The Beetheory introduces a novel approach to modelling waves by considering localized functions, such as $\Psi(R, t) = A \cdot e^{-\alpha(RA – A_0)} \cdot e^{i\omega_1 t}$. This function uniquely combines spatial localization (through a Gaussian-like envelope) with temporal oscillations (at a frequency $\omega_1$). While traditional wave modelling often relies on Fourier decomposition into plane waves, the Beetheory extends this by focusing on localized wave modes that are better suited for representing spatially confined phenomena.

This article explores the foundations of this approach, draws analogies with Fourier series decomposition, and demonstrates how it can generalize to represent any spatial wave. It also highlights the scientific motivations and applications of this methodology.

Foundations of Fourier Series Decomposition

Fourier series decomposition is a classical method for representing periodic functions as a sum of sinusoidal components. For a periodic function $f(x)$ of period $T$, the Fourier series is given by:

$$f(x) = a_0 + \sum_{n=1}^\infty \left( a_n \cos\left(\frac{2\pi n x}{T}\right) + b_n \sin\left(\frac{2\pi n x}{T}\right) \right),$$

where the coefficients $a_n$ and $b_n$ capture the contributions of cosine and sine terms, respectively. Fourier analysis is crucial for describing oscillatory phenomena but has limitations when applied to non-periodic or spatially confined functions.

The Beetheory builds upon the Fourier decomposition by addressing these limitations. Rather than representing a wave as an infinite summation of plane waves, it introduces localized wave modes that are better suited for capturing spatially confined oscillations.

Generalizing the Concept: Localized Wave Decomposition

Traditional Plane Wave Representation

In classical wave theory, any spatially varying function $\Psi(R, t)$ can be represented as a superposition of plane waves:

$$\Psi(R, t) = \int_{-\infty}^\infty \Phi(k, t) \, e^{i k R} \, dk,$$

where:

• $k$ is the wave vector or spatial frequency,
• $\Phi(k, t)$ is the spectral amplitude, representing the contribution of the wave vector $k$,
• $e^{i k R}$ is the plane wave oscillation corresponding to $k$.

This decomposition is widely used but assumes that the waves extend infinitely in space, which is unrealistic in most physical systems. The Beetheory proposes an alternative based on localized wave modes.

Localized Wave Representation

Instead of relying solely on plane waves, the Beetheory introduces localized wave functions that combine a spatial envelope and oscillatory components. A single localized wave mode can be expressed as:

$$\phi(R, k) = e^{-\alpha(R – R_0)} \cdot e^{i k R},$$

where:

• $e^{-\alpha(R – R_0)}$ is a spatial envelope that localizes the wave around $R_0$,
• $e^{i k R}$ represents the oscillatory component of the wave,
• $\alpha$ controls the degree of localization.

The full wave function is then constructed as a superposition of these localized modes:

$$\Psi(R, t) = \int_{-\infty}^\infty \int_{-\infty}^\infty C(k, R_0) \, e^{-\alpha(R – R_0)} \cdot e^{i k R} \, dk \, dR_0,$$

where $C(k, R_0)$ specifies the amplitude of the localized mode with wave vector $k$ and center $R_0$.

Spectral Analysis of Localized Functions

For the specific case of $\Psi(R, t) = A \cdot e^{-\alpha(RA – A_0)} \cdot e^{i\omega_1 t}$, the spatial component $e^{-\alpha(RA – A_0)}$ is a Gaussian function. Its Fourier transform yields:

$$\Phi(k) = A \cdot \frac{\sqrt{\pi}}{\alpha} \cdot e^{-\frac{(k – k_0)^2}{4\alpha^2}},$$

where $k_0$ represents the central spatial frequency. This result demonstrates that the function $\Psi(R, t)$ can be viewed as a superposition of plane waves, but with weights distributed in a Gaussian profile around $k_0$.

Unlike a purely oscillatory wave (e.g., $e^{i k R}$), which has infinite spatial extent, this localized wave is confined to a region of space, making it more representative of physical phenomena.

Connection to Beetheory: Beyond Fourier Analysis

The Beetheory extends Fourier analysis by emphasizing spatial and frequency localization. While Fourier series or transforms decompose a function into infinite, non-localized components, Beetheory incorporates the following key innovations:

1. Localized Envelopes: Gaussian-like spatial envelopes $e^{-\alpha(R – R_0)}$ ensure that wave modes are spatially confined, capturing real-world phenomena like wave packets or confined fields.

2. Superposition of Localized Modes: Instead of relying exclusively on plane waves, the theory allows the combination of spatially confined modes, enabling the modelling of complex, non-periodic systems.

3. Temporal Dynamics: By integrating temporal oscillations $e^{i\omega t}$, Beetheory seamlessly connects the spatial and temporal domains, making it applicable to dispersive or non-linear wave phenomena.

Applications and Implications

1. Quantum Mechanics: In quantum systems, localized functions like $\Psi(R, t)$ are essential for describing wave packets, which represent particles with a definite position and momentum distribution.

2. Optics: Beetheory can be applied to model spatially confined laser beams or light fields, where the Gaussian envelope plays a crucial role.

3. Signal Processing: The decomposition into localized modes can aid in analyzing signals that are non-periodic or confined to specific regions of space or time.

4. Wave Propagation in Media: By modelling waves with spatial localization, Beetheory provides insights into phenomena such as waveguides, localized vibrations, or acoustic fields.

Conclusion

The Beetheory redefines wave modelling by bridging the gap between traditional Fourier analysis and the physical reality of spatially localized waves. By introducing localized modes and generalizing the concept of wave decomposition, it offers a versatile framework for understanding complex wave phenomena across disciplines. This approach, rooted in functions like $\Psi(R, t)$, opens new possibilities for representing and analyzing waves in both classical and quantum domains.