The Interplay of Phase Shifts, Time Distortions, and Energy Changes in Wave Systems:
Soumendra Nath Thakur
22-04-2024
Description:
The text provides a detailed examination of equations and relationships governing phase shifts, time distortions, and energy changes in wave systems. It begins by establishing fundamental equations linking phase shift T(deg), time distortion (Δt), and energy change (ΔE), subsequently extending these equations to calculate energy changes based on frequency and Planck's constant.
A key concept introduced is the generalization of time distortion to incorporate phase shift (x), allowing for a more versatile analysis of wave systems with different phase shifts. The conclusion drawn asserts a direct relationship between phase shift measured in degrees T(deg) and the degree of phase shift (x).
Furthermore, an expression explicitly integrating phase shift (x) into the equation for energy change is derived, confirming relationships between Planck's constant, frequency, and the period of the wave.
Overall, the text provides a coherent and comprehensive exploration of wave system dynamics, offering insights into phase shifts, time distortions, and energy changes.
Soumendra
Nath Thakur
ORCID
iD: 0000-0003-1871-7803
Tagore’s
Electronic Lab, West Bengal, India
Email:
postmasterenator@gmail.com,
postmasterenator@telitnetwork.in
The equations and relationships concerning phase shifts, time distortions, and energy changes in wave systems are:
T(deg) = (1/360f₀) = Δt
ΔE = hf₀Δt
ΔE = (2πhf₀/360) × T(deg)
ΔE = (2πh/360) × T(deg) × (1/Δt)
These equations establish the relationship between phase shift T(deg) and time distortion (Δt), where f₀ represents the primary or initial frequency of the wave. It essentially quantifies the time distortion (Δt) caused by a phase shift.
The equations subsequently calculate the infinitesimal loss of wave energy (ΔE) based on Planck's constant (h), frequency (f₀), and phase shift T(deg) or time distortion (Δt). These equations allow for the determination of energy changes given certain variables.
Δtₓ = x (1/360f₀)
The above equation generalizes the concept of time distortion (Δt) with respect to phase shift (x), where x represents the degree of phase shift. The conclusion drawn is that T(deg) = x, which essentially states that the degree of phase shift (x) is directly related to T(deg), the phase shift measured in degrees.
The conclusion is valid because throughout the analysis, the relationships between phase shift, time distortion, and energy changes are consistently tied together, culminating in the assertion that the degree of phase shift (x) corresponds to the phase shift measured in degrees T(deg).
The expression:
(h/360) 2πf₀x
It is derived from the basic equation ΔE = hf₀Δt, explicitly including the phase shift (x).
T = 360
f₀ = 1/T
hf₀ = h/360
The statement T = 360 implies that the period of the wave (T) is equivalent to 360 units of time. This aligns with the concept that a complete cycle of a wave occurs over 360 degrees.
f₀ =1/T indicates that the primary or initial frequency (f₀) of the wave is inversely proportional to its period (T). If the period is 360 units of time, then the frequency is indeed 1/360 cycles per unit time.
Combining these, hf₀ = h/360 follows from substituting f₀ = 1/T into the equation hf₀, yielding h/360.
This statement essentially confirms the relationship between Planck's constant (h), frequency (f₀), and the period of the wave (T), providing a basis for further derivations and calculations involving energy changes and phase shifts.
The above statements present a consistent exploration of the equations and relationships governing phase shifts, time distortions, and energy changes in wave systems. Let's dissect the consistency:
Establishing Equations and Relationships: The initial equations T(deg) = (1/360f₀) = Δt and ΔE=hf₀ Δt lay the foundation for understanding the interplay between phase shift (T(deg), time distortion (Δt), and energy change (ΔE).
Calculation of Energy Changes: The subsequent equations ΔE = (2πhf₀/360) × T(deg) and ΔE = (2πh/360) × T(deg) × (1/Δt) provide methods for calculating energy changes based on various variables such as phase shift, time distortion, frequency, and Planck's constant.
Generalization of Time Distortion: The equation Δtₓ = x × (1/360f₀) extends the concept of time distortion (Δt) to incorporate phase shift (x), offering a versatile tool for analysing wave systems with different phase shifts.
Conclusion: The conclusion that T(deg)= x solidifies the relationship between phase shift measured in degrees T(deg) and the degree of phase shift (x), emphasizing their direct correspondence.
Derivation and Confirmation of Relationships: The expression (h/360) × 2πf₀ × x is derived from the fundamental equation ΔE = hf₀Δt, explicitly integrating the phase shift (x). The subsequent statements regarding T = 360, f₀ = 1/T, and hf₀ = h/360 confirm the relationships between Planck's constant, frequency, and the period of the wave, providing a coherent basis for further analysis.
The above statements offer a comprehensive and consistent exploration of the dynamics governing wave systems, providing a robust framework for understanding and analysing phase shifts, time distortions, and energy changes.
Applicable for:
DOIs:
· http://dx.doi.org/10.13140/RG.2.2.28013.97763
· http://dx.doi.org/10.35248/2161-0398.23.13.365
URLs:
PDF:
· https://www.longdom.org/open-access/phase-shift-and-infinitesimal-wave-energy-loss-equations.pdf
