4.150. How long will it take sound waves to travel the distance 1
between the points A and B if the air temperature between them
varies linearly from Ti to T2? The velocity of sound propagation in
air is equal to v = a T, where a is a constant.
4.151. A plane harmonic wave with frequency o propagates at a
velocity v in a direction forming angles a, with the x, y, z axes.
Find the phase difference between the oscillations at the points of
medium with coordinates x1, yi, z1 and x2, y2, z2.
4.152. A plane wave of frequency o propagates so that a certain
phase of oscillation moves along the x, y, z axes with velocities v1,
v2, v3 respectively. Find the wave vector k, assuming the unit vectors
ex, ey, e2 of the coordinate axes to be assigned.
4.153. A plane elastic wave
= a cos (cot — kx) propagates in
a medium K. Find the equation of this wave in a reference frame K'
moving in the positive direction of the x axis with a constant velocity V relative to the medium K. Investigate the expression obtained.
4.154. Demonstrate that any differentiable function f (t
ax),
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
= 60 cos (1800t — 5.3x), where t 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;
(c) 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 = a cos (cot — kx) propagates in a homogeneous elastic medium. For the moment t = 0 draw
(a) the plots of
avat, and OVOx vs x;
(b) the velocity direction of the particles of the medium at the
points where = 0, for the cases of longitudinal and transverse waves;
(c) the approximate plot of density distribution p (x) of the medium
for the case of longitudinal waves.
4.157. A plane elastic wave = ae -Yx cos (cot — kx), where a, 7,
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 1 1 = 1.0%, if Y == 0.42 m -1 and the wavelength is X, = 50 cm.
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 al
and a2. The damping of the wave is negligible, the medium is homogeneous.
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between the points A and B if the air temperature between them
varies linearly from Ti to T2? The velocity of sound propagation in
air is equal to v = a T, where a is a constant.
4.151. A plane harmonic wave with frequency o propagates at a
velocity v in a direction forming angles a, with the x, y, z axes.
Find the phase difference between the oscillations at the points of
medium with coordinates x1, yi, z1 and x2, y2, z2.
4.152. A plane wave of frequency o propagates so that a certain
phase of oscillation moves along the x, y, z axes with velocities v1,
v2, v3 respectively. Find the wave vector k, assuming the unit vectors
ex, ey, e2 of the coordinate axes to be assigned.
4.153. A plane elastic wave
= a cos (cot — kx) propagates in
a medium K. Find the equation of this wave in a reference frame K'
moving in the positive direction of the x axis with a constant velocity V relative to the medium K. Investigate the expression obtained.
4.154. Demonstrate that any differentiable function f (t
ax),
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
= 60 cos (1800t — 5.3x), where t 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;
(c) 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 = a cos (cot — kx) propagates in a homogeneous elastic medium. For the moment t = 0 draw
(a) the plots of
avat, and OVOx vs x;
(b) the velocity direction of the particles of the medium at the
points where = 0, for the cases of longitudinal and transverse waves;
(c) the approximate plot of density distribution p (x) of the medium
for the case of longitudinal waves.
4.157. A plane elastic wave = ae -Yx cos (cot — kx), where a, 7,
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 1 1 = 1.0%, if Y == 0.42 m -1 and the wavelength is X, = 50 cm.
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 al
and a2. The damping of the wave is negligible, the medium is homogeneous.
189