Show that if the displacement of the waves on the membrane...

Show that if the displacement of the waves on the membrane of width $b$ of Figure $9.3$ is given by the superposition $$ \mathbf{z}=A_{1} \mathrm{e}^{\left.\mathrm{i} \mid \Delta t-\left(\boldsymbol{k}_{1} x+k_{2} y\right)\right]}+A_{2} \mathrm{e}^{\mathrm{i}\left[\Delta t-\left(k_{1} x-k_{2} y\right)\right]} $$ with the boundary conditions $$ z=0 \quad \text { at } \quad y=0 \quad \text { and } \quad y=b $$ then the real part of $\mathbf{z}$ is $$ z=+2 A_{1} \sin k_{2} y \sin \left(\omega t-k_{1} x\right) $$ where $$ k_{2}=\frac{n \pi}{b} $$

Show that if the displacement of the waves on the membrane of width $b$ of Figure $9.3$ is given by the superposition
$$
\mathbf{z}=A_{1} \mathrm{e}^{\left.\mathrm{i} \mid \Delta t-\left(\boldsymbol{k}_{1} x+k_{2} y\right)\right]}+A_{2} \mathrm{e}^{\mathrm{i}\left[\Delta t-\left(k_{1} x-k_{2} y\right)\right]}
$$
with the boundary conditions
$$
z=0 \quad \text { at } \quad y=0 \quad \text { and } \quad y=b
$$
then the real part of $\mathbf{z}$ is
$$
z=+2 A_{1} \sin k_{2} y \sin \left(\omega t-k_{1} x\right)
$$
where
$$
k_{2}=\frac{n \pi}{b}
$$

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Key Concepts

Superposition Principle

The superposition principle states that complex waveforms can be constructed by adding together simpler wave solutions. In the context of membrane vibrations, two traveling waves interfere to form a resultant wave, which can exhibit phenomena such as standing waves, depending on how their phases and directions combine.

Complex Representation of Waves and Euler's Formula

Using complex exponentials to represent waves simplifies the handling of oscillatory functions. Euler’s formula, which relates complex exponentials to sines and cosines, allows the extraction of the physical (real) part of a wave function. This approach is particularly useful in deriving standing wave patterns from the combination of traveling waves.

Boundary Conditions and Quantization

Boundary conditions restrict the allowed solutions of wave equations by enforcing specific constraints at the edges of a medium like a membrane. For instance, requiring the displacement to vanish at the boundaries leads to quantization of the wave number. This results in discrete values for parameters such as k?, ensuring that the standing wave pattern fits within the physical dimensions of the system.

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