Consider now the extension of Problem 9.2 where the waves...

Consider now the extension of Problem $9.2$ where the waves are reflected at the rigid edges of the rectangular membrane of sides length $a$ and $b$ as shown in the diagram. The final displacement is the result of the superposition $$ \begin{aligned} \mathbf{z}=& A_{1} \mathrm{e}^{\mathrm{i}\left[\omega t-\left(k_{1} x+k_{2} y\right)\right]} \\ &+A_{2} \mathrm{e}^{\left.\mathrm{i} \omega t-\left(k_{1} x-k_{2} y\right)\right]} \\ &+A_{3} \mathrm{e}^{\left.\mathrm{i} \mid \mu t-\left(-k_{1} x-k_{2} y\right)\right]} \\ &+A_{4} \mathrm{e}^{\left.\mathrm{i} \| t-\left(-k_{1} x+k_{2} y\right)\right]} \end{aligned} $$ with the boundary conditions $$ z=0 \quad \text { at } \quad x=0 \quad \text { and } \quad x=a $$

Consider now the extension of Problem $9.2$ where the waves are reflected at the rigid edges of the rectangular membrane of sides length $a$ and $b$ as shown in the diagram. The final displacement is the result of the superposition
$$
\begin{aligned}
\mathbf{z}=& A_{1} \mathrm{e}^{\mathrm{i}\left[\omega t-\left(k_{1} x+k_{2} y\right)\right]} \\
&+A_{2} \mathrm{e}^{\left.\mathrm{i} \omega t-\left(k_{1} x-k_{2} y\right)\right]} \\
&+A_{3} \mathrm{e}^{\left.\mathrm{i} \mid \mu t-\left(-k_{1} x-k_{2} y\right)\right]} \\
&+A_{4} \mathrm{e}^{\left.\mathrm{i} \| t-\left(-k_{1} x+k_{2} y\right)\right]}
\end{aligned}
$$
with the boundary conditions
$$
z=0 \quad \text { at } \quad x=0 \quad \text { and } \quad x=a
$$

Key Concepts

Boundary Conditions

Boundary conditions are constraints imposed on the solutions of differential equations that model physical systems, such as vibrations and waves. For a membrane with fixed edges, the requirement that the displacement be zero at the boundaries restricts the possible forms of the solution, leading to quantization of the allowed wave numbers and frequencies.

Standing Wave Modes

Standing wave modes are patterns that emerge from the interference of two waves traveling in opposite directions. In the context of a membrane with fixed edges, these modes exhibit nodes at the rigid boundaries and correspond to specific resonant frequencies, which are determined by the boundary conditions and the geometry of the system.

Superposition Principle

The superposition principle is a fundamental concept in wave theory stating that when two or more waves overlap, the resulting displacement is the sum of the individual displacements. This principle allows complex wave patterns to be analyzed as the sum of simpler, constituent waves, each with its own amplitude, frequency, and phase.

Wave Reflection

Wave reflection occurs when a traveling wave encounters a boundary or an obstacle and bounces back into the medium. In systems with rigid boundaries, such as a membrane with fixed edges, the reflected wave can interfere with incoming waves, leading to patterns of constructive and destructive interference.

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