Irodov Solutions → Oscillations and waves → Elastic Waves. Acoustics

4.157. A plane elastic wave ξ = ae^-γx(ωt - kx), where a, γ, ω, and k 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%, if γ = 0.42 m^-1 and the wavelength is λ = 50 cm.
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4.158. Find the radius vector defining the position of a point source of spherical waves if that source is known to be located on the straight line between the points with radius vectors r₁ and r₂ at which the oscillation amplitudes of particles of the medium are equal to a₁ and a₂. The damping of the wave is negligible, the medium is homogeneous.
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4.163. A point isotropic source with sonic power P = 0.10 W is located at the centre of a round hollow cylinder with radius R = 1.0 m and height h = 2.0 m. Assuming the sound to be completely absorbed by the walls of the cylinder, find the mean energy flow reaching the lateral surface of the cylinder.
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4.164. The equation of a plane standing wave in a homogeneous elastic medium has the form ξ = a cos kx * cos ωt. Plot:
(a) ξ and ∂ξ/∂x as functions of x at the moments t = 0 and t = T/2, where T is the oscillation period;
(b) the distribution of density ρ(x) of the medium at the moments t = 0 and t = T/2 in the case of longitudinal oscillations;
(c) the velocity distribution of particles of the medium at the moment t = T/4; indicate the directions of velocities at the antinodes, both for longitudinal and transverse oscillations.
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4.165. A longitudinal standing wave ξ = a cos kx * cos ωt is maintained in a homogeneous medium of density ρ. Find the expressions for the space density of
(a) potential energy w_p (x, t);
(b) kinetic energy w_k (x, t).
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4.174. A source of sonic oscillations with frequency ν₀ = 1000 Hz moves at right angles to the wall with a velocity u = 0.17 m/s. Two stationary receivers R₁ and R₂ are located on a straight line, coinciding with the trajectory of the source, in the following succession: R₁-source-R₂-wall. Which receiver registers the beatings and what is the beat frequency? The velocity of sound is equal to v = 340 m/s.
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4.176. A receiver and a source of sonic oscillations of frequency ν₀ = 2000 Hz are located on the x axis. The source swings harmonically along that axis with a circular frequency ω and an amplitude a = 50 cm. At what value of ω will the frequency bandwidth registered by the stationary receiver be equal to Δν = 200 Hz? The velocity of sound is equal to v = 340 m/s.
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4.179. A stationary source sends forth monochromatic sound. A wall approaches it with velocity u = 33 cm/s. The propagation velocity of sound in the medium is v = 330 m/s. In what way and how much, in per cent, does the wavelength of sound change on reflection from the wall?
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4.185. A plane longitudinal harmonic wave propagates in a medium with density ρ. The velocity of the wave propagation is v. Assuming that the density variations of the medium, induced by the propagating wave, Δρ << ρ, demonstrate that
(a) the pressure increment in the medium Δp = -ρv²(∂ξ/∂x), where ∂ξ/∂x is the relative deformation;
(b) the wave intensity is defined by Eq. (4.3i).
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4.186. A ball of radius R = 50 cm is located in the way of propagation of a plane sound wave. The sonic wavelength is λ = 20 cm, the frequency is ν = 1700 Hz, the pressure oscillation amplitude in air is (Δp)_m = 3.5 Pa. Find the mean energy flow, averaged over an oscillation period, reaching the surface of the ball.
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4.188. At a distance r = 100 m from a point isotropic source of sound of frequency 200 Hz the loudness level is equal to L = 50 dB. The audibility threshold at this frequency corresponds to the sound intensity I₀ = 0.10 nW/m². The damping coefficient of the sound wave is γ = 5.0*10^-4 m^-1. Find the sonic power of the source.
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