Posts
1.
General
expression for disturbance of a harmonic vibratory rope with angular frequency ω and circular wave-number $k={\omega
}/{c}\;$ is given by: $g(x,t)=A\cos \{(kx-\omega t)+{{\phi }_{0}}\}+B\cos
\{(kx+\omega t)+{{\phi }_{1}}\}$, the so-called standing waves.
The
boundary conditions in this case, wherein the rope is driven at x = 0 and fixed at x = L, are:
At
x = 0: $g(0,t)={{A}_{0}}\cos \omega
t$
At
x = L: $g(L,t)=0$
The
complex representation of the disturbance on the rope is: $G(x,\omega
)=C{{e}^{+ikx}}+D{{e}^{-ikx}}$, here, C and
D are complex constants.
a.
Give
C and D in terms of real constants: A,
B, ${{\phi }_{0}}$, and ${{\phi
}_{1}}$.
b.
Give
the boundary conditions for G(x,ω) at x = 0 and at x = L. Define
C and D in this case.
c.
‘Eigen-modes’
of the rope are possible for a defined angular-frequencies ωr. For these frequencies, with the corresponding values
of $k={{{\omega }_{r}}}/{c}\;$, as A0
is moving to 0 a finite solution of g(x,t)
is exist (except at x = 0 and at x = L). Define the value of k and ωr for possible eigen-modes with the help of the
boundary conditions: $G(0,\omega )=G(L,\omega )=0$.
d.
Define
for ω ≠ ωr with the values of C and D from (b), the complex expression for disturbance
G(x,ω) in terms of A0, k and L.
e.
Define
g(x,t).
Solution:
1. A rectangular membrane is fixed
on its four sides. The coordinates of its corner are: (0,0); (a,0); (a,b); and (0,b). The disturbance g(x,y) on the membrane
satisfies the wave equation:
$\frac{{{\partial
}^{2}}g}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}g}{\partial
{{y}^{2}}}=\frac{1}{{{c}^{2}}}\frac{{{\partial }^{2}}g}{\partial {{t}^{2}}}$
On its edge applies that g = 0. We assume that the membrane vibrates
harmonically with angular frequency ω.
a. The
complex amplitude G(x,y)
satisfies a differential equation. Please derive this differential equation.
b.
Assume
that the complex amplitude G(x,y) can be written as a product of:
$G(x,y)={{G}_{0}}{{H}_{1}}(x){{H}_{2}}(y)={{G}_{0}}({{e}^{i{{k}_{x}}x}}+p{{e}^{-i{{k}_{x}}x}})({{e}^{i{{k}_{y}}y}}+q{{e}^{-i{{k}_{y}}y}})$
i. Define
the boundary conditions for H1(x) and H2(y).
ii. Define
from (i) first the conditions for p
and q, then show that for kx and ky the conditions: ${{k}_{x}}=\frac{\pi m}{a}$ and ${{k}_{y}}=\frac{\pi
n}{b}$ with m = 0,1,2,3,... and n = 0,1,2,3,... are valid.
c. For
which angular frequency ω = ωnm in the wave equation is
valid? Also, give the lowest frequency f0
whereby the wave equation is valid.
d. Define
the disturbance g(x,y,t) as a function of the coordinates
when the membrane vibrates harmonically with the lowest frequency f0 and the maximum
disturbance at the center of the membrane is g0.
Solution:
