Re
exp i k z t
a
ik
j
k
z
t
j
dk
0
0
~
exp
k k
.
dk
0
Comparing the last expression with the initial form of the wave packet,
ikz
~
q ( )
0 Re a e j dk Re exp
exp
,
(6.35)
j
k
ik z 0 j ak ikzj dk
and taking into account that the phase factors before the integrals in the last forms of Eqs. (34) and (35) do not affect its envelope, we see that in this approximation, the envelope sustains its initial form and propagates along the system with the so-called group velocity
d
v
.
(6.36) Group
gr
k k
dk
0
velocity
Except for the acoustic wave limit (31), this velocity, which characterizes the propagation of the waveform’s envelope, is different from the phase velocity (29), which describes the propagation of the carrier wave, e.g., the spatial position of one of its zeros – see the red and blue arrows in Fig. 6.11
Next, for our particular dispersion relation (30), the difference between v ph and v gr increases as
approaches max, with the group velocity (36) tending to zero, while the phase velocity stays almost constant. The physics of such a maximum frequency available for the wave propagation may be readily understood by noticing that according to Eq. (30), at = max, the wave number k equals n/ d, where n 11 Taking into account the next term in the Taylor expansion of the function ( q), proportional to d 2/ dq 2, we would find that the dispersion leads to a gradual change of the envelope’s form. Such changes play an important role in quantum mechanics, so that they are discussed in detail in the QM part of these lecture notes.
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is an odd integer, and hence the phase shift kd is an odd multiple of . Plugging this value into Eq.
(28), we see that at = max, the oscillations of two adjacent particles are in anti-phase, for example: q ( t)
a
i t
q t
a
i i t
q t
(6.37)
0
Re
exp
,
( )
1
Re
exp
( ).
0
It is clear, especially from Fig. 4b for longitudinal oscillations, that at such a phase shift, all the springs are maximally stretched/compressed (just as in the hard mode of the two coupled oscillators analyzed in Sec. 1), so that it is natural that this mode has the highest possible frequency.
This fact invites a natural question: what happens with the system if it is agitated at a frequency
> max, say by an external force exerted on its boundary? Reviewing the calculations that have led to the dispersion relation (30), we see that they are all valid not only for real but also for any complex values of k. In particular, at > max it gives
2 n 1 i
d
k
,
where n ,
1 ,
2 ,...,
3
Λ
(6.38)
d
Λ
cosh
2
1
- max .
/
Plugging this relation into Eq. (28), we see that the wave’s amplitude becomes an exponential function of the particle’s position:
q
j Im
a e
kd
z
.
(6.39)
j
exp
/
j
Physically this means that penetrating into the structure, the wave decays exponentially (from the excitation point), dropping by a factor of e 3 at the so-called penetration depth . (According to Eq.
(38), at ~ max this depth is of the order of the distance d between the adjacent particles, and decreases but rather slowly as the frequency is increased beyond max.) Such a limited penetration is a very common property of waves, including the electromagnetic waves penetrating into various plasmas and superconductors, and the quantum-mechanical de Broglie waves penetrating into classically-forbidden regions of space. Note that this effect of “wave expulsion” from the medium’s bulk does not require any energy dissipation.
Finally, one more fascinating feature of the dispersion relation (30) is its periodicity: if the relation is satisfied with some wave number k 0(), it is also satisfied with any kn() = k 0() + 2 n/ d, where n is an integer. This property is independent of the particular dynamics of the system and is a common property of all systems that are d-periodic in the usual (“direct”) space. It has especially important implications for the quantum de Broglie waves in periodic systems – for example, crystals – leading, in particular, to the famous band/gap structure of their energy spectrum.12
6.4. Acoustic waves
Now let us return to the limit of low-frequency, dispersion-free acoustic waves, with << 0, propagating with the frequency-independent velocity (31). Such waves are the general property of any elastic continuous medium and obey a simple (and very important) partial differential equation. To derive it, let us note that in the acoustic wave limit, kd << 1,13 the phase shift kd is very close to 12 For more detail see, e.g., QM Sec. 2.5.
13 Strictly speaking, per the discussion at the end of the previous section, in this reasoning k means the distance of the wave number from the closest point 2 n/ d – see Fig. 5 again.
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2 n. This means that the differences qj+1( t) – qj( t) and qj( t) – qj-1( t), participating in Eq. (24), are relatively small and may be approximated with q/ j q/( z/ d) d( q/ z), with the derivatives taken at middle points between the particles: respectively, z+ ( zj+1 – zj)/2 and z- ( zj – zj-1)/2. Let us now consider z as a continuous argument, and introduce the particle displacement q( z, t) – a continuous function of space and time, satisfying the requirement q( zj, t) = qj( t). In this notation, in the limit kd 0, the sum of the last two terms of Eq. (24) becomes – d[ q/ z( z+) – q/ z( z-)], and hence may be approximated as – d 2(2 q/ z 2), with the second derivative taken at point ( z+ – z-)/2 zj, i.e. exactly at the same point as the time derivative. As the result, the whole set of ordinary differential equations (24), for different j, is reduced to just one partial differential equation 2
2
q
q
2
m
d
0 .
(6.40a)
2
ef
2
t
z
Using Eq. (31), we may rewrite this 1D wave equation in a more general form
1
2
2
1D wave
q( z, t) 0
.
(6.40b) equation
2
2
2
v t
z
The most important property of the wave equation (40), which may be verified by an elementary substitution, is that it is satisfied by either of two traveling wave solutions (or their linear superposition): q
,
(6.41)
z, t f t z / v,
q z, t f t z / v
where f are any smooth functions of one argument. The physical sense of these solutions may be revealed by noticing that the displacements q do not change at the addition of an arbitrary change t to their time argument, provided that it is accompanied by an addition of the proportional addition of v t to their space argument. This means that with time, the waveforms just move (respectively, to the left or the right), with the constant speed v, retaining their form – see Fig. 7. 14
q z, t
z v t
t t
t t t
Fig. 6.7. Propagation of a
0
0
traveling
wave
in
a
0
z
dispersion-free 1D system.
Returning to the simple model shown in Fig. 4, let me emphasize that the acoustic-wave velocity v is different for the waves of two types: for the longitudinal waves (with ef = , see Fig. 4b), 1/ 2
v v d ,
(6.42)
l
m
while for the transverse waves (with ef = T/ d, see Fig. 4c):
14 From the point of view of Eq. (40), the only requirement to the “smoothness” of the functions f is to be doubly differentiable. However, we should not forget that in our case the wave equation is only an approximation of the discrete Eq. (24), so that according to Eq. (30), the traveling waveform conservation is limited by the acoustic wave limit condition << max, which should be fulfilled for all Fourier components of these functions.
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1/ 2
1/ 2
1/ 2
T
T d
T
v v
,
(6.43)
t
d
md
m
where the constant m/ d has a simple physical sense of the particle chain’s mass per unit length.
Evidently, these velocities, in the same system, may be rather different.
The wave equation (40), with its only parameter v, may conceal the fact that any wave-supporting system is characterized by one more key parameter. In our current model (Fig. 4), this parameter may be revealed by calculating the forces F( z, t) accompanying any of the traveling waves (41) of particle displacements. For example, in the acoustic wave limit kd 0 we are considering now, the force exerted by the j th particle on its right neighbor may be approximated as q
F( z , t)
( )
( )
,
(6.44)
j
ef q
t
q
t
j
j1
d
ef
z z z j
where, as was discussed above, ef is equal to for the longitudinal waves, and to T/ d for the transverse waves. But for the traveling waves (41), the partial derivatives q/ z are equal to f / v
(where the dot
means the differentiation over the full arguments of the functions f), so that the corresponding forces are equal to
d
ef
F
f
,
(6.45)
v
i.e. are proportional to the particle’s velocities u = q/ t in these waves,15 u = f , for the same z and t.
This means that the ratio
F
z, t
q
/ z
f/ v d
ef
(6.46)
u
d
d
z, t
,
ef
q
/
ef
t
f
v
depends only on the wave propagation direction, but is independent of z and t, and also of the propagating waveform. Its magnitude,
Wave
F z, t
d
ef
1/
Z
,
(6.47)
impedance
u
m
z, t
ef 2
v
characterizing the dynamic “stiffness” of the system for the propagating waves, is called the wave impedance.16 Note that the impedance is determined by the product of the system’s generic parameters
ef and m, while the wave velocity (31) is proportional to their ratio, so that these two parameters are completely independent, and both are important. According to Eq. (47), the wave impedance, just as the wave velocity, is also different for the longitudinal and transverse waves:
d
Z
m
Z
.
(6.48)
l
1/2,
T
t
T1/2
v
v
l
t
15 Of course, the particle’s velocity u (which is proportional to the wave amplitude) should not be confused with the wave’s velocity v (which is independent of this amplitude).
16 This notion is regretfully missing from many physics (but not engineering!) textbooks.
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(Note that the first of these expressions for Z coincides with the one used for a single oscillator in Sec.
5.6. In that case, Z may be also recast in a form similar to Eq. (46), namely, as the ratio of the force and velocity amplitudes at free oscillations.)
One of the wave impedance’s key functions is to scale the power carried by a traveling wave:
q q
d
Traveling
P F z t u z t d
f Zf .
(6.49)
, ,
ef
2
2
ef
z t
v
wave’s
power
Two remarks about this important result. First, the sign of P depends only on the direction of the wave propagation, but not on the waveform. Second, the instant value of the power does not change if we move with the wave in question, i.e. measure P at points with z vt = const. This is natural because in the Hamiltonian system we are considering, the wave energy is conserved. Hence, the wave impedance Z characterizes the energy transfer along the system rather than its dissipation.
Another important function of the wave impedance notion becomes clear when we consider waves in nonuniform systems. Indeed, our previous analysis assumed that the 1D system supporting the waves (Fig. 4) is exactly periodic, i.e. macroscopically uniform, and extends all the way from – to +.
Now let us examine what happens when this is not true. The simplest, and very important example of such nonuniform systems is a sharp interface, i.e. a point (say, z = 0) at which system parameters experience a jump while remaining constant on each side of the interface – see Fig. 8.
f
f '
f
0
z
Fig. 6.8. Partial reflection of a
wave from a sharp interface.
d',v',Z '
d, v, Z
In this case, the wave equation (40) and its partial solutions (41) are is still valid for z < 0 and z > 0 – in the former case, with primed parameters. However, the jump of parameters at the interface leads to a partial reflection of the incident wave from the interface, so that at least on the side of the incidence (in the case shown in Fig. 8, for z 0), we need to use two such terms, one describing the incident wave and another one, the reflected wave:
f ' t z / v'
for
,
z ,
0
q( z, t)
(6.50)
f t z / v f t z / v for
,
z 0.
To find the relations between the functions f–, f+, and f– ’ (of which the first one, describing the incident wave, may be considered known), we may use two boundary conditions at z = 0. First, the displacement q 0( t) of the particle at the interface has to be the same whether it is considered a part of the left or right sub-system, and it participates in Eqs. (50) for both z 0 and z 0. This gives us the first boundary condition:
f '
.
(6.51)
t
f t f t
On the other hand, the forces exerted on the interface from the left and the right should also have equal magnitude, because the interface may be considered as an object with a vanishing mass, and any Chapter 6
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nonzero net force would give it an infinite (and hence unphysical) acceleration. Together with Eqs. (45) and (47), this gives us the second boundary condition:
Z' f '
.
(6.52)
t
Z f t f t
Integrating both parts of this equation over time, and neglecting the integration constant (which describes a common displacement of all particles rather than their oscillations), we get Z'f '
t
Z f t f t .
(6.53)
Now solving the system of two linear equations (51) and (53) for f+( t) and f+ ’( t), we see that both these functions are proportional to the incident waveform:
f
R
(6.54)
t
f t,
f ' t T f t,
with the following reflection ( R ) and transmission ( T ) coefficients: Reflection
Z Z'
2 Z
and
R
,
T
.
(6.55)
transmission
Z Z'
Z Z'
coefficients
Later in this series, we will see that with the appropriate re-definition of the impedance, these relations are also valid for waves of other physical nature (including the de Broglie waves in quantum mechanics) propagating in 1D continuous structures, and also in continua of higher dimensions, at the normal wave incidence upon the interface.17 Note that the coefficients R and T give the ratios of wave amplitudes, rather than their powers. Combining Eqs. (49) and (55), we get the following relations for the powers – either at the interface or at the corresponding points of the reflected and transmitted waves: 2
Z Z'
4 ZZ'
P
P ,
P '
(6.56)
Z Z' -
Z Z' P .
2
-
Note that P– + P+ = P– ’, again reflecting the wave energy conservation.
Perhaps the most important corollary of Eqs. (55)–(56) is that the reflected wave completely vanishes, i.e. the incident wave is completely transmitted through the interface (P+ ’ = P+), if the so-called impedance matching condition Z’ = Z is satisfied, even if the wave velocities v (32) are different on the left and the right sides of it. On the contrary, the equality of the acoustic velocities in the two continua does not guarantee the full transmission of their interface. Again, this is a very general result.
Finally, let us note that for the important particular case of a sinusoidal incident wave:18
f
(6.57)
t Re
i t
a e
,
that
so
f t Re
i t
Ra e
,
where a is its complex amplitude, the total wave (50) on the right of the interface is
i t z / v
i t z / v
q z, t
Re a e
R a e
Re
ikz
ikz
i t
a e R e e , for z 0, (6.58)
17 See, e.g. the corresponding parts of this series: QM Sec. 2.3 and EM Sec. 7.3.
18 In the acoustic wave limit, when the impedances Z and Z’, and hence the reflection coefficient R, are real, R
and Z may be taken from under the Re operators in Eqs. (57)-(59). However, in the current, more general form of these relations they are also valid for the case of arbitrary frequencies, ~ max, when R and Z may be complex.
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while according to Eq. (45), the corresponding force distribution is
F z, t F z, t F z, t Zf t z / v f Z t z / v
Re
ikz
ikz
i t
i Za e
R e
e
. (6.59)
These expressions will be used in the next section.
6.5.Standing waves
Now let us consider the two limits in which Eqs. (55) predicts a total wave reflection ( T = 0): Z’/ Z (when R = –1) and Z’/ Z 0 (when R = +1). According to Eq. (53), the former limit corresponds to f-( t) + f+( t) q(0, t) = 0, i.e. to vanishing oscillations at the interface. This means that this particular limit describes a perfectly rigid boundary, not allowing the system’s end to oscillate at all. In this case, Eqs. (58)-(59) yield
q z, t Re a e ikz
e ikz
e i t 2Re ae i tsin kz,
(6.60)
ikz
ikz
i
t
i
t
/ 2
F z, t
Re i Za
e
e
e
2 Z
Re a e
cos kz
.
(6.61)
These equalities mean that we may interpret the process on the right of the interface using two mathematically equivalent, but physically different languages: either as the sum of two traveling waves (the incident one and the reflected one, propagating in opposite directions), or as a single standing wave.
Note that in contrast with the traveling wave (Fig. 9a, cf. Fig. 7), in the standing sinusoidal wave (Fig.
9b) all particles oscillate in time with the same phase.
(a)
v t
q
t
t t
0
z
q
/ 2
t
t t
(b) Fig. 6.9. The time evolution of
(a) a traveling sinusoidal wave,
and (b) a standing sinusoidal
z
0
wave at a rigid boundary.
Note also that the phase of the force oscillations (61) is shifted, both in space and in time, by /2
relatively to the particle displacement oscillations. (In particular, at the rigid boundary the force amplitude reaches its maximum.) As a result, the average power flow vanishes, so that the average energy of the standing wave does not change, though its instant energy still oscillates, at each spatial point, between its kinetic and potential components – just as at the usual harmonic oscillations of one particle. A similar standing wave, but with a maximum of the displacement q, and with a zero (“node”) of the force F, is formed at the open boundary, with Z’/Z 0, and hence R = +1.
Now I have to explain why I have used the sinusoidal waveform for the wave reflection analysis.
Let us consider a 1D wave system, which obeys Eq. (40), of a finite length l, limited by two rigid walls (located, say, at z = 0 and z = l), which impose the corresponding boundary conditions, Chapter 6
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q( ,
0 t) q( l, t) 0 ,
(6.62)
on its motion. Naturally, a sinusoidal traveling wave, induced in the system, will be reflected from both ends, forming the standing wave patterns of the type (60) near each of them. These two patterns are compatible if l is exactly equal to an integer number (say, n) of /2, where 2/ k is the wavelength:
l n
n .
(6.63)
2
k
This requirement yields the following spectrum of possible wave numbers:
k n ,
(6.64)
n
l
where the list of possible integers n may be limited to non-negative values: n = 1, 2, 3,… (Indeed, negative values give absolutely similar waves (60), while n = 0 yields kn = 0, and the corresponding wave vanishes at all points: sin(0 z) 0.) In the acoustic wave limit we are discussing, Eq. (31), = vk, may be used to translate this wave-number spectrum into an equally simple spectrum of possible standing-wave frequencies:19
v
vk n
,
with n ,
1 ,
2 ,...
3
(6.65)
n
n
l
Now let us notice that this spectrum, and the corresponding standing-wave patterns, 20
n
q z, t 2
Re a exp
sin
,
0
for ,
(6.66)
n
i tn k z
z l
n
may be calculated in a different way, by a direct solution of the wave equation (41) with the boundary conditions (62). Indeed, let us look for the general solution of this partial differential equation in the so-called variable-separated form21
q( z, t) Z z T t ,
(6.67)
n n
n
where each partial product Zn( z) Tn( t) is supposed to satisfy the equation on its own. Plugging such partial solution into Eq. (40), and then dividing all its terms by the same product, ZnTn, we may rewrite the result as
2
2
1 1 d Tn
1 d Z
n
.
(6.68)
2
2
2
v T dt
Z
dz
n
n
Here comes the punch line of the variable separation method: since the left-hand side of the equation may depend only on t, while its right-hand side, only on z, Eq. (68) may be valid only if both its sides are constant. Denoting this constant as – k 2
n , we get two similar ordinary differential equations,22
2
2
d Z
d T
n
2
n
2
2
2
2
k Z ,
0
T ,
0
where v k ,
(6.69)
2
n
n
2
n n
n
n
dz
dt
19 Again, negative values of may be dropped, because they give similar real functions q( z, t).
20 They describe, in particular, the well-known transverse standing waves on a guitar string.
21 This variable separation method is very general and is discussed in all parts of this series, especially in EM
Chapter 2.
22 The first of them is the 1D form of what is frequently called the Helmholtz equation.
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with well-known (and similar) sinusoidal solutions
Z c cos k z s sin k z,
T u cos t v sin z Re
exp
,
(6.70)
n
n
n
n
n
n
n
n
n
n
an i tn
where cn, vn, un, and vn (or, alternatively, an un + ivn) are constants. The first of these relations, with all kn different, may satisfy the boundary conditions only if for all n, cn = 0, and sin knl = 0, giving the same wave number spectrum (64) and hence the own frequency spectrum (65), so that the general solution (67) of the so-called boundary problem, given by Eqs. (40) and (62), takes the form q z, t Re a exp sin
,
(6.71)
n
i tn k zn
n
where the complex amplitudes an are determined by the initial conditions.
Hence such sinusoidal standing waves (Fig. 10a) are not just an assumption, but a natural property of the 1D wave equation. It is also easy to verify that the result (71) is valid for the same system with different boundary conditions, though with a modified wave number spectrum. For example, if the rigid boundary condition ( q = 0) is implemented at z = 0, and the so-called open boundary condition ( F = 0, i.e. q/ z = 0) is imposed at z = l, the spectrum becomes
1
k
(6.72)
n
n ,
with n ,
1 ,
2 ,...,
3
2 l
so that the lowest standing waves look like Fig. 10b shows.23
q
(a)
n
1
0
n 3
n
z
2
l
Fig. 6.10. The lowest standing
q
(b)
n 2
n 1
wave modes for the 1D
systems with (a) two rigid
l
z
boundaries, and (b) one rigid
0
n 3
and one open boundary.
Note that the difference between the sequential values of kn is still a constant:
k
k ,
(6.73)
n1
n
l
the same one as for the spectrum (64). This is natural because in both cases the transfer from the n th mode to the ( n + 1)th mode corresponds just to an addition of one more half-wave – see Fig. 10. (This conclusion is valid for any combination of rigid and free boundary conditions.) As was discussed above, for the discrete-particle chain we have started with (Fig. 4), the wave equation (40), and hence the above derivation of Eq. (71), are only valid in the acoustic wave limit, i.e. when the distance d between the particles is much less than the wavelengths n 2/ kn of the mode under analysis. For a chain of length l, this means that the number of particles, N ~ l/ d, has to be much larger than 1. However, a remarkable 23 The lowest standing wave of the system, with the smallest kn and n, is usually called its fundamental mode.
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property of Eq. (71) is that it remains valid, with the same wave number spectrum (64), not only in the acoustic limit but also for arbitrary N > 0. Indeed, since sin knz (exp{+ iknz} – exp{ -iknz})/2, each n th term of Eq. (71) may be represented as a sum of two traveling waves with equal but opposite wave vectors. As was discussed in Sec. 3, such a wave is a solution of equation (24) describing the discrete-particle system for any kn, with the only condition that its frequency obeys the general dispersion relation (30), rather than its acoustic limit (65).
Moreover, the expressions for kn (with appropriate boundary conditions), such as Eq. (64) or Eq.
(72), also survive the transition to arbitrary N, because their derivation above was based only on the sinusoidal form of the standing wave. The only new factor arising in the case of arbitrary N is that due to the equidistant property (73) of the wave number spectrum, as soon as n exceeds N, the waveforms (71), at particle locations zj = jd, start to repeat. For example,
sin k
z sin k N k
jd sin
.
(6.74)
k N jd sin k z jN sin k z
n N
j
n
n
n j
n
j
d
Hence the system has only N different (linearly-independent) modes. But this result is in full compliance with the general conclusion made in Sec. 2, that any system of N coupled oscillators has exactly N own frequencies and corresponding oscillation modes. So, our analysis of a particular system shown in Fig. 4, just exemplifies this general conclusion. Fig. 11 below illustrates this result for a particular finite value of N; the curve connecting the points shows exactly the same dispersion relation as was shown in Fig. 5, but now it is just a guide for the eye, because for a system with a finite length l, the wave number spectrum is discrete, and the intermediate values of k and do not have an immediate physical sense.24 Note that the own frequencies of the system are generally not equidistant, while the wave numbers are.
n 2
n
n 1
n N n N 1
Fig. 6.11. The wave numbers and
2 / l
own frequencies of a chain of a
finite number N of particles in a
chain with one rigid and one open
0
boundary – schematically.
kn
2 / d
This insensitivity of the spacing (73) between the adjacent wave numbers to the particular physics of a macroscopically uniform system is a very general fact, common for waves of any nature, and is broadly used for analyses of systems with a very large number of particles (such as human-size crystals, with N ~ 1023). For N so large, the effect of the boundary conditions, e.g., the difference between the spectra (64) and (72) is negligible, and they may be summarized as the following rule for the number of different standing waves within some interval k >> / l: 24 Note that Fig. 11 shows the case of one rigid and one open boundary (see Fig. 10b), where l = Nd; for a conceptually simpler system with two rigid boundaries (Fig. 10a) we would need to take l = ( N + 1) d, because neither of the end points can oscillate.
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k
l
standing
N
k
.
(6.75a)
standing
k
k
n 1
n
For such analyses, it is frequently more convenient to work with traveling waves rather than the standing ones. In this case, we have to take into account that (as was just discussed above) each standing wave (66) may be decomposed into two traveling waves with wave numbers kn, so that the interval k doubles, and Eq. (75a) becomes25
l
Traveling
N
k
.
(6.75b) wave
traveling
2
number
Note that this counting rule is valid for waves of just one type. As was discussed above, for the model system we have studied (Fig. 4), there are 3 types of such waves – one longitudinal and two transverse, so that if we need to count them all, N should be multiplied by 3.
6.6 Wave decay and attenuation
Now let us discuss the effects of energy dissipation on the 1D waves, on the example of the same uniform system shown in Fig. 4. The simplest description of this effect is the linear drag that may be described, as it was done for a single oscillator in Sec. 5.1, by adding the term dqj/ dt, to Eq. (24) for each particle:
q
m q ( q
q ) ( q q ) 0 .
(6.76)
j
j
ef
j 1
j
ef
j
j 1
(In a uniform system, the drag coefficient should be similar for all particles, though it may be different for the longitudinal and transverse oscillations.)
To analyze the dissipation effect on the standing waves, we may again use the variable separation method, i.e. look for the solution of Eq. (76) in the form similar to Eq. (67), naturally re-adjusting it for our current discrete case:
q( z , t)
Z z T t .
(6.77)
j
n j n
n
After dividing all terms by mZn( zj) Tn( t) and separating the time-dependent and space-dependent terms, we get
T
T
Z z
Z z
n
n
n
ef
j 1
n j 1
.
(6.78)
T
m T
m
Z z
Z z
n
n
n j
n j
2 const
As we know from the previous section, the resulting equation for the function Zn( zj) is satisfied if the variable separation constant is equal to – 2
n , where n obeys the dispersion relation (30) for the wave
number kn, properly calculated for the dissipation-free system, with the account of the given boundary conditions – see, e.g. Eqs. (62) and (72). Hence for the function Tn( t) we are getting the following ordinary differential equation:
25 Note that this simple, but very important relation is frequently derived using the so-called Born-Carman boundary condition q 0( t) qN( t), which implies bending the system of interest into a closed loop. For a 1D system with N >> 1, such mental exercise may be somehow justified, but for systems of higher dimension, it is hardly physically plausible – and is unnecessary.
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T 2 T
2
T ,
0
with
,
(6.79)
n
n
n
n
2 m
which is absolutely similar to Eq. (5.6b) for a single linear oscillator, which was studied in Sec. 5.1. As we already know, it has the solution (5.9) describing the free oscillation decay with the relaxation time given by (5.10), = 1/, and hence similar for all modes.26
Hence, the above analysis of the dissipation effect on free standing waves has not brought any surprises, but it gives us a hint of how their forced oscillations, induced by some external forces Fj( t) exerted on the particles, may be analyzed. Indeed, representing each of the forces as a sum over the system’s modes (spatial harmonics),
F t
( )
f z , t
j
f t Z z ,
(6.80)
j
n
n j
m
n
and using the variable separation (77), we arrive at the natural generalization of Eq. (79): T
T
2
2
T f
,
(6.81)
n
n
n n
n t
which is identical to Eq. (5.13b) for a single oscillator. This fact enables us to use Eq. (5.27), with G()
Gn(), for the calculation of each Tn( t). Now finding the functions fn( t) from Eq. (80) by the usual reciprocal Fourier transform, and plugging these results into Eq. (77), we get the following generalization of Eq. (5.27):
N
q( z , t) f z t G z z d
G z z
G Z z Z z . (6.82)
j
, j' , ,
,
where
, ,
j
j'
j j'
n
n j n j'
j' 1
0
n
(Here the mutually orthogonal functions Zn( zj) are assumed to be normalized, i.e. the sums of their squares over j = 1, 2,…, N to equal 1.) Such G( zj, zj’,) is called the spatial-temporal Green’s function of the system – in our current case, of a discrete, 1D set of N particles located at points zj = jd. The reader is challenged to spell out this function for at least one of the particular cases discussed above and use it to solve at least one forced-oscillation problem.
Now let us discuss the dissipation effects on the traveling waves, where they may take a completely different form of attenuation. Let us discuss it on a simple example when one end (located at z = 0) of a very long chain ( l ) is externally forced to perform sinusoidal oscillations of a certain frequency and a fixed amplitude A 0. In this case, it is natural to look for a particular solution to Eq.
(76) in a form very different from Eq. (77):
i
q( z , t)
t
Re
,
(6.83)
j
c e
j
with time-independent but generally complex amplitudes cj. As our discussion of a single oscillator in Sec. 5.1 implies, this is not the general, but rather a partial solution, which describes the forced 26 Even an elementary experience with acoustic guitars shows that for their strings this particular conclusion of our theory is not valid: higher modes (“overtones”) decay substantially faster, leaving the fundamental mode oscillations for a slower decay. This is a result of another important energy dissipation (i.e. the wave decay) mechanism, not taken into account in Eq. (76) – the radiation of the sound into the guitar’s body through the string supports, mostly through the bridge. Such radiation may be described by a proper modification of the boundary conditions (62), in terms of the ratio of the wave impedance (47) of the string and those of the supports.
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oscillations in the system, to that it settles after some initial transient process. (At non-zero damping, we may be sure that free oscillations fade after a finite time, and thus may be ignored for most purposes.) Plugging Eq. (83) into Eq. (76), we reduce it to an equation for the amplitudes cj,
2
m i 2 c c
c
,
(6.84)
ef
0
j
ef
j 1
ef
j 1
which is a natural generalization of Eq. (25). As a result, partial solutions of the set of these equations (for j = 0, 1, 2,…) may be looked for in the form (26) again, but now, because of the new, imaginary term in Eq. (84), we should be ready to get a complex phase shift , and hence a complex wave number k / d.27 Indeed, the resulting characteristic equation for k,
2
kd
2
2
sin
i
(6.85)
2
2
2
max
max
(where max is defined by Eq. (30), and the damping coefficient is defined just as in a single oscillator,
/2 m), does not have a real solution even at < max. Using the well-known expressions for the sine function of a complex argument,28 Eq. (85) may be readily solved in the most important low-damping limit << . In the linear approximation in , it does not affect the real part of k, but makes its imaginary part different from zero:
2
1
2
1
k
sin
i
sin
i
,
for Re k
,
(6.86)
d
d
max
max
v
max
with a periodic extension to other periods – see Fig. 5. Just as was done in Eq. (28), due to two values of the wave number, generally we have to take cj in the form of not a single wave (26), but of a linear superposition of two partial solutions:
c
c exp
i Re kz z ,
(6.87)
j
j
j
v
where the constants c should be found from the boundary conditions. In our particular case, when c 0
= A 0 and c = 0, only one of these two waves, namely the wave exponentially decaying at its penetration into the system, is different from zero: c+ = A 0, c– = 0. Hence our solution describes a single wave, with the real amplitude and the oscillation energy decreasing as
2
δ
2
Wave
A c A exp z ,
E A exp
,
with
,
(6.88)
j
j
0
j
j
j
zj
attenuation
v
v
i.e. with a frequency-independent attenuation constant = 2/ v,29 so that the spatial scale of wave penetration into a dissipative system is given by ld 1/. Certainly, our simple solution (88) is only valid for a system of length l >> ld; otherwise, we would need the second term in the sum (87) to describe the wave reflected from its opposite end.
27 As a reminder, we have already met such a situation in the absence of damping, but at > max – see Eq. (38).
28 See, e.g., MA Eq. (3.5).
29 I am sorry to use for the attenuation the same letter as for the phase shift in Eq. (26) and a few of its corollaries, but both notations are traditional.
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6.7 Nonlinear and parametric effects
Now let me discuss (because of the lack of time, very briefly, and on a semi-quantitative level), the new nonlinear and parametric phenomena that appear in oscillatory systems with more than one degree of freedom – cf. Secs. 5.4-5.8. One important new effect here is the mutual phase locking of (two or more) weakly coupled self-excited oscillators with close frequencies: if the own frequencies of the oscillators are sufficiently close, their oscillation frequencies “stick together” to become exactly equal.
Though its dynamics of this process is very close to that of the phase locking of a single oscillator by an external signal, which was discussed in Sec. 5.4, it is rather counter-intuitive in view of the results of Sec. 1, and in particular, the anticrossing diagram shown in Fig. 2. The analysis of the effect using the van der Pol method (which is left for the reader’s exercise) shows that the origin of the difference is the oscillators’ nonlinearity, which makes oscillation amplitudes virtually independent of the phase evolution – see Eq. (5.68) and its discussion.
One more new effect is the so-called non-degenerate parametric excitation. It may be illustrated on the example of just two coupled oscillators – see Sec. 1 above. Let us assume that the coupling constant participating in Eqs. (5) is not constant, but oscillates in time – say with some frequency p.
In this case, the forces acting on each oscillator from its counterpart, described by the right-hand side of Eqs. (5), will be proportional to q 2,1(1 + cosp t). Assuming that the oscillations of q 1 and q 2 are close to sinusoidal ones, with certain frequencies 1,2, we see that the force exerted on each oscillator contains the so-called combinational frequencies
.
(6.89)
p
2 1
,
If one of these frequencies is close to the own oscillation frequency of the oscillator, we can expect a substantial parametric interaction between the oscillators (on top of the constant coupling effects discussed in Sec. 1). According to Eq. (89), this may happen in two cases:
Parametric
,
(6.90a)
p
1
2
interaction
conditions
.
(6.90b)
p
1
2
The quantitative analysis (also highly recommended to the reader) shows that in the case (90a), the parameter modulation indeed leads to energy “pumping” into the oscillations.30 As a result, a sufficiently large , at sufficiently small damping coefficients 1,2 and the effective detuning
( ),
(6.91)
p
1
2
may lead to a simultaneous self-excitation of two frequency components 1,2. These frequencies, while being approximately equal to the corresponding own frequencies 1,2 of the system, are related to the pumping frequency p by the exact relation (90a), but otherwise are arbitrary, e.g., may be incommensurate (Fig. 12a), thus justifying the term non-degenerate parametric excitation.31 (The parametric excitation of a single oscillator, which was analyzed in Sec. 5.5, is a particular, degenerate case of such excitation, with 1 = 2 = p/2.) On the other hand, for the case described by Eq. (90b), the parameter modulation always extracts energy from the oscillations, effectively increasing the system’s damping.
30 Hence the common name of p – the pumping frequency.
31 Note that in some publications, the term parametric down-conversion (PDC) is used instead.
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Somewhat counter-intuitively, this difference between the two cases (90) may be simpler interpreted using the basic notions of quantum mechanics. Namely, the equality p = 1 + 2 enables a decay of an external photon of energy p into two photons of energies 1 and 2 of the oscillators.
On the contrary, the complementary relation (90b), meaning that 1 = p + 2, results in a pumping-induced decay of photons of frequency 1.
(a)
(b)
1
2
2
1
2
0
0
1
2
p
1
2
frequency
1
2
frequency
Fig. 6.12. Spectra of oscillations at (a) the non-degenerate parametric excitation, and (b) the four-wave mixing. The arrow directions symbolize the energy flows into and out of the system.
Note that even if the frequencies 1 and 2 of the parametrically excited oscillations are incommensurate, the oscillations are highly correlated. Indeed, the quantum-mechanical theory of this effect32 shows that the generated photons are entangled. This fact makes the parametric excitation very popular for a broad class of experiments in several currently active fields including quantum computation and encryption, and the Bell inequality/local reality studies.33
Proceeding to nonlinear phenomena, let us note, first of all, that the simple reasoning that accompanied Eq. (5.108) in Sec. 5.8, is also valid in the case when oscillations consist of two (or more) sinusoidal components with incommensurate frequencies. Replacing the notation 2 with p, we see that the non-degenerate parametric excitation of the type (90a) is possible in a system of two coupled oscillators with a quadratic nonlinearity (of the type q 2), “pumped” by an intensive external signal at frequency p 1 + 2. In optics, it is often more convenient to have all three of these frequencies within the same, relatively narrow range. A simple calculation, similar to the one made in Eqs. (5.107)-
(5.108), shows that this may be done using the cubic nonlinearity34 of the type q 3, which allows a similar parametric energy exchange at the frequency relation shown in Fig. 12b:
Four-
2 ,
with .
(6.92a) wave
1
2
1
2
mixing
This process is often called the four-wave mixing, because it may be interpreted quantum-mechanically as the transformation of two externally-delivered photons, each with energy , into two other photons of energies 1 and 2. The word “wave” in this term stems from the fact that at optical frequencies, it is hard to couple a sufficient volume of a nonlinear medium with lumped-type resonators.
It is much easier to implement the parametric excitation (as well as other nonlinear phenomena such as the higher harmonic generation) of light in distributed systems of a linear size much larger than the involved wavelengths. In such systems, the energy transfer from the incoming wave of frequency to 32 Which is, surprisingly, not much more complex than the classical theory – see, e.g., QM Sec.5.5.
33 See, e.g., QM Secs. 8.5 and 10.3, respectively.
34 In optics, such nonlinearity is implemented using transparent crystals such as lithium niobate (LiNbO3), with the cubic-nonlinear dependence of the electric polarization on the applied electric field: P E + E 3.
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generated waves of frequencies 1 and 2 is gradually accumulated at their joint propagation along the system. From the analogy between Eq. (85) (describing the evolution of the wave’s amplitude in space), and the usual equation of the linear oscillator (describing its evolution in time), it is clear that this energy transfer accumulation requires not only the frequencies but also the wave numbers k be in similar relations. For example, the four-wave mixing requires that not only the frequency balance (92a) but also a similar relation
2 k k k ,
(6.92b)
1
2
to be fulfilled. Since all three frequencies are close, this relation is easy to arrange. Unfortunately, due to the lack of time/space, for more discussion of this very interesting subject, called nonlinear optics, I have to refer the reader to special literature.35
It may look like a dispersion-free media, with / k = v = const, is the perfect solution for arranging the nonlinear/parametric interaction of waves, because in such media, for example, Eq. (92b) automatically follows from Eq. (92a). However, in such a medium, not only the desirable three parametrically interacting waves but also all their harmonics, have the same velocity. At these conditions, the energy transfer rates between all harmonics are of the same order. Perhaps the most important result of such a multi-harmonic interaction is that intensive incident traveling waves, interacting with a nonlinear medium, may develop sharply non-sinusoidal waveforms, in particular those with an almost instant change of the field at a certain moment. Such shock waves, especially those of mechanical nature, are of large interest for certain applications – some of them not quite innocent, e.g., the dynamics of explosion in the usual (chemical) and nuclear bombs.36
To conclude this chapter, let me note that the above discussion of 1D acoustic waves will be extended, in Sec. 7.7, to elastic 3D media. There we will see that generally, the waves obey a more complex equation than the apparently natural generalization of Eq. (40):
1 2
2
qr, t 0
,
(6.93)
2
2
v t
where 2 is the 3D Laplace operator. This fact adds to the complexity of traveling-wave and standing-wave phenomena in higher dimensions. Moreover, in multi-dimensional systems, including such pseudo-1D systems as thin rods and pseudo-2D systems such as thin membranes, even static elastic deformations may be very nontrivial. An introduction to the general theory of small deformations, with a focus on elastic continua, will be the subject of the next chapter.
6.8 Exercise problems
For each of the systems specified in Problems 6.1-6.6:
(i) introduce convenient generalized coordinates qj of the system,
35 See, e.g., N. Bloembergen, Nonlinear Optics, 4th ed., World Scientific, 1996, or a more modern treatment by R. Boyd, Nonlinear Optics, 3rd ed., Academic Press, 2008. This field is currently very active. As just a single example, let me mention the recent experiments with parametric amplification of ultrashort (~20-fs) optical pulses to peak power as high as ~51012 W – see X. Zeng et al., Optics Lett. 42, 2014 (2017).
36 The classical (and perhaps still the best) monograph on the subject is Ya. Zeldovich, Physics of Shock Waves and High-Temperature Phenomena, Dover, 2002.
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(ii) calculate the frequencies of its small harmonic oscillations near the equilibrium,
(iii) calculate the corresponding distribution coefficients, and
(iv) sketch the oscillation modes.
6.1. Two elastically coupled pendula, confined to a vertical plane, with
l
l
the parameters shown in the figure on the right (see also Problems 1.8 and 2.9).
g
m
m
l
6.2. The double pendulum, confined to a vertical plane containing the support point (considered in Problem 2.1), with m’ = m and l = l’ – see the figure on the right.
m
l
g
m
l'
6.3 The chime bell considered in Problem 4.12 (see the figure on the right), for the
particular case l = l’.
M
l
g
6.4. The triple pendulum shown in the figure on the right, with the motion
l
m
confined to a vertical plane containing the support point.
l
m
Hint: You may use any (e.g., numerical) method to calculate the characteristic g l
equation’s roots.
m
6.5. A symmetric system of three particles, shown in the figure on 1
2
3
the right, where the connections between the particles not only act as usual
elastic springs (giving potential energies U = ( l)2/2) but also resist m m'
m
bending, giving additional potential energy U’ = ’( l) 2/2, where is the l
l
(small) bending angle.37
m
, l 0
R
6.6. Three similar beads of mass m, which may slide along a circle of radius R without friction, connected with similar springs with elastic constants
,
m
l
and equilibrium lengths l
0
0 (generally not equal to 3 R) – see the figure on the
right.
, l 0
m
37 This is a good model for small oscillations of linear molecules such as the now-infamous CO2.
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6.7. On the example of the model considered in Problem 1, explore free oscillations in a system of two similar and weakly coupled linear oscillators.
6.8. A small body is held by four similar elastic springs as shown
in the figure on the right. Analyze the effect of rotation of the system as a
2
whole about the axis normal to its plane, on the body’s small oscillations
m
within this plane. Assume that the oscillation frequency is much higher
than the angular velocity of the rotation. Discuss the physical sense of
0
/ 2
/ 2
your results, and possible ways of using such systems for measurement
of the rotation.
2
6.9. An external longitudinal force F( t) is
F t
applied to the right particle of the system shown in
m
m
Fig. 1, with L = R = ’ and m 1 = m 2 m (see the
figure on the right), and the response q 1( t) of the left
'
q
q
κ'
particle to this force is being measured.
1
2
(i) Calculate the temporal Green’s function for this response.
(ii) Use this function to calculate the response to the following force:
F t ,
0
for
t ,
0
F sin t,
0
for t,
0
with constant amplitude F 0 and frequency .
6.10. Use the Lagrangian formalism to re-derive Eqs. (24) for both the longitudinal and the transverse oscillations in the system shown in Fig. 4a.
6.11. Calculate the energy (per unit length) of a sinusoidal traveling wave propagating in the 1D
system shown in Fig. 4a. Use your result to calculate the average power flow created by the wave, and compare it with Eq. (49) in the acoustic wave limit.
6.12. Calculate spatial distributions of the kinetic and potential energies in a standing, sinusoidal, 1D acoustic wave, and analyze their evolution in time.
6.13. The midpoint of a guitar string of length l has been slowly pulled off by distance h << l from its equilibrium position, and then let go. Neglecting dissipation, use two different approaches to calculate the midpoint’s displacement as a function of time.
cos2 m
1
2
Hint: You may like to use the following series:
1
,
0
for
.
2
m1
2 m
1
8
/ 2
6.14. Spell out the spatial-temporal Green’s function (82) for waves in a 1D uniform system of N
points, with the rigid boundary conditions (62). Explore the acoustic limit of your result.
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6.15. Calculate the dispersion law ( k) and the maximum
and minimum frequencies of small longitudinal waves in a long g
l
l
l
chain of similar, spring-coupled pendula – see the figure on the
right.
m
m
m
6.16. Calculate and analyze the dispersion relation
( k) for small waves in a long chain of elastically coupled
particles with alternating masses – see the figure on the m
m'
m
m'
right. In particular, discuss the dispersion relation’s period
k, and its evolution at m’ m.
d
d
d
6.17. Analyze the traveling wave’s reflection from a
“point inhomogeneity”: a single particle with a different
mass m
m
m
m
m
0 m, within an otherwise uniform 1D chain – see
0
the figure on the right.
d
d
d
6.18.*
(i) Explore an approximate way to analyze waves in a continuous 1D system with
parameters slowly varying along its length.
(ii) Apply this method to calculate the frequencies of transverse standing waves on
a freely hanging heavy rope of length l, with a constant mass per unit length – see the l
g
figure on the right.
(iii) For the three lowest standing wave modes, compare the results with those obtained in the solution of Problem 4 for the triple pendulum.
Hint: The reader familiar with the WKB approximation in quantum mechanics (see, e.g., QM
Sec. 2.4) is welcome to adapt it for this classical application. Another possible starting point is the van der Pol approximation discussed in Sec. 5.3, which should be translated from the time domain to the space domain.
6.19. A particle of mass m is attached to an infinite string, of mass per unit length, stretched with tension T. The particle is confined to move along the x-axis normal to the string (see the figure below), in an additional potential U( x) with a minimum at x = 0. Assuming that the waves on the string are excited only by the motion of the particle (rather than any external source), reduce the system of equations describing the system to an ordinary differential equation for the small displacements x( t). For the case of a linear oscillator, when U( x) = m2 x 2/2, calculate its Q-factor due to the effective drag caused by the string.
x, q
m
0
z
T
T
6.20.* Use the van der Pol method to analyze the mutual phase locking of two weakly coupled self-oscillators with the dissipative nonlinearity, for the cases of:
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(i) the direct coordinate coupling described by Eq. (5), and
(ii) a bilinear but otherwise arbitrary coupling of two similar oscillators.
Hint: In Task (ii), describe the coupling by a linear operator, and express the result via its Fourier image.
6.21.* Extend Task (ii) of the previous problem to the mutual phase locking of N similar self-oscillators. In particular, explore the in-phase mode’s stability for the case of the so-called global coupling via a single force F contributed equally by all oscillators.
6.22.* Find the condition of non-degenerate parametric excitation in a system of two coupled oscillators described by Eqs. (5), but with a time-dependent coupling: (1 + cosp t), with p
1 + 2, and / m << 2 – 1 .
Hint: Assuming the modulation depth , the static coupling coefficient , and the detuning
p – (1+ 2) sufficiently small, use the van der Pol method for each of the coupled oscillators.
6.23. Show that the cubic nonlinearity of the type q 3 indeed enables the parametric interaction (“four-wave mixing”) of oscillations with incommensurate frequencies related by Eqs. (92a).
6.24. In the first nonvanishing approximation in small oscillation amplitudes, calculate their effect on the own frequencies of the same double-pendulum system that was the subject of Problem 1.
6.25. Calculate the velocity of small transverse waves propagating on a thin, planar, elastic membrane, with mass m per unit area, pre-stretched with force per unit width.
6.26. A membrane discussed in the previous problem is
T
T
stretched on a thin but firm plane frame of area a a. Calculate the frequency spectrum of small transverse standing waves in the system; d
m
T
sketch a few lowest wave modes. Compare the results with those for a
discrete-point analog of this system, with four particles of equal masses d
T
m, connected with light flexible strings that are stretched, with equal tensions T, on a similar frame – see the figure on the right. (The frames d
do not allow the membrane edges/string ends to deviate from their
planes.)
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Chapter 6
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Chapter 7. Deformations and Elasticity
The objective of this chapter is a discussion of small deformations of 3D continua, with a focus on the elastic properties of solids. The reader will see that such deformations are nontrivial even in the absence of their evolution in time, so that several key problems of statics will need to be discussed before proceeding to such dynamic phenomena as elastic waves in infinite media and thin rods.
7.1. Strain
As was already discussed in Chapters 4-6, in a continuum, i.e. a system of particles so close to each other that the system discreteness may be neglected, particle displacements q may be considered as a continuous function of not only time but also space. In this chapter, we will consider only small deviations from the rigid-body approximation discussed in Chapter 4, i.e. small deformations. The deformation smallness allows us to consider the displacement vector q as a function of the initial (pre-deformation) position of the particle, r, and time t – just as was done in Chapter 6 for 1D waves.
The first task of the deformation theory is to exclude from consideration the types of motion considered in Chapter 4, namely the body’s translation and rotation, unrelated to deformations. This means, first of all, that the variables describing deformations should not depend on the displacement’s part that is independent of the position r (i.e. is common for the whole media), because that part corresponds to a translational shift rather than to a deformation (Fig. 1a). Moreover, even certain nonuniform displacements do not contribute to deformation. For example, Eq. (4.9) (with dr replaced with dq to comply with our current notation) shows that a small displacement of the type
dqrotation = dr,
(7.1)
where d = dt is an infinitesimal vector common for the whole continuum, corresponds to its elementary rotation of the body about the direction of that vector, and has nothing to do with its deformation (Fig. 1b).
(a)
(b)
O
dq
q
const
n
translatio
rotation = dr
Fig.
7.1.
Two
types
of
displacement vector distributions
that are unrelated to deformation:
(a) translation and (b) rotation.
This is why to develop an adequate quantitative characterization of deformation, so far for fixed t, we should start with finding suitable functions of the spatial distribution of displacements, q(r), that exist only due to deformations. One of such measures is the change of the distance dl dr between two close points:
© K. Likharev
CM: Classical Mechanics
3
3
2
2
( dl)
– ( dl)
( dr
dq )
( dr ) ,
(7.2)
n
deformatio
after
n
deformatio
before
2
j
j
2
j
j1
j1
where dqj is the j th Cartesian component of the difference dq between the displacements q of these close points. If the deformation is small in the sense dq<< dl, we may keep, in this expression, only the terms proportional to the first power of the infinitesimal vector dq:
3
3
2
2
( dl)
– ( dl)
2 dr dq
( dq )
2
dr dq .
(7.3)
n
deformatio
after
2
n
deformatio
before
j
j
j
j j
j1
j1
Since qj is a function of three independent scalar arguments rj, its full differential (at fixed time) may be represented as
3 q
dq
dr
(7.4)
j
j
.
j'
j' 1
rj'
The coefficients qj/ rj’ may be considered as elements of a tensor providing a linear relation between the vectors dr and dq.1 Plugging Eq. (4) into Eq. (2), we get
3
q
2
2
( dl)
– ( dl)
2
j dr dr .
(7.5)
n
deformatio
after
n
deformatio
before
j
j'
j, j'1 rj'
The convenience of the tensor qj/ rj’ for characterizing deformations is that it automatically excludes the translation displacement (Fig. 1a), which is independent of rj. Its drawback is that its particular elements are still affected by the rotation of the body – even though the sum (5) is not. Indeed, according to the vector product’s definition, Eq. (1) may be represented in Cartesian coordinates as dq
d r d r
,
(7.6)
j rotation
j' j"
j" j'
jj'j"
where jj’j” is the Levi-Civita symbol . Differentiating Eq. (6) over a particular Cartesian coordinate of vector r, and taking into account that this partial differentiation () is independent of (and hence may be swapped with) the differentiation ( d) over the common rotation angle , we get the amounts
q
q
j
d
d
and
j'
d
d
d ,
(7.7)
jj'j"
j"
j'jj"
j"
jj'j"
j"
r
r
j' rotation
j rotation
which may differ from 0. However, notice that the sum of these two differentials equals zero for any d, which is possible only if2
q
q
j'
j
0,
for j j'
.
(7.8)
r
r
j
j' rotation
This is why it is convenient to rewrite Eq. (5) in a mathematically equivalent form,
1 Since both dq and dr are legitimate physical vectors (whose Cartesian components are properly transformed as the transfer between reference frames), the 33 matrix with elements qj/ rj’ is indeed a legitimate physical tensor
– see the discussion in Sec. 4.2.
2 As a result, the full sum (5), which includes three partial sums (8), is not affected by rotation – as we already know.
Chapter 7
Page 2 of 38
CM: Classical Mechanics
3
2
2
( dl)
- ( dl)
2
s dr dr ,
(7.9a)
n
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after
n
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before
jj' j j'
j, j' 1
where sjj’ are the elements of the so-called symmetrized strain tensor, defined as Strain
1 q
q
tensor
s
j
j' .
(7.9b)
jj'
2 r
r
j'
j
(Note that this modification does not affect the diagonal elements sjj = qj/ rj. ). So, the advantage of the symmetrized tensor (9b) over the initial tensor with elements qj/ rj’ is that according to Eq. (8), at pure rotation, all elements of the symmetrized strain tensor vanish.
Now let us discuss the physical meaning of this tensor. As was already mentioned in Sec. 4.2, any symmetric tensor may be diagonalized by an appropriate selection of the reference frame axes. In such principal axes, sjj’ = sjj jj’, so that Eq. (4) takes a simple form: q
j
dq
dr s dr .
(7.10)
j
j
jj
j
r
j
We may use this expression to calculate the change of each side of an elementary cuboid (parallelepiped) with its sides dqj parallel to the principal axes:
dr
dr
dq s dr ,
(7.11)
j
n
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after
j
n
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before
j
jj
j
and of the cuboid’s volume dV = dr 1 dr 2 dr 3:
3
3
3
dV
dV
( dr s dr )
dr
dV
s
(7.12)
j
jj
j
j
n
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after
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before
1 jj 1,
j 1
j 1
j 1
