Soumendra Nath Thakur | ORCiD: 0000-0003-1871-7803 |
Tagore’s Electronic
Sections:1. Frequency as a Function of Phase Shift and Time Delay: Beyond the Standard Periodic Definition:2. Phase-Time Relationship at Constant Frequency:3. Reduction of Oscillation Frequency Due to Partial Phase Progression:4. Phase-Time-Frequency Interdependence in Partial Oscillation Dynamics:5. Progressive Frequency Decay Through Phase-Time Scaling: Foundations for ECM Frequency Shift Dynamics:6. Redshift Interpretation Across Phase-Time Frequency Decay Cases:7. Electromagnetic Kinetic Energy in ECM: Frequency Decay, Energy Reduction, and Apparent Mass Displacement:
1. Frequency as a Function of Phase Shift and Time Delay: Beyond the Standard Periodic Definition:
Frequency - Time period relationship:
Frequency (f) is inversely proportional to period (T). When presented in degrees:
f = 1/T = 1/360°
We derive the frequency (f) formula in relation to the phase shift in degrees (x°) and the corresponding time delay (Δt) expressed as:
f = Phase shift (in degrees)/(360° × Δt) or,
f = x°/(360° × Δt) or,
A phase shift, represented as x° or Φ°, is directly related to the time delay (Δt) between two waves or a wave and a reference point. But we cannot say that frequency is inversely proportional to time delay (Δt), since, frequency (f) depends on both, the magnitude of the phase shift (x°) and the corresponding time delay (Δt), when x° varies:
f = 1/Δt × (x°/360°), because the value of (x°/360°) is not equals to (≠1) when x° varies.
The expression defines frequency as a function of two variables: time delay (Δt) and phase shift (x°). When the phase shift varies, frequency no longer reflects the repetition of a full cycle, but rather a partial progression through it. The term (x°/360°) acts as a scaling factor that adjusts the frequency according to what fraction of the cycle is observed.
If x° is small, the phase shift represents only a small portion of the full wave, so the computed frequency reflects an extrapolated or scaled-down version of the actual repetition rate. Conversely, when x° approaches 360°, the full cycle is represented, and the expression converges to the standard definition of frequency as the reciprocal of the time period.
Thus, frequency in this form is interpreted as the projected full-cycle rate, derived from the time it takes to complete a specified fraction (x°) of a cycle. It is especially useful when the system under observation only reveals a partial phase progression over a known time interval. This model captures how frequency varies proportionally with the phase angle and inversely with the associated time delay.
2. Phase-Time Relationship at Constant Frequency:
f = x°/(360° × Δt)
Implies x° ∝ Δt (proportional change) when frequency f is constant, x° ≠ 0 (not equals to).
When the frequency of oscillation remains unchanged, any change in the phase shift measured in degrees directly corresponds to a proportional change in time delay. This means that as the phase shift increases, the time delay increases in equal proportion, provided the frequency is constant and the phase shift is not zero. In this context, phase shift and time delay are linearly dependent — a larger phase angle reflects a longer time displacement, and vice versa. This relationship highlights the synchronized behaviour between angular displacement and temporal progression in systems where frequency remains stable.
3. Reduction of Oscillation Frequency Due to Partial Phase Progression:
When an oscillation originally characterized by frequency f₁ is only partially observed — that is, a phase shift less than a full 360° occurs over a measurable time delay — the effective frequency reduces. This reduced frequency, f₂, reflects the incomplete cycle behaviour and represents a lower number of oscillations completed per second.
An oscillation with initial frequency f₁ completes a full cycle in a certain period. If, instead of observing a full 360° cycle, only a smaller phase shift x° is observed over a time delay Δt, the frequency must be adjusted to reflect this partial behaviour.
In this context, the frequency becomes a scaled version of f₁, determined by the ratio x°/360°. This means the oscillation appears to complete fewer full cycles per unit time than initially — effectively reducing the frequency from f₁ to a lower value f₂.
Mathematically, f₂ results from multiplying the inverse of the observed time delay by the phase fraction.
f₂ = x°/(360° × Δt)
So, as either the phase shift x° becomes smaller or the time delay Δt becomes longer, the resulting frequency f₂ decreases further. This reflects that less of the oscillatory motion is occurring (or being completed) per unit of time.
Thus, the transition f₁ → f₂ represents a loss or reduction in the number of full cycles per second, caused directly by a decrease in observed phase progression and/or an increase in the time interval over which that partial phase shift occurs.
4. Phase-Time-Frequency Interdependence in Partial Oscillation Dynamics:
An initial oscillation frequency f₁ represents the number of full cycles completed per second. When instead of observing a full cycle, only a partial phase shift x° > 1° is measured over a corresponding time delay Δt, the frequency must be rescaled to reflect only that partial progression through the cycle.
In this case, the observed frequency f₂ becomes a fraction of f₁, determined by how much of the cycle was completed (expressed as x°/360°) within the measured time delay. As a result, f₁ reduces to f₂, where f₂ < f₁, since only a portion of the oscillatory motion is occurring per unit of time.
This reduction implies that fewer full cycles are completed per second than originally, not because the wave itself slows down, but because the observed portion (given by x°) over Δt represents only a fraction of its full periodic behaviour. Hence, changes in x° > 1° and Δt jointly determine how the full-cycle frequency f₁ transitions into a lower apparent or effective frequency f₂.
The expression:
f = x°/(360° × Δt)
Assuming two different phase shift and time delay conditions (both satisfying x° > 1°), let:
f = x°/(360° × Δt)
The Expression:
x° = Δt × Δf × 360° ; x° > 1° ; Δt > 0.
This expression shows how frequency f is calculated from a known phase shift x°, expressed in degrees, and the time delay Δt over which that phase shift occurs. When both the phase shift is greater than one degree and the time delay is greater than zero, this relation gives a defined value for frequency. If the phase and time values change, the frequency will also change accordingly, resulting in a difference between the initial and the new frequency — represented by the term Δf, which reflects the change caused by that variation.
The Expression:
Δf = x°/(360° × Δt)
This expression gives a way to directly compute the frequency change Δf from a known phase shift and its corresponding time delay. It shows that Δf is determined by dividing the phase shift by the product of 360 degrees and the time delay. This means the larger the phase shift or the shorter the time delay, the greater the frequency change.
Conversely, the phase shift itself can be found if both the frequency change and the time delay are known. In this case, the phase shift equals the product of Δt, Δf, and 360 degrees. This makes phase shift a result of how much the frequency has changed over a given time.
Together, these equations describe how changes in frequency, time, and phase are interrelated. A variation in one directly affects the others, forming a coherent framework for analysing oscillatory behaviour using phase-time dynamics.
5. Progressive Frequency Decay Through Phase-Time Scaling: Foundations for ECM Frequency Shift Dynamics:
To understand how oscillation frequency varies under different phase-time conditions, we begin with a reference point: the original frequency f, defined in the absence of any phase shift — that is, when x° = 0° and the time delay Δt₀ = 0. In this case, the system completes full cycles without temporal distortion, and frequency remains unchanged. However, when a phase shift greater than zero is introduced — such as x° = 1°, or even extended beyond a full cycle — the corresponding time delay increases, and the frequency is no longer equal to f. Instead, we observe progressively reduced frequencies f₁, f₂, f₃,..., each arising from the combination of a particular phase shift and the time required to complete it. The following discussion outlines this progression and the resulting implications for dynamic systems governed by phase-time-frequency scaling.
Consistent Presentation:
For a phase shift of x°:
Time delay per degree of phase:
Tdeg = x°/(360° × f) = Δt
Case 1:
x° = 0° (no phase shift)
Time delay: Δt₀ = 0
Frequency remains: f = f
Case 2:
x° = 1° (small phase shift)
Time delay increases: Δt₁ > 0
Observed Frequency reduces: f₁ < f
From frequency equation:
f = x°/(360° × Δt)
Then for x° = 1°, we write:
f > f₁ since Δt₀ < Δt₁
So the frequency difference becomes:
Δf = f − f₁ = 1°/(360° × Δt₁)
Case 3:
For extended phase shift x° > 360°:
At longer Δt = Δt₂, the frequency loss is scaled:
Δf = f − f₁ = x°/(360° × Δt₂)
Case 4:
x° > 360° × n, where n is a positive integer (n = 1, 2, 3, …)
When the phase shift exceeds a full cycle and continues through multiple full rotations — such as 720°, 1080°, or more — it implies that the system is either oscillating for longer durations or being observed across multiple repeated cycles.
In such cases, the time delay, denoted Δt₂, reflects the extended duration over which multiple full-phase rotations are observed. The frequency loss becomes significantly scaled relative to the initial frequency f, and the frequency reduction Δf is now determined by the full accumulated phase shift across time, according to:
Δf = f − f₁ = x°/(360° × Δt₂)
This highlights that when the phase shift accumulates beyond a single cycle, the frequency continues to decline proportionally — not simply due to slower oscillation, but due to the distribution of phase progression across an extended time frame. The frequency thus perceived, f₂, represents the apparent full-cycle rate reconstructed from the observation of multiple partial or complete phase sequences across time. This reinforces the interpretation of progressive frequency decay and establishes the basis for generalizing time-scaled frequency in ECM where energy and mass relationships are also influenced by multi-cycle dynamics.
Interpretation:
• As phase shift increases and more time elapses to
complete that shift, the apparent frequency decreases.
• This leads to a progressive frequency decay f → f₁ → f₂ as x° and Δt increase.
• The general relationship holds:
Δf = x°/(360° × Δt),
where higher phase angle over longer delay implies greater frequency shift.
The presentation outlines how a phase shift, measured in degrees, relates to a corresponding time delay and how both influence the frequency of oscillation.
It begins with the idea that for any given phase shift, the associated time delay is directly proportional to the size of that shift and inversely related to the system’s frequency. When there is no phase shift (0°), the time delay is zero, and the original frequency remains unchanged.
As a small phase shift occurs — such as 1° — a measurable time delay greater than zero is introduced. This delay results in a reduction of the observed frequency compared to the original. The frequency is no longer a reflection of a complete oscillation per unit time, but instead, it becomes a scaled measure determined by the fraction of the phase completed during the delay.
As the phase shift increases further — particularly beyond 360°, corresponding to more than one full cycle — the scaling becomes more significant. Here, the phase angle exceeds one full rotation and can be represented as a multiple of 360°, where x° > 360° × n, and n is a positive integer (1, 2, 3, ...). The accompanying time delay, Δt = Δt₂, captures the extended duration required to complete such multiple phase rotations.
Under these conditions, the change in frequency Δf, defined by the difference between the original frequency f and the reduced observed frequency f₁, becomes:
Δf = x°/(360° × Δt₂
This expression indicates that the larger the total accumulated phase shift (even across multiple cycles), the greater the potential frequency change, particularly when that shift occurs over a finite or increasing time delay. It reveals that frequency decay is not strictly limited to sub-cycle observations but continues progressively as the observed system completes more than one cycle — especially when phase accumulation exceeds 360°, 720°, and so on.
Thus, this extended framework shows that frequency is sensitive not just to fractional phase evolution, but also to the number of full rotations and the corresponding temporal expansion. This scaling behaviour lays the foundation for interpreting oscillatory dynamics in terms of frequency dilution or decay — a concept that becomes essential in Extended Classical Mechanics (ECM) when associating time-evolving phase progression with mass-energy redistribution, reversible kinetic behaviour, and apparent mass transitions.
6. Redshift Interpretation Across Phase-Time Frequency Decay Cases:
In Extended Classical Mechanics (ECM), redshift is interpreted not merely as a function of wavelength elongation, but more fundamentally as a frequency decay phenomenon resulting from progressive phase shift over time. This framework allows a deeper understanding of cosmological redshift as a consequence of oscillatory energy reduction mediated by cumulative phase-time behaviour, rather than velocity-induced Doppler effect alone.
Building on the previous section (5), we interpret redshift across various phase-time cases as follows:
Case 1: Zero Phase Shift (x° = 0°)
Time delay: 0 seconds
Frequency shift: 0 Hz
Redshift (z): 0
Interpretation: There is no displacement in phase or time; the observed frequency equals the source frequency. No redshift is observed.
Case 2: Small Phase Shift (x° = 1°)
Time delay: Greater than 0 seconds
Frequency shift: Small (inversely proportional to time delay)
Redshift (z): Negligible to minimal
Interpretation: A single-degree phase displacement produces an observable time delay. However, the frequency shift remains minute, resulting in no significant spectral redshift.
Case 3: Moderate Phase Shift (x° ≥ 360°)
Time delay: Measurable and positive
Frequency shift: Still relatively small over long durations
Redshift (z): Marginal under cosmological scales
Interpretation: One full cycle of phase shift is spread across large time delays (e.g., millions of years). Frequency shift is detectable but does not yield appreciable redshift unless observed at high sensitivity.
Case 4: Large Phase Shift (x° ≥ 360° × n, where n ranges from 100 to 100 million)
Time delay: Ranges from 1 million to 10 billion years
Frequency shift: Can vary from approximately 10⁻¹⁴ Hz to 10⁻¹² Hz or higher
As confirmed in the revised ECM Frequency Shift Table:
For x° = 3.6 × 10⁴ and time delay = 1 billion years, frequency shift ≈ 3.17 × 10⁻¹⁵ Hz
For x° = 3.6 × 10⁶ and time delay = 1 billion years, frequency shift ≈ 3.17 × 10⁻¹³ Hz
Interpretation: Even with phase shift representing thousands to millions of cycles, the effective redshift is negligible over billion-year delays. This demonstrates that cosmologically significant redshift requires either extremely high frequency change or much shorter delay time for a given phase shift.
Case 5: High Redshift (z ≥ 1)
Required frequency shift: Must be a large fraction of the original frequency
Interpretation in ECM: Only achievable if either:
Phase shift is astronomically large within a given time delay
Or the time delay is extremely short (such as in high-energy, early-universe events)
This reframes redshift not as a passive observation but as an energy dynamic, where frequency loss maps to cumulative phase displacement. Therefore:
z = Δf / f = x° / (360° × f × Δt)
This equation shows that redshift is governed by phase-time structure. As such, ECM introduces a new mechanism of redshift causality grounded in reversible or irreversible frequency decay from oscillatory energy redistribution.
Conclusion:
Through these cases, ECM offers a deterministic, precision-based view of redshift as emerging from cumulative phase evolution over measurable temporal intervals. The scaling of phase shift x°, particularly when considered across cosmological time delays, provides a powerful interpretive tool for understanding redshift without invoking expansion of space alone. This opens the path to reinterpret photon energy loss, apparent mass transitions, and kinetic energy redistribution within a unified oscillatory framework.
7. Electromagnetic Kinetic Energy in ECM: Frequency Decay, Energy Reduction, and Apparent Mass Displacement
In Extended Classical Mechanics (ECM), electromagnetic energy is not only described through traditional quantum relations such as E = hf, but is also reinterpreted in terms of frequency decay, mass-energy redistribution, and phase-time dynamics. Frequency, in ECM, is both a measure of oscillatory behaviour and a carrier of kinetic energy — tightly linked to apparent mass transformations and displacement across time.
Building upon the frequency decay framework established in Section 5: Progressive Frequency Decay Through Phase-Time Scaling, this section explores how kinetic energy is expressed in frequency-domain terms and how various redshift or frequency loss cases can be understood through ECM dynamics.
We start with Planck's equation:
E = hf
where:
• E is the energy associated with a photon or oscillation,
• h is Planck’s constant (6.62607015 × 10⁻³⁴ J·s),
• f is the frequency of oscillation.
From this, a reduction in frequency (Δf) directly implies a reduction in energy (ΔE):
ΔE = hΔf
In ECM, such ΔE is not simply radiated or lost, but reinterpreted as a shift in kinetic state and apparent mass.
Additionally, in scenarios involving cumulative or time-integrated oscillatory behaviour, where the wave undergoes continuous displacement over a measurable delay without resetting its frequency, ECM allows another formulation:
ΔE = h f Δt
This relation captures the integrated wave energy displaced over time, particularly useful in analysing phase shift over cosmological durations where Δf may be extremely small but Δt is enormously large. It highlights how wave energy can accumulate or be displaced even when frequency loss per unit time is nearly negligible.
Case-Based Energy-Kinetics Mapping (Derived from Section 5):
Case 1: x° = 0°, Δf = 0, ΔE = 0
• No phase shift or frequency change.
• No kinetic displacement.
• Mass remains unchanged: ΔMᴍ = 0
• No KEᴇᴄᴍ manifestation.
Case 2: x° = 1°, small Δf
• Small energy shift: ΔE ≈ hΔf (very low)
• Minimal reversible energy movement.
• Tiny apparent mass loss: ΔMᴍ ≡ −Mᵃᵖᵖ (infinitesimal)
• Weak kinetic activity, but measurable under high-resolution dynamics.
Case 3: x° ≥ 360°
• Phase shift spans full cycle; Δf becomes relevant.
• Kinetic energy drops more appreciably: ΔE = hΔf
(measurable)
• Reversible apparent mass displacement: ΔMᴍ ≡ −Mᵃᵖᵖ
• KEᴇᴄᴍ associated with this frequency drop implies a direct reduction in effective energy per unit wave.
Case 4: x° ≥ 360° × n (n ≫ 1)
• Massive phase accumulation (thousands to millions of
cycles lost).
• Energy shift becomes significant even if Δf is small due
to long Δt.
• Mass displacement becomes layered: ΔMᴍ ≡ −2Mᵃᵖᵖ or more, due to cumulative
frequency loss.
• ECM interprets this as large-scale kinetic transition,
with oscillatory energy decaying into phase-shift-stored displacement.
• Observed KEᴇᴄᴍ decreases progressively with time, while the system retains the ‘lost’ energy in displaced apparent mass form.
Over such vast durations, ΔE = h f Δt serves to estimate cumulative wave energy displaced.
Case 5: High Redshift (z ≥ 1), Δf ≈ f
• Frequency decays to a small fraction of original.
• Energy loss: ΔE ≈ hf
• Apparent mass loss corresponds to complete wave-energy to mass redistribution.
• ΔMᴍ ≡ −2Mᵃᵖᵖ or beyond (system-wide transformations).
• KEᴇᴄᴍ is now nearly zero in the observed frame, but stored as displaced mass-energy across spacetime.
Summary of Key Relations in ECM Energy Framework:
• E = hf: Classical quantum relation retained.
• ΔE = hΔf: Energy shift from frequency decay.
• ΔE = h f Δt: Cumulative energy displaced over measurable
phase-time evolution.
• KEᴇᴄᴍ = ½Mᵉᶠᶠv²: Kinetic energy expression linked to Mᵉᶠᶠ that includes displaced mass.
• ΔMᴍ ≡ −Mᵃᵖᵖ: Fundamental apparent mass loss interpretation.
• ΔMᴍ ≡ −2Mᵃᵖᵖ: Advanced kinetic collapse or transition condition.
Conclusion:
In ECM, electromagnetic energy and kinetic motion are
unified through frequency and phase shift behaviour. Every reduction in
frequency maps to a corresponding energy shift, which is not discarded but
re-encoded into apparent mass or stored displacement. These transitions —
whether reversible, cumulative, or redistributive — underlie a broader
understanding of mass-energy motion and decay not as loss, but as
transformation. This enables ECM to reinterpret redshift and KE reduction not
merely as observational effects, but as dynamic, quantifiable, and reversible
phenomena of oscillatory systems.
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