. Similarly, were the wave motion to include a region of discontinuity in dψ/dx, the corresponding arbitrarily small element would experience a finite force, and thus arbitrarily large ...

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