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The scalar Klein-Gordon equation describes wave motion in a waveguide with acut-off. For example, the displacement of an elastic cord anchored to a solidbase by elastic elements can be described by the scalar Klein-Gordon equation.We analyse this equation using the concept of analytical continuation ofdispersion diagram. Particularly, it is shown that the dispersion diagram istopologically equivalent to a tube analytically embedded in two-dimensionalcomplex space. The corresponding Fourier integral is studied on this tube usingthe Cauchy's theorem. The basic properties of the scalar Klein-Gordon equationare established.

Scalar Klein--Gordon equation and its analytically continued dispersion diagram