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Elastic Waves. Acoustics.

Problem 4.157

A plane elastic wave ξ=ae−γxcos⁡(ωt−kx),\xi=a e^{-\gamma x} \cos (\omega t-k x), where a,γ,a, \gamma, ω,\omega, and kk are constants, propagates in a homogeneous medium. Find the phase difference between the oscillations at the points where the particles' displacement amplitudes differ by η=1.0%,\eta=1.0 \%, if γ=\gamma= =0.42 m−1=0.42 \mathrm{~m}^{-1} and the wavelength is λ=50 cm\lambda=50 \mathrm{~cm}

Reveal Answer
Δφ=−2πγλln⁡(1−η)≈2πηγλ=0.3rad.\Delta \varphi=-\frac{2 \pi}{\gamma \lambda} \ln (1-\eta) \approx \frac{2 \pi \eta}{\gamma \lambda}=0.3 \mathrm{rad} .
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