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Elastic Waves. Acoustics.

Problem 4.160

4.160.4.160 . Two plane waves propagate in a homogeneous elastic medium, one along the xx axis and the other along the yy axis: ξ1=\xi_{1}= =acos⁡(ωt−kx),ξ2=acos⁡(ωt−ky).=a \cos (\omega t-k x), \quad \xi_{2}=a \cos (\omega t-k y) . Find the wave motion pattern of particles in the plane xyx y if both waves (a) are transverse and their oscillation directions coincide; (b) are longitudinal.

Reveal Answer
4.160. (a) See Fig. 34a34 a. The particles of the medium at the points lying on the solid straight lines (y=x±nλ,n=0,1,2,…)(y=x \pm n \lambda, n=0,1,2, \ldots) oscillate with maximum amplitude, those on the dotted lines do not oscillate at all. (b) See Fig. 34b34 b. The particles of the medium at the points lying on the straight lines y=x±nλ,y=x±(n±1/2)λy=x \pm n \lambda, y=x \pm(n \pm 1 / 2) \lambda and y=y= =x±(n±1/4)λ=x \pm(n \pm 1 / 4) \lambda oscillate respectively along those lines, at right angles to them, or move along the circles (here n = 0, 1, 2, . . .). At all other points the particles move along the ellipses.
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