# 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. Free solution >> 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. Free solution >> 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. Free solution >> 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. Free solution >> 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 wp (x, t); (b) kinetic energy wk (x, t). Free solution >> 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. Free solution >> 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. Free solution >> 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? Free solution >> 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). Free solution >> 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. Free solution >> 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. Free solution >>