– Equations of plane and spherical waves:
\[
t=a \cos (\omega t-k x), \quad t=\frac{c}{r} \cos (\omega t-k r) .
\]

In the case of a homegeneous absorbing medium the factors $e^{-7 x}$ and $e^{-7}$ res. pectively appear is the formulas, whete $\gamma$ is the wave camping coefficient.
– Wave equation:
– Phase velocity of longitudinal waves in as elastie medium $\left(v_{\mu}\right)$ and transverse wave in a string $\left(v_{2}\right)$ :
\[
n_{n}=\sqrt{E / p}, \quad n_{4}=V \overline{F / p_{h}} .
\]
shere $E$ is Youg’, modulus, $p$ is the desaity of the medism, $r$ is the tensien of the string, $h$ is its linear density.
– Volutine density of energy of an elastic wave:
– Energy flow denaity, or the Umov vector for a travelling wave:
– Standibg vave equation:
\[
t=a \cos k r \operatorname{tos} \omega t .
\]
– Acoustical Doppler effect:
\[
\mathrm{v}=\mathrm{v}_{0} \frac{v+v_{0}}{v-v_{4}} .
\]
– Leodness level (is bels):
\[
t=\log (I I I) \text {. }
\]
– Relation between the intensity $I$ of a sousd wave and the pressure oscillation amplitude $(\Delta p)$ s:
\[
I=(\Delta p) \mathrm{a} / 2 p \mathrm{~s}
\]
4.150. How long will it take sound waves to travel the distance $l$ between the points $A$ and $B$ if the air temperature between them varies linearly from $T_{1}$ to $T_{2}$ ? The velocity of sound propagation is air is equal to $v=a V \bar{T}$, where $\alpha$ is a constant.
4.151. A plane harmonic wave with frequency a propagates at a velocity $v$ in a direction forming angles $\alpha, \beta, \gamma$ with the $x, y, z$ axes. Find the phase diflerence between the oseillations at the points of medium with coondiates $x_{1}, y_{1}, z_{1}$ and $x_{2}, y_{2}, x_{2}$.
4.152. A plane wave of frequency $\omega$ propagates so that a certain phase of oscillation moves along the $x, y, z$ axes with velocities $v_{b}$. $V_{4}, v_{2}$ respectively. Find the wave vector $k$, assuming the anit vectors $e_{n}, e_{c} e_{\text {, }}$, of the coordinate axes to be assigned.
4.153. A plane elastic wave $\mathrm{f}-\mathrm{a} \cos (\omega t-k x)$ propagates in a medium $\boldsymbol{K}$. Find the equation of this wave in a reference frame $K^{\prime \prime}$ moving in the positive direction of the $x$ axis with a constant velocity $Y$ relative to the medium $K$. Investigate the expression obtained.
4.154. Demonstrate that any diflerentiable function $f(t+a x)$. where $a$ is a constant, provides a solution of wave equation. What is the physical meaning of the constant a?
4.155. The equation of a travelling plane sound wave has the form $\mathrm{z}=60 \cos (1800 t-5.3 x)$, where $\mathrm{f}$ is expressed in micrometres, $t$ in seconds, and $x$ in metres. Find:
(a) the ratio of the displacement amplitude, with which the particles of medium oscillate, to the wavelength;
(b) the velocity oscillation amplitude of particles of the medium and its ratio to the wave propagation velocity:
(e) the oscillation amplitude of relative deformation of the medium and its relation to the velocity oscillation amplitude of particles of the medium.
4.156. A plane wave $\mathrm{g}-a \cos (\omega t-k x)$ propagates in a homoseneous elastic medlum. For the moment $t=0$ draw
(a) the plots of $\mathrm{E}$. $\partial \mathrm{J} / \partial t$, and $\partial \mathrm{J} / \partial \mathrm{xs} \mathrm{x}$;
(b) the velocity direction of the particles of the mediam at the points where $\xi=0$, for the cases of longitudinal and transverse waves;
(e) the approximate plot of density distribution $p(x)$ of the medium for the case of longitudinal waves.
4.157. A plane elastie wave $\mathrm{g}-e^{-\gamma^{x}} \cos (\omega t-k x)$, where $a, \gamma$. o, and $k$ are constants, propagates in a homogeneous medium. Find the phase dillerence between the oscillations at the points where the particles’ displacement amplitudes differ by $\eta=1.0 \%$, if $\gamma=$ $=0.42 \mathrm{~m}^{-1}$ and the wavelength is $\lambda=50 \mathrm{~cm}$.
4.15s. Find the radius vecter 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 $r_{1}$ and $r_{1}$ at which the oscillation amplitudes of particles of the medium are equal to $a_{1}$ and $a_{2}$. The damping of the wave is negligible, the medium is homogeneous.
4.159. A point isotropic source generates sound oscillations with frequency $\mathrm{v}=1.45 \mathrm{kHz}$. At a distance $r_{4}=5.0 \mathrm{~m}$ from the source the displacement amplitude of particles of the medium is equal to $a_{9}=50 \mu \mathrm{m}$, and at the point $A$ located at a distance $r=10.0 \mathrm{~m}$ from the source the displacement anplitude is $\eta=3.0$ times less than $a_{4}$. Find:
(a) the damping coefficient $\gamma$ of the wave;
(b) the velocity oscillation amplitude of particles of the medium at the point $\boldsymbol{A}$.
4.160. Two plane waves propagate in a homogeneons elastic medium, one along the $x$ axis and the other along the $y$ axis: $z_{1}=$ – a $\cos (e t-k x), \quad \xi_{1}=a \cos (a t-k y)$. Find the wave motion pattern of particles in the plane $x y$ if both waves
(a) are transverse and their escillation directions coineide;
(b) are longitudiaal.
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 displacement maximum.
4.162. A point isotropic sound source is located on the perpendicular to the plane of a ring drawn through the centre $O$ of the ring. The distance between the point $O$ and the source is $l=1.00 \mathrm{~m}$, the radius of the ring is $\boldsymbol{R}=0.50 \mathrm{~m}$. Find the mean energy flow across the area enclosed by the ring if at the point $O$ the intensity of sound is equal to $I_{0}-30 \mu \mathrm{W} / \mathrm{m}^{2}$. The damping of the waves is negligible.
4.163. A point isotrepic source with sonic power $P=0.10 \mathrm{~W}$ is located at the centre of a round hollow cylinder with radius $A$. $=1.0 \mathrm{~m}$ and height $h=2.0 \mathrm{~m}$. 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 $t=a \cos k x \cdot \cos \omega t$. Plot:
(a) $t$ and $\partial t / \partial x$ as functions of $x$ at the mements $t=0$ and $t=r / 2$. where $T$ is the oscillation period;
(b) the distribution of density $\rho(x)$ of the medinm at the mements $t=0$ 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 escillations.
4.165. A longitudinal standing wave $\mathrm{t}=a \cos k x-\cos$ et is maintained in a homegeneous medium of density $\rho$. Find the expressions for the space density of
(a) potential energy $w,(x, t)$;
(b) kinetie energy on $(x, n)$.

Plot the space density distribution of the total energy wo in the space between the displacement nodes at the moments $t=0$ and $t=T / 4$, where $T$ is the oscillation period.
190
4.166. A string $120 \mathrm{~cm}$ in length sustains a standing wave, with the points of the string at which the displacement amplitude is equal to $3.5 \mathrm{~mm}$ being separated by $15.0 \mathrm{~cm}$. Find the maximum displacement amplitude. To which overtone do these oscillations correspond?
4.167. Find the ratio of the fundamental tone frequencies of two identical strings after one of them was stretched by $\eta_{1}=2.0 \%$ and the ether, by $\eta_{1}=4.0 \%$. The tension is assumed to be proportional to the elongation.
4.168. Determine in what way and how many times will the fundamental tone frequency of a stretched wire change if its length is shortened by $35 \%$ and the tension increased by $70 \%$.
4.169. To determine the sound propagation velocity in air by acoustic resonance techaique one can use a pipe with a piston and a sonie membrane elosing one of its ends. Find the velecity of sound if the distance between the adjacent positions of the pisten at which resonance is observed at a frequency $\mathrm{v}-2000 \mathrm{~Hz}$ is equal to $t=$ $=8.5 \mathrm{~cm}$.
4.170. Find the number of possible natural oscillations of air column in a pipe whose frequencies lie below $\mathrm{v}_{0}=1250 \mathrm{~Hz}$. The length of the pipe is $l=85 \mathrm{~cm}$. The velocity of sound is $v=340 \mathrm{~m} / \mathrm{s}$. Consider the two cases:
(a) the pipe is closed from one end;
(b) the pipe is opened from both ends.
The open ends of the pipe are assumed to be the antinodes of displacement.
4.171. A copper rod of length $l=50 \mathrm{~cm}$ is elamped at its midpoint. Find the number of natural longitudinal oscillations of the rod in the frequency range from 20 to $50 \mathrm{kHz}$. What are those frequencies equal to?
4.172. A string of mass $m$ is fixed at both ends. The fundamental tone oscillations are excited with circular frequency 0 and maximum displacement amplitude $a_{m o x}$. Find:
(a) the maximum kinetie energy of the string:
(b) the mean kinetic energy of the string averaged over one oscil. lation period.
4.173. A standing wave $t=a$ sin $k x \cdot \cos \omega t$ is maintained in a homogeneous rod with cross-sectional area $S$ and density $p$. Find the total mechanical energy confined between the sections corresponding to the adjacent displacement nodes.
4.174. A source of sonic oscillations with frequency $\mathrm{v}_{4}=1000 \mathrm{~Hz}$ moves at right angles to the wall with a velocity $u=0.17 \mathrm{~m} / \mathrm{s}$. Two stationary receivers $\boldsymbol{R}_{1}$ and $\boldsymbol{R}_{2}$ are located on a straight line. coinciding with the trajectory of the source, in the following succes: sion: $R_{t}$-source- $R_{r}$ wall. Which receiver registers the beatings and what is the beat frequency? The velocity of sound is equal to $v=$ $-340 \mathrm{~m} / \mathrm{s}$.
4.175. A stationary observer receives sonic oscillations from two tuning forks one of which approaches, and the other recedes with