4.149. A wooden core (Fig. 4.36) supports two coils: coil 1 with
inductance L 1 and short-circuited coil 2 with active resistance R
and inductance L 2. The mutual inductance of the coils depends on
the distance x between them according to the law L12 (x). Find the
mean (averaged over time) value of the interaction force between
the coils when coil 1 carries an alternating current I, = / 0 cos wt.
4.3. Elastic Waves. Acoustics
- Equations of plane and spherical waves:
=a cos (o)t— kx), t=-- —ao
r cos (cot—kr). (4.3a)
In the case of a homogeneous absorbing medium the factors e-yx and e-vr res-
pectively appear in the formulas, where y is the wave damping coefficient.- Wave equation:
, 32
(1) ,2 ±—+—=--- ay 2 (^) az 2 v2^ a t2 • (4.3b
- Phase velocity of longitudinal waves in an elastic medium (v 11 ) and trans-
verse waves in a string (v 1 ):
ull = -"VP' v1 = T/P1' (4.3c)
where E is Young's modulus, p is the density of the medium, T is the tension of
the string, Pi is its linear density.- Volume density of energy of an elastic wave:
w = peep sine (0t — kx), (w) = 1/ 2pa^2 (0^2.^ (4.3d) - Energy flow density, or the Umov vector for a travelling wave:
j = wv, (j)= ii2pa^2 (0^2 v.^ - Standing wave equation:
= a cos kx•cos cot. (4.3f) - Acoustical Doppler effect:
v+
vo (^) v —v (^) • (4.3g)
vob
s
- Loudness level (in bels):
L = log (///o). (4.3h) - Relation between the intensity I of a sound wave and the pressure oscil-
lation amplitude (4p)m:
(4.3e)I = (4p)m/2pv. (4.3i)