`4.159. A point isotropic source generates sound oscillations with`

frequency v = 1.45 kHz. At a distance r, = 5.0 m from the source

the displacement amplitude of particles of the medium is equal to

a, = 50 pm, and at the point A located at a distance r = 10.0 m

from the source the displacement amplitude is = 3.0 times less

than a 0. Find:

(a) the damping coefficient Y of the wave;

(b) the velocity oscillation amplitude of particles of the medium

at the point A.

4.160. Two plane waves propagate in a homogeneous elastic me-

dium, one along the x axis and the other along the y axis:^1 =

= a cos (cot — kx), 2 = a cos (wt ky). Find the wave motion

pattern of particles in the plane xy if both waves

(a) are transverse and their oscillation directions coincide;

(b) are longitudinal.

4.161. A plane undamped harmonic wave propagates in a medium.

Find the mean space density of the total oscillation energy (w),

if at any point of the medium the space density of energy becomes

equal to w, one-sixth of an oscillation period after passing the dis-

placement maximum.

4.162. A point isotropic sound source is located on the perpendicu-

lar to the plane of a ring drawn through the centre 0 of the ring.

The distance between the point 0 and the source is 1 = 1.00 m, the

radius of the ring is R = 0.50 m. Find the mean energy flow across

the area enclosed by the ring if at the point (^0) the intensity of sound

is equal to / 0 = 30 μW/m 2. The damping of the waves is negligible.

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.

4.164. The equation of a plane standing wave in a homogeneous

elastic medium has the form = a cos kx• cos wt. Plot:

(a) and OVax as functions of x at the moments t = 0 and t = T/2,

where T is the oscillation period;

(b) the distribution of density p (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 mo-

ment t = T/4; indicate the directions of velocities at the antinodes,

both for longitudinal and transverse oscillations.

4.165. A longitudinal standing wave = a cos kx- cos wt is main-

tained in a homogeneous medium of density p. Find the expressions

for the space density of

(a) potential energy wp (x, t);

(b) kinetic energy wk (x, t).

Plot the space density distribution of the total energy w in the space

between the displacement nodes at the moments t = 0 and t = T14,

where T is the oscillation period.

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