Question 9.1: Determine the angular frequency of the first four modes of a...
Determine the angular frequency of the first four modes of a square membrane of sides a if the speed of propagation is v . What is their degeneracy? Determine the nodal lines of the modes u_{12} and u_{21} . Determine the nodal lines of the linear superposition u_{12} + u_{21} if they have the same amplitude and the same phase.
The normal frequencies are given by [9.22], i.e
k_1=m\pi /a, \ \ \ \ \ k_2=n\pi /b \ \ \ \ \ \text {and} \ \ \ \ \ \omega _{m,n}=\pi v\sqrt{m^2/a^2+n^2/b^2} [9.22]
\omega _{11}=\sqrt{2} \pi v/a, \ \ \ \ \ \omega _{12}=\omega _{21}=\sqrt{5} \pi v/a, \ \ \ \ \ \omega _{22}=\sqrt{8}\pi v/a, \ \ \ \ \ \omega _{13}=\omega _{31}=\sqrt{10} \pi /a.\omega _{11} corresponds to one wave function u_{11} , and \omega _{22} corresponds to one wave function u_{22} ; thus, they are not degenerate. The angular frequency \omega _{12}=\omega _{21} corresponds to two different wave functions u_{12} and u_{21} ; thus, it has degeneracy 2. Any linear superposition of these modes is also a standing wave with the same angular frequency \omega _{12} . Similarly, \omega _{13}=\omega _{31} has degeneracy 2.
Taking the origin of coordinates at the vertex O , we may write:
u_{12}=A \sin(\pi x/a) \sin(2\pi y/a) \cos(\omega _{12}t+\alpha ) \\ u_{21}=A \sin(2\pi x/a) \sin(\pi y/a) \cos(\omega _{12}t+\alpha ) \\ u_{12}+u_{21}=4A \sin(\pi x/a) \sin(\pi y/a) \cos\left[\pi (x+y)/2a\right] \cos\left[\pi (x-y)/2a\right] \cos(\omega _{12}t+\alpha ).There is one nodal line (other than the periphery) y=a/2 \ \text {for} \ u_{12}, the line x=a/2 for u_{21} and the line x+y=a for u_{12}+u_{21} (Figure 9.4).
