Question

In class, we discussed the application of the two-dimensional wave equation $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} - \frac{1}{v^2} \frac{\partial^2 u}{\partial t^2} = 0$ (1) in polar coordinates $(r, \varphi)$ to vibrations of a circular membrane of radius $a$, as sketched in Fig. 1(a). We found the spatial shapes of various normal modes of vibrations and their characteristic frequencies $\omega$. /a (a) (b) /a /a (c) Figure 1: Schematically: (a) Circular membrane, (b) Semi-circular membrane, (c) Quarter- circular membrane. (a) In this problem you are asked to analyze the normal modes of vibrations of semi-circular and quarter-circular membranes of the same radius $a$, as sketched in Figs. 1(b) and (c), respectively. The membranes are rigidly supported along the thick lines in Fig. 1. (b) In your analysis, clearly delineate the differences with the classroom example of Fig. 1(a). Specify the spatial shapes of various normal modes and their frequencies $\omega$. (c) Compare, in particular, the lowest possible vibrational frequencies for all membranes of Fig. 1 and order them from lower to higher values.

          In class, we discussed the application of the two-dimensional wave equation
$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} - \frac{1}{v^2} \frac{\partial^2 u}{\partial t^2} = 0$
(1)
in polar coordinates $(r, \varphi)$ to vibrations of a circular membrane of radius $a$, as sketched in
Fig. 1(a). We found the spatial shapes of various normal modes of vibrations and their
characteristic frequencies $\omega$.
/a
(a) (b)
/a
/a
(c)
Figure 1: Schematically: (a) Circular membrane, (b) Semi-circular membrane, (c) Quarter-
circular membrane.
(a) In this problem you are asked to analyze the normal modes of vibrations of semi-circular
and quarter-circular membranes of the same radius $a$, as sketched in Figs. 1(b) and (c),
respectively. The membranes are rigidly supported along the thick lines in Fig. 1.
(b) In your analysis, clearly delineate the differences with the classroom example of Fig. 1(a).
Specify the spatial shapes of various normal modes and their frequencies $\omega$.
(c) Compare, in particular, the lowest possible vibrational frequencies for all membranes of
Fig. 1 and order them from lower to higher values.

In class, we discussed the application of the two-dimensional wave equation
(∂^2 u)/(∂ x^2) + (∂^2 u)/(∂ y^2) - (1)/(v^2)(∂^2 u)/(∂ t^2) = 0
(1)
in polar coordinates (r, φ) to vibrations of a circular membrane of radius a, as sketched in
Fig. 1(a). We found the spatial shapes of various normal modes of vibrations and their
characteristic frequencies ω.
/a
(a) (b)
/a
/a
(c)
Figure 1: Schematically: (a) Circular membrane, (b) Semi-circular membrane, (c) Quarter-
circular membrane.
(a) In this problem you are asked to analyze the normal modes of vibrations of semi-circular
and quarter-circular membranes of the same radius a, as sketched in Figs. 1(b) and (c),
respectively. The membranes are rigidly supported along the thick lines in Fig. 1.
(b) In your analysis, clearly delineate the differences with the classroom example of Fig. 1(a).
Specify the spatial shapes of various normal modes and their frequencies ω.
(c) Compare, in particular, the lowest possible vibrational frequencies for all membranes of
Fig. 1 and order them from lower to higher values.

Added by Michael M.

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