Soumendra
Nath Thakur
ORCiD:
0000-0003-1871-7803
Dated
Abstract:
This study delves into the intricate interrelations among phase shift, angular frequency, and time in wave mechanics through rigorous mathematical analysis. By establishing a comprehensive framework and deriving key equations, it elucidates the dynamics of these relationships. The definition of parameter 'n' facilitates precise calculations of phase shift, while adjustments for time changes and degree representation enhance clarity. Correlations between phase shift representation in degrees and frequency deepen understanding, alongside discussions on phase shift values, time delay, and angular frequency changes. The study concludes with equations determining resulting angular and frequency changes, providing valuable insights into wave propagation dynamics.
Keywords: Phase Shift, Angular Frequency, Time, Wave Mechanics, Mathematical Analysis
Tagore's
Electronic Lab, WB.
Correspondence:
postmasterenator@gmail.com
postmasterenator@telitnetwork.in
Declaration:
The Author declares no conflict of interest.
List of Sections in Mathematical Presentation:
1. Introduction to Phase
Shift and Its Relation with Angular Frequency and Time
2. Adjusting Phase Shift
Equation for Time Changes and Degree Representation Using Parameter 'n'
3. Correlation between
Phase Shift Representation in Degrees and Frequency
4. Relationship between
Phase Shift Values in Degrees and Time Delay/Time Distortion
5. Correlations between
Phase Shift, Time, and Angular Frequency Changes
6. Relationship between
Phase Shift, Time Distortion, and Frequency Changes
7. Determination of
Resulting Angular and Frequency Changes
8. Adjustments for Phase
Shift, Time, and Frequency Changes
9. Derivation of Resulting
Frequency Equation
10. Equation for Resulting
Frequency in Terms of Time Shift
Introduction:
Understanding the intricate relationships between phase shift and frequency is fundamental to comprehending wave mechanics. In this exploration, we delve into the nuanced interplay among phase shift, angular frequency, and time, aiming to provide a comprehensive elucidation of these crucial concepts. Through a systematic analysis and mathematical formulations, we unravel the complexities inherent in phase shift calculations and their implications for frequency alterations. By delineating the correlation between phase shift representation in degrees and frequency, we elucidate the mechanisms governing wave behaviour. Moreover, we investigate the dynamic relationship between phase shift, time distortion, and frequency changes, shedding light on the interconnected nature of these phenomena. This study endeavours to offer a robust framework for understanding phase shift and frequency dynamics, paving the way for deeper insights into wave mechanics.
Method:
Literature Review: Conduct a comprehensive review of existing literature on wave mechanics, phase shift, and frequency relationships. Identify key concepts, theories, and equations relevant to the study.
Conceptual Framework Development: Develop a conceptual framework outlining the fundamental principles of phase shift and frequency relationships in wave mechanics. Define key terms and parameters, such as phase shift, angular frequency, time delay, and frequency modulation.
Equation Derivation: Derive mathematical equations describing the relationship between phase shift, time variations, and frequency changes. Incorporate parameters such as 'n' to represent phase shift in degrees relative to a full cycle
Correlation Analysis: Analyse the correlation between phase shift representation in degrees and frequency variations. Explore how changes in phase shift affect frequency modulation and vice versa.
Simulation Studies: Conduct simulation studies using mathematical models derived from the equations to simulate various scenarios of phase shift and frequency changes. Investigate the impact of different parameters on wave propagation and frequency modulation.
Experimental Validation: Validate the derived equations and simulation results through experimental studies. Perform experiments using wave generators and measuring devices to observe phase shift, time delay, and frequency changes in real-world wave phenomena.
Data Analysis: Analyse the data collected from simulations and experiments to identify patterns, trends, and correlations between phase shift, time variations, and frequency changes. Quantify the relationship between these variables using statistical analysis techniques.
Discussion and Interpretation: Discuss the findings in the context of existing literature and theoretical frameworks. Interpret the results to gain insights into the underlying mechanisms governing phase shift and frequency relationships in wave mechanics.
Conclusion and Implications: Summarize the key findings of the study and their implications for understanding wave propagation and frequency modulation. Discuss the potential applications of the research findings in various fields, such as telecommunications, signal processing, and acoustics.
Future Research Directions: Identify areas for further research and exploration based on the limitations and gaps identified in the study. Propose potential avenues for extending the research to advance our understanding of phase shift and frequency relationships in wave mechanics.
Mathematical Presentation:
1. Introduction to Phase Shift and Its Relation with Angular Frequency and Time:
Introduction to Phase Shift" and "Definition of Parameter 'n'.
This presentation aims to provide a comprehensive understanding of phase shift (Φ) concerning angular frequency (ω), and time (t), along with the convenient representation of phase shift in degrees 'T(deg)'.
In wave mechanics, the phase shift in frequency 'f' is commonly denoted by the symbol 'Φ' (phi). The equation for phase shift 'Φ' for a frequency 'f' is expressed as:
Φ = ω⋅t
Where:
• Φ (phi) represents the
phase shift,
• ω (omega) denotes the
angular frequency, measured in radians per unit time,
• t signifies the time.
1.1. Definition of Parameter 'n' in Phase Shift Calculations:
This definition aims to provide a comprehensive understanding of phase shift representation, incorporating the equation n =−360 + x°. 'n' serves as a parameter used to express the degree of phase shift (x°) in a manner that accounts for a full cycle (360°). It's defined as:
n = −360 + x°
Where:
• n is the parameter representing the phase shift.
• x° is the degree of phase shift, ranging from zero to a full cycle.
This equation ensures that phase shifts are properly aligned relative to a full cycle. The constant term -360 adjusts the phase shift to maintain consistency, as 360° represents one complete cycle or period.
The mathematical presentation of the definition clarifies the role of the equation:
• When the phase shift from 'x' to 'n' is −360°, and 'n' is positioned at '0' - one cycle before 'x' - then a phase shift of 'x' is 0° at the origin of the oscillation or wave.
• When 'n' is at 0° phase shift from the origin of 'x', positioned at '0' - one cycle before 'x' - then 'x' represents a phase shift of 360° of the oscillation or wave.
• When 'n' is at 90° phase shift from the origin of the oscillation or wave, then 'x' represents a phase shift of 450°.
• When 'n' is at 360° phase shift from the origin of the oscillation or wave, then 'x' represents a phase shift of 720°.
Thus, the definition ensures that 'n' properly accounts for the phase shift relative to a full cycle and maintains mathematical consistency in phase shift calculations.
Additionally, we can express the relationship between phase shift (x°), parameter 'n', and the phase shift represented in degrees (T) as follows:
x° - n = T
This expression highlights the equivalence between the phase shift represented by 'x°' and the parameter 'n', which in turn determines the phase shift 'T' in degrees.
2. Adjusting Phase Shift Equation for Time Changes and Degree Representation Using Parameter 'n':
When there is a change in time, such as a time delay or time distortion (Δt), the equation is adjusted to incorporate this change.
Φ = 360 · f · Δt
This equation represents the phase shift 'Φ' in terms of the frequency 'f' and the time shift Δt, allowing for the calculation of phase shift when there is a change in time.
Alternatively, the phase shift in frequency 'f' can be expressed in degrees using the symbol T(deg). While ω typically represents angular frequency in radians per unit time, T(deg) allows for the representation of phase shift in degrees rather than radians.
The derived equations for expected values of x° represented through 'n' in terms of 'x°', we can use the relation 'n = −360 + x°'. This equation indicates that 'n' is equal to the degree of phase shift 'x°', adjusted by a constant (-360). We can rewrite this equation to express 'x°' in terms of 'n' as follows:
x° = n + 360
Now, substituting this expression for 'x°' into the equation 'T(deg) = (x/360)⋅T', we get:
Simplifying this equation:
T(deg) = (n/360) ⋅ T + T
This equation represents the phase shift 'T(deg)' in degrees in relation to the parameter 'n', which represents the specific phase shift '(-360 + x°)' affecting the frequency change. It captures the expected values of 'x°' through the parameter 'n'.
3. Correlation between Phase Shift Representation in Degrees and Frequency:
The phase shift in frequency 'f' is conveniently represented by the symbol T(deg) to indicate phase shift in degrees. As the time interval or period (T) of the frequency (f) is represented by the expression T = 360°, T(deg) is also a representation of the degree of phase shift in T. The following expressions describe the relationship between phase shift and the degree of phase shift:
Time interval or period T = 360°
For 1° of phase shift, T(deg) = (1/360) ⋅ T
For x° of phase shift, T(deg) = (x/360) ⋅ T
The numerator '1' in the equation T(deg) = (1/360) ⋅ T increases linearly with the degree of phase shift, denoted as x°, which progresses as 1°, 2°, 3°... and so forth. Given that T = 1/f, for 1° of phase shift, T(deg) = (1/f) ⋅ (1/360), and for x° of phase shift, T(deg) = (x/f) ⋅ (1/360).
This statement elucidates the relationship between phase shift T(deg) and the degree of phase shift x for a given frequency f. It highlights the direct correlation between the numerator '1' in the equation "T(deg) = (1/360) ⋅T " and the degree of phase shift. Moreover, by substituting T = 1/f into the equation, expressions for both 1° and x° of phase shift are derived.
Since the equation T(deg) = 1/360 ⋅ T can be expressed as:
T(deg) = (x/f) ⋅ (1/360)
Where:
•
T(deg): Represents the phase shift in degrees.
• x:
Represents the degree of phase shift.
• f:
Represents the frequency.
•
1/360: This term is a constant factor that converts the phase shift from
radians to degrees.
• Here, x represents the ratio of the degree of phase shift to the frequency, indicating how the phase shift changes relative to the frequency.
This single equation efficiently captures both scenarios: for 1° of phase shift and for x° of phase shift.
4. Relationship between Phase Shift Values in Degrees and Time Delay/Time Distortion:
The time interval T(deg) for a 1° phase shift is inversely proportional to the frequency (f), resulting in a corresponding time shift (Δt). Utilizing the previously derived equations, we establish that for an x° phase shift, T(deg) = (x/360) ⋅ T. Since T = 1/f, this equation can be expressed as T(deg) = (x/f) ⋅ (1/360), or alternatively as (n/360) ⋅ T + T. This relationship corresponds to the time shift (Δt). The expression is:
T(deg) = (x/f) ⋅ (1/360) = (n/360)⋅T+T = Δt.
This presentation discusses the relationship between phase shift values represented in degrees (T(deg)) and the corresponding time delay or time distortion (Δt). It begins by highlighting that the time interval T(deg) for a 1° phase shift is inversely proportional to the frequency (f), leading to a time shift Δt. The statement then proceeds to use derived equations to express T(deg) for an x° phase shift.
Given that T = 1/f, the equation T(deg) = (x/360) ⋅ T can be re-expressed as T(deg) = (x/f) ⋅ (1/360), demonstrating the relationship between phase shift, frequency, and time. Additionally, it introduces the parameter 'n,' representing the specific phase shift (-360 + x°), which allows for another representation of T(deg) as (n/360) ⋅ T + T. This equation showcases the relationship between phase shift values and the resulting time delay.
The concluding part of the statement reaffirms the equivalence of the two expressions for T(deg) and highlights their relationship to the time delay, denoted as Δt. Therefore, the statement clarifies how phase shift values correspond to time distortions, providing a clear understanding of their relationship.
5. Correlations between Phase Shift, Time, and Angular Frequency Changes.
From the equations discussed previously, it's evident that changes in Phase Shift (Φ) or Time (t) correspond to alterations in Angular Frequency (ω). Considering a change in angular frequencies, denoted as Δω, it reflects the difference between two angular frequencies: the original angular frequency (ω₀) and the resulting angular frequency (ω₁). Therefore, the change in angular frequency is expressed as
Δω = (ω₀ - ω₁)
6. Relationship between Phase Shift, Time Distortion, and Frequency Changes:
Similarly, from the previously discussed equations, it's apparent that changes in Phase Shift 'T(deg)' or Time Delay/Time Distortion (Δt) lead to adjustments in frequency (f). Now, let's examine the change in frequencies, denoted as Δf. Δf represents the disparity between two frequencies: the original frequency (f₀) and the resulting frequency (f₁). Thus, the change in frequencies is expressed as:
Δf = (f₀ - f₁)
7. Determination of Resulting Angular and Frequency Changes:
The resulting Angular frequency ω₁ can be determined either when the original/source Angular frequency ω₀ is known or measured, or Time (t) is known or measured.
Similarly, the resulting frequency f₁ can be determined either when the original/source frequency f₀ is known or measured, and any one of either phase shift in x° or time shift (Δt) is known or measured.
8. Adjustments for Phase Shift, Time, and Frequency Changes:
The above mentioned equations express the correlations between phase shift, time, angular frequency, and frequency changes in wave mechanics. It begins by noting that changes in phase shift (Φ) or time (t) induce alterations in angular frequency (ω) and frequency (f). The difference between two angular frequencies, namely the original (ω₀) and resulting (ω₁) frequencies, is expressed as Δω = (ω₀ - ω₁), signifying a change in angular frequency.
Similarly, adjustments in phase shift 'T(deg)' or time delay/time distortion 'Δt' lead to changes in frequency 'f'. The disparity between the original (f₀) and resulting (f₁) frequencies is represented by Δf = (f₀ - f₁), indicating a change in frequency.
Furthermore, the text discusses how the resulting angular frequency (ω₁) can be determined when the original angular frequency (ω₀) and time (t) are known. Likewise, the resulting frequency (f₁) can be determined when the original frequency (f₀) is known, along with either the phase shift in x° or the time shift (Δt).
This following equation represents the resulting angular frequency ω₁ as the original angular frequency ω₀ minus the product of the angular frequency ω and time t.
ω₁ = 2πf₀ - (ω·t)
Breakdown of the components:
•
ω₁: This represents the resulting angular frequency, which is the frequency of
a wave after some change has occurred.
•
2πf₀: This term represents the original angular frequency ω₀ converted from frequency
f₀ using the formula ω = 2πf. It denotes the angular frequency before any
change or alteration.
• (ω•t): This term represents the product of the angular frequency ω and time t. It accounts for the change or alteration in the angular frequency due to the passage of time.
The equation describes the equation for the resulting angular frequency ω₁ in terms of the original angular frequency ω₀ and the passage of time t.
9. Derivation of Resulting Frequency Equation:
This equation defines the resulting frequency f₁ in relation to the original frequency f₀ and the phase shift x° (in degrees).
f₁ = (f₀ - x) / {T(deg) · 360}
Breakdown of the components:
•
f₁: Represents the resulting frequency, indicating the frequency of a wave
post-change
•
f₀: Denotes the original frequency, representing the frequency before any
alteration.
• x:
Signifies the degree of phase shift, indicating the extent of change in the
wave's phase.
•
T(deg): Represents the phase shift in degrees relative to the wave's period.
• 360: Represents one complete cycle of the wave, equivalent to 360° or 2π radians.
This equation illustrates how the resulting frequency f₁ varies with the original frequency f₀ and the degree of phase shift x°. The phase shift in degrees T(deg) adjusts the frequency, considering the extent of phase shift relative to the wave's period. Overall, the equation effectively captures the interplay between phase shift, frequency, and the resulting frequency following a change.
10. Equation for Resulting Frequency in Terms of Time Shift:
This equation represents the resulting frequency f₁ in terms of the time shift Δt.
f₁ = 1/(Δt×360)
Breakdown of the components:
•
f₁: Represents the resulting frequency, which is the frequency of a wave after
some change has occurred
•
Δt: Represents the time shift, also known as time distortion. It is the
difference in time between the original wave and the resulting wave, causing a
change in frequency. When Δt is positive, it implies a delay in time, leading
to a decrease in frequency. Conversely, when
• Δt is negative, it signifies an advancement in time, resulting in an increase in frequency.
Therefore, the equation f₁ = 1/(Δt×360) succinctly expresses how the resulting frequency f₁ is determined by the reciprocal of the product of the time shift Δt and 360. This equation captures the relationship between frequency changes and time shifts in the context of wave mechanics, providing a mathematically consistent representation.
Discussion:
The study provides a comprehensive exploration of phase shift and frequency relationships within the context of wave mechanics. It delves into various mathematical equations, derivations, and concepts to elucidate the intricate connections between phase shift, time, angular frequency, and frequency changes. Here, we discuss key points and insights drawn from the study:
Definition of Phase Shift and Parameter 'n': The presentation begins by defining phase shift (Φ) in terms of angular frequency (ω) and time (t). It introduces parameter 'n' as a numerical value representing the degree of phase shift, ensuring consistency in calculations and interpretations. The inclusion of 'n' facilitates a clear understanding of phase shift relative to a full cycle, enhancing mathematical precision.
Adjustments for Time Changes and Degree Representation: The text explores adjustments to the phase shift equation to incorporate time changes, such as time delay or distortion. It demonstrates how phase shift can be represented in degrees using parameter 'n', allowing for convenient interpretation and calculation. The derived equations provide a systematic approach to account for time variations and their effects on phase shift.
Correlation between Phase Shift and Frequency: A significant aspect discussed is the correlation between phase shift representation in degrees T(deg) and frequency. By expressing phase shift in degrees, the text establishes a direct relationship between phase shift values and the degree of phase shift for a given frequency. This correlation aids in understanding how changes in phase shift impact frequency alterations.
Relationship with Time Delay and Frequency Changes: The presentation elucidates the relationship between phase shift values in degrees and time delay or distortion. It demonstrates how variations in phase shift correspond to changes in frequency, highlighting the inverse proportionality between time interval and frequency for a given phase shift. This relationship provides valuable insights into the dynamic nature of wave phenomena.
Derivation of Resulting Frequency Equation: A key highlight is the derivation of the resulting frequency equation, which relates the original frequency, phase shift, and period of the wave. The equation captures the interplay between these factors, offering a concise representation of how frequency changes following a phase shift. This derivation enhances understanding of frequency modulation in wave propagation.
Equation for Resulting Frequency in Terms of Time Shift: Finally, the presentation introduces an equation for the resulting frequency in terms of time shift. This equation provides a mathematical framework for analysing frequency changes resulting from time distortions, further enriching the understanding of wave dynamics.
The discussed study offers a rigorous mathematical treatment of phase shift and frequency relationships, shedding light on fundamental principles underlying wave mechanics. By exploring the intricacies of phase shift, time variations, and frequency alterations, the presentation contributes to a deeper understanding of wave phenomena and their mathematical representations.
Conclusion:
In conclusion, the comprehensive discussion on phase shift and frequency relationships presented in the quoted text offers valuable insights into the mathematical foundations of wave mechanics. Through meticulous exploration and derivation of equations, the text illuminates the intricate interplay between phase shift, time variations, and frequency changes. By defining parameters, adjusting equations, and deriving relationships, it provides a systematic framework for understanding how phase shift influences frequency modulation and vice versa.
The introduction of parameter 'n' facilitates consistent and precise calculations, ensuring accurate representation of phase shift relative to a full cycle. This parameterization enhances the interpretability of phase shift in degrees, enabling convenient analysis of wave phenomena. Furthermore, the correlation between phase shift representation in degrees and frequency deepens our understanding of their dynamic relationship, elucidating how changes in phase shift impact frequency alterations.
The derivation of resulting frequency equations and expressions for frequency changes in terms of time shift enriches our understanding of wave dynamics, offering mathematical tools for analysing frequency modulation. These equations capture the complex interactions between phase shift, time distortions, and frequency variations, providing a comprehensive framework for studying wave propagation.
Overall, the discussion presented in the quoted text contributes significantly to the field of wave mechanics by providing a rigorous mathematical treatment of phase shift and frequency relationships. By elucidating fundamental principles and deriving key equations, it advances our understanding of wave phenomena and lays the groundwork for further research in this domain.
References:
[1] Thakur, S. N., &
Bhattacharjee, D. (2023a). Phase shift and infinitesimal wave energy loss equations.
ResearchGate.
https://www.researchgate.net/publication/378462690_Phase_Shift_and_Infinitesimal_Wave_Energy_Loss_Equations
[2] Thakur, S. N. (2023j).
Wave Dynamics -Interplay of phase, frequency, time, and energy. ResearchGate.
https://doi.org/10.13140/RG.2.2.16473.70242
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Decoding Time Dynamics: The Crucial Role of Phase Shift Measurement amidst
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https://doi.org/10.32388/mrwnvv
[4] Thakur, S. N., Samal,
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Relationships made easy: Time Intervals, Phase Shifts, and Frequency in
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