432
8 Morphogenesis
The particular solution is found by plugging the Fourier series
¯
v( ¯
x) =
∞
n=1
¯
v n sin
nπ ¯
x
¯
L
¯
M g ( ¯
x) =
∞
n=1
¯
M gn sin
nπ ¯
x
¯
L
(8.52)
into Eq. (8.50) to get
¯
v n =
¯
M gn
nπ
¯
L
2
nπ
¯
L
4
+ ¯
N
nπ
¯
L
2
+ ¯
k
,
n= 1, 2, . . . ,
(8.53)
where
¯
M gn =
2
¯
L
¯
e+ ¯
d
¯
e
¯
M g ( ¯
x) sin
nπ ¯
x
¯
L
d ¯
x
= −
4a
nπ
cos
nπ
¯
L
( ¯
d + ¯
e) − cos
nπ ¯
e
¯
L
,
n= 1, 2, . . . .
Because each term in the series for ¯
v satisfies all four boundary conditions, the
complementary solution is not needed.
To illustrate fundamental behavior, results are shown for a = 0.2, L = 10h,
d = 0.1L, and 1000 terms in the series. The series representation for M g (Fig. 8.17a)
captures the discontinuities in growth quite accurately. To isolate the effects of
tension, the deflection is first plotted for k = 0 and increasing values of N
(Fig. 8.17b, c). For N = 0, the curvature changes only in the region of cell wedging
at the middle of the plate. Because the supports do not resist rotation, the rest of the
plate deflects downward but remains straight. As N increases, the straight segments
are pulled upward (like pulling a rope), curving them and turning the wedging region
into a local invagination that becomes shallower with increasing tension.
The effects of the foundation are shown for N = 0 (Fig. 8.17d). The effects of
increasing k appear similar to those caused by increasing N , i.e., the invagination
becomes shallower and more local. However, whereas the deflection remains
downward everywhere with increasing tension, the compressed springs in the
foundation push the invaginating cells upward, causing upward deflection outside
the invagination. Consistent with the behavior illustrated in Fig. 8.17c, d, it can be
shown for N > 0 and k = 0 that the solution to Eq. (8.49) decays exponentially
with distance from a local disturbance, while setting N = 0 and k > 0 leads to an
exponentially decaying oscillation (see Problem 8.3).
8 Morphogenesis
The particular solution is found by plugging the Fourier series
¯
v( ¯
x) =
∞
n=1
¯
v n sin
nπ ¯
x
¯
L
¯
M g ( ¯
x) =
∞
n=1
¯
M gn sin
nπ ¯
x
¯
L
(8.52)
into Eq. (8.50) to get
¯
v n =
¯
M gn
nπ
¯
L
2
nπ
¯
L
4
+ ¯
N
nπ
¯
L
2
+ ¯
k
,
n= 1, 2, . . . ,
(8.53)
where
¯
M gn =
2
¯
L
¯
e+ ¯
d
¯
e
¯
M g ( ¯
x) sin
nπ ¯
x
¯
L
d ¯
x
= −
4a
nπ
cos
nπ
¯
L
( ¯
d + ¯
e) − cos
nπ ¯
e
¯
L
,
n= 1, 2, . . . .
Because each term in the series for ¯
v satisfies all four boundary conditions, the
complementary solution is not needed.
To illustrate fundamental behavior, results are shown for a = 0.2, L = 10h,
d = 0.1L, and 1000 terms in the series. The series representation for M g (Fig. 8.17a)
captures the discontinuities in growth quite accurately. To isolate the effects of
tension, the deflection is first plotted for k = 0 and increasing values of N
(Fig. 8.17b, c). For N = 0, the curvature changes only in the region of cell wedging
at the middle of the plate. Because the supports do not resist rotation, the rest of the
plate deflects downward but remains straight. As N increases, the straight segments
are pulled upward (like pulling a rope), curving them and turning the wedging region
into a local invagination that becomes shallower with increasing tension.
The effects of the foundation are shown for N = 0 (Fig. 8.17d). The effects of
increasing k appear similar to those caused by increasing N , i.e., the invagination
becomes shallower and more local. However, whereas the deflection remains
downward everywhere with increasing tension, the compressed springs in the
foundation push the invaginating cells upward, causing upward deflection outside
the invagination. Consistent with the behavior illustrated in Fig. 8.17c, d, it can be
shown for N > 0 and k = 0 that the solution to Eq. (8.49) decays exponentially
with distance from a local disturbance, while setting N = 0 and k > 0 leads to an
exponentially decaying oscillation (see Problem 8.3).