December 25, 20205
Soumendra Nath Thakur
A phase shift is not merely a geometric or angular quantity. Physically, a phase shift represents a fractional loss of completed oscillatory cycles in a propagating wave.
Because frequency is defined as the number of cycles completed per unit time, any loss of cycles immediately implies a reduction in effective frequency. This establishes the first physical link:
Phase shift → fractional cycle loss → frequency reduction.
Through Planck’s relation (E = hf), frequency directly determines the energy carried by an oscillatory quantum. Therefore, a fractional loss of cycles produces a proportional loss of Planck energy:
Δf/f₀ = Δt/T = x°/360 → ΔE = hf₀(x°/360).
This establishes the complete physical bridge:
phase → time distortion → frequency shift → energy loss→ redshift.
In Extended Classical Mechanics (ECM), this Planck-quantified energy loss is not an abstract bookkeeping change. It corresponds to a real physical conversion of stored potential structure into dynamical and mass-like manifestations:
-ΔPEᴇᴄᴍ ↔ ΔKEᴇᴄᴍ ↔ ΔMᴍ ⟶ observable Planck energy loss.
Thus, phase drift is the physical trigger by which oscillatory energy is removed from the wave, converted into Negative Apparent Mass (NAM) and associated kinetic and mass manifestations, and finally observed as redshifted radiation.
This provides a direct, causal, and Planck-consistent bridge between wave phase dynamics and ECM’s mass–energy conversion framework.
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Phase Shift Calculations and Example:
To illustrate the practical application of the phase–time relation T(deg), an example is presented.
Example 1 — 1° Phase Shift on a 5 MHz Wave
The time shift associated with a phase change is given by
T(deg) = x°/360f
For x = 1° and f = 5 MHz = 5 × 10⁶ Hz:
Now, plug in the frequency (f) into the equation for T(deg):
T(deg) = 1/(360 × 5 × 10⁶) = 5.556 × 10⁻¹⁰ s
T(deg) ≈ 555 picoseconds (ps)
Thus, a 1° phase shift on a 5 MHz wave corresponds to a time shift of approximately 555 ps.
This calculation demonstrates how to determine the time shift caused by a 1° phase shift on a 5 MHz wave. It involves substituting the known frequency (f = 5 MHz) into the equation for T(deg).
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Phase Shift Equation 1.1 — General Form
For a x° phase shift on a f₀ Hz Wave:
T(deg) = x°/360f₀
By plugin the values of frequency (f₀) and phase shift (x°) into the equation, the calculated value of T(deg):
T(deg) = x°/360f₀ ≈ Δt
So, a x° phase shift on a f₀ Hz wave corresponds to a time shift of approximately Δt s.
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Infinitesimal Loss of Wave Energy Equations:
These equations relate to the infinitesimal loss of wave energy (ΔE) due to various factors, including phase shift:
The Planck energy-frequency equation:
• E = hf
So for a small change,
• ΔE = hΔf.
We write this in fractional form relative to the source frequency f₀:
• ΔE = hf₀(Δf/f₀) → hf₀(x°/360)
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Or, if phase–time coupling:
• ΔE = hf₀(Δt/T)
because only fraction of a cycle changes energy.
Derivation of hf₀(Δt/T):
A wave with period T has:
f₀ = 1/T
A phase shift means that the wave is no longer completing full cycles.
If the time shifts by Δt, the fractional cycle loss is:
Δt/T.
The fractional cycle loss is exactly fractional frequency loss:
Δf/f₀ = Δt/T = x°/360
This is the definition of frequency as cycles per unit time.
• [ΔE = hf₀(Δf/f₀) = hf₀(Δt/T) = hf₀(x°/360)]
This expression states:
Phase drift → fractional cycle loss → frequency reduction → Planck-quantified energy loss.
In Extended Classical Mechanics (ECM), this lost oscillatory energy is not abstract. It corresponds to a real conversion:
• -ΔPEᴇᴄᴍ → ΔKEᴇᴄᴍ → ΔMᴍ ⟶ Planck-quantified energy loss.
with the measurable manifestation appearing as the Planck energy deficit
• ΔE = hΔf.
Thus, phase drift directly generates Negative Apparent Mass (NAM) through the loss of oscillatory existence.
This is one of the key bridges between Planck physics and ECM’s NAM–phase–redshift mechanism.
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This equation determines the infinitesimal loss of wave energy (ΔE) from Planck’s constant (h) when the source frequency (f₀) and either the phase shift (x°) or the corresponding time shift (Δt) are known. It represents Planck energy scaled by the fractional loss of oscillatory phase.
When the phase shift in degrees (x°) is known, the infinitesimal energy loss is
• ΔE = hf₀(x°/360).
Since a phase shift corresponds to a fractional time shift (Δt) of one oscillation period (T), the energy loss may equivalently be written as
• ΔE = hf₀(Δt/T).
Dimensionally, (T) is the time duration of one oscillation cycle, whereas (360°) is the angular phase span of one cycle; the two are related by the fractional-cycle identity, not by numerical substitution.
These expressions form the foundation for analyzing phase shift, time distortion, frequency change, and the resulting infinitesimal loss of wave energy. They apply to both theoretical and practical wave analyses and align directly with the ECM interpretation of phase drift → energy loss → redshift → ΔMᴍ (NAM mapping).
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Loss of Wave Energy Calculations and Example:
Loss of Wave Energy Example 1:
To illustrate the practical applications of the derived equations of loss of wave energy, example calculation is presented:
To determine the energy (E₀) and infinitesimal loss of energy (ΔE) of an oscillatory wave with a frequency (f₀) of 5 MHz and a phase shift x° = 0°, use the following equations:
Oscillation frequency 5 MHz, when 0° Phase shift in frequency.
Calculate the energy (E₀) of the oscillatory wave:
E₀ = hf₀
Where:
h is Planck's constant ≈ 6.626 × 10⁻³⁴ Js .
f₀ is the frequency of the wave, which is 5 MHz (5 × 10⁶ Hz).
E₀ = (6.626 × 10⁻³⁴ Js) × (5 × 10⁶) = 3.313 × 10⁻²⁷ J.
So, the energy (E₀) of the oscillatory wave is approximately 3.313 × 10⁻²⁷ Joules.
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Calculate the time distortion T(deg) of the oscillatory wave when phase shift x° = 0°:
For an oscillatory wave of frequency f₀ = 5 MHz with zero phase shift,
T(deg) = x°/360f₀ = Δt
Since x° = 0°,
T(deg) = Δt = 0.
Thus, there is no time distortion because no phase shift has occurred. ECM-consistent chain: Phase → time distortion → energy change.
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Calculate the infinitesimal loss of wave energy (ΔE₀) when both Δf₀ and Δt are zero:
The infinitesimal energy change is given by
ΔE₀ = hΔf₀.
Since Δf₀ = 0,
ΔE₀ = 6.626 × 10⁻³⁴ × 0 = 0.
Therefore, the infinitesimal loss of wave energy (ΔE₀) is 0 joules because there is no time distortion (Δt = 0), no phase shift (x° = 0°), no frequency shift (Δf₀ = 0), meaning there is no infinitesimal loss of wave energy during this specific time interval.
Conclusion for the zero-phase-shift case
These calculations demonstrate that for an oscillatory wave of frequency f₀ = 5 MHz with x° = 0°:
• the time distortion Δt = 0,
• the frequency change Δf₀ = 0,
• and the infinitesimal energy loss ΔE₀ = 0.
The wave therefore retains its full Planck energy
E₀ = hf₀ = 3.313 × 10⁻²⁷ J.
The energy (E₀) of the oscillating wave with a frequency 5 MHz and no phase shift (x° = 0°) is approximately 3.313 × 10⁻²⁷ joules. Due to the absence of a phase shift, there is no time distortion (Δt) and no infinitesimal energy loss (ΔE) of the wave during this specific time interval.
This establishes the correct reference state against which phase-drift, redshift, and ECM-based energy conversion can be measured.
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Loss of Wave Energy — Example 2
To illustrate the practical application of the derived equations for wave-energy loss, the following example is presented.
Consider an oscillatory wave with an original frequency
f₀ = 5 MHz
that undergoes a phase shift of
x° = 1°.
This x° phase shift produces a slightly reduced oscillation frequency f₁ and a corresponding infinitesimal loss of wave energy ΔE.
This example demonstrates how to determine:
• the new wave energy E₁,
• the infinitesimal energy loss ΔE, and
• the resulting shifted frequency f₁,
relative to the original frequency f₀, when the wave experiences a phase shift
x° = 1°.
To determine the energy E₁, the energy loss ΔE, and the resulting frequency f₁ for a wave with a (1°) phase shift from the original frequency f₀ = 5 MHz, proceed as follows:
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Calculate the energy E₁ of the oscillatory wave with the shifted frequency f₁:
Using Planck’s energy relation,
E₁ = hf₁
where
h is Planck’s constant ≈ 6.626 × 10⁻³⁴ J·s,
f₁ is the frequency after the phase shift.
Determine the frequency change Δf produced by a phase shift of x° = 1°:
A phase shift represents a fractional displacement of one oscillation cycle.
Therefore, the corresponding fractional change in frequency is:
Δf/f₀ = x°/360°
so,
Δf = (x°/360°)f₀
For x° = 1° and f₀ = 5 MHz = 5 × 10⁶ Hz,
Δf = (1/360) 5 × 10⁶ = 13,888.89 Hz
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The shifted frequency is therefore
f₁ = f₀ - Δf
for a red-shifting (energy-losing) phase drift in ECM.
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Now that the frequency shift Δf has been determined, the shifted frequency f₁ is:
f₁ = f₀ - Δf
Substituting the values,
f₁ = f₀ - Δf
f₁ = (5.0 × 10⁶) - (13,888.89) = 4,986,111.11 Hz
Thus, the resulting frequency of the oscillatory wave after a 1° phase shift is approximately
f₁ = 4.98611111 × 10⁶ Hz
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This correctly implements the ECM rule:
Δf/f₀ = x°/360°
So a 1° phase drift produces a (1/360) fractional frequency reduction — and therefore a proportional energy and mass decrement, exactly as required by ECM-consistent chain: phase-drift → ΔE → ΔMᴍ mapping.
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Calculate the energy (E₁) using the new frequency (f₁):
E₁ = hf₁
E₁ ≈ (6.626 × 10⁻³⁴) × (4.98611111 × 10⁶) Hz.
E₁ ≈ 3.3048 × 10⁻²⁷ J
Thus, the energy of the oscillatory wave with a frequency of approximately 4,986,111.11 Hz and a x° = 1° phase shift is approximately E₁ ≈ 3.3048 × 10⁻²⁷ Joules.
This reflects the ECM relation
ΔE/E₀ = Δf/f₀ = x°/360°
so a 1° phase drift produces a real, proportional Planck-energy loss, exactly as ECM's phase-drift → energy-loss → redshift → ΔMᴍ chain requires.
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To determine the infinitesimal loss of energy (ΔE) due to the phase shift, use the formula:
ΔE = hΔf = hf₀(Δt/T) = h(f₀)²Δt
Where:
h = 6.626 × 10⁻³⁴ Js is Planck's constant.
f₀ = 5 Mhz = 5 × 10⁶ Hz
f₁ ≈ 4.98611111 × 10⁶ Hz
Δf = f₀ - f₁ = 13,888.89 Hz = 0.01388889 × 10⁶ MHz
Δt ≈ 555 ps = 5.55 × 10⁻¹⁰ s, corresponding to a 1° phase shift on f₀.
ECM-consistent chain: Phase drift (x°) → Δt → Δf → ΔE = hΔf → ΔMᴍ
So,
ΔE = hΔf = 9.2036 × 10⁻³⁰ J
or,
ΔE = hf₀(Δt/T) = 9.2036 × 10⁻³⁰ J, where T = 1/f₀ = 2.0 × 10⁻⁷ s
or,
ΔE = h(f₀)²Δt = 9.2036 × 10⁻³⁰ J
Thus, the infinitesimal loss of energy (ΔE) due to the 1° phase shift is approximately 9.2036 × 10⁻³⁰ Joules.
Resolved, the energy (E₁) of this oscillatory wave is ≈ 3.3048 × 10⁻²⁷ Joules.
Resolved, the infinitesimal loss of energy (ΔE) due to the 1° phase shift is approximately 9.2036 × 10⁻³⁰ Joules.
Resolved, the resulting frequency (f₁) of the oscillatory wave with a 1° phase shift is ≈ 4.98611111 × 10⁶ Hz.
