Learning Materials
Features
Discover
Did you know that there can be two exact waves, where the only difference is that one of them has been shifted from a certain point of reference? A wave is a spatial and temporal process in which energy is transported. A periodic wave is a wave that repeats as a function of position and time. Mathematically, periodic waves are used to describe oscillations and simple harmonic motion, which describes the movement of spring-mass systems. This type of wave is described by two characteristics: a magnitude and a phase. In this article, we will discuss the concept of the phase angle in a periodic wave.
Millions of flashcards designed to help you ace your studies
Sign up for freeThe phase angle is the ___ component of a periodic wave.
The phase angle is not part of the solution to the differential equations that describes the oscillatory motion for spring-mass systems.
The phase angle is the angular component of a periodic wave, such that it is defined as the argument of the sine function:
By choosing \(\phi_0\), we specify the oscillating object's ___ to make sure we have the correct equation with the oscillator's position, no matter where it may have been located at \(t=0\).
What is the initial phase if the object's initial position and amplitude are equal to \(1\;\mathrm m\)?
To find the initial phase, we must use the following equation:
Consider the function \(x=3\sin\left(\pi t+\frac\pi2\right)\). What is the position of the object at \(t=0\)?
The following expressions yield the same outputs:
A negative initial phase will shift the function horizontally to the ___, while a positive initial phase will shift the function horizontally to the ___.
What type of functions are used to describe oscillating systems like spring-mass systems?
Consider the function \(f(x)=\sin\left(x-\frac\pi4\right)\). The sine function has shifted horizontally to the ___ by an amount of \(\frac\pi4\).
The phase angle is the ___ component of a periodic wave.
The phase angle is not part of the solution to the differential equations that describes the oscillatory motion for spring-mass systems.
The phase angle is the angular component of a periodic wave, such that it is defined as the argument of the sine function:
By choosing \(\phi_0\), we specify the oscillating object's ___ to make sure we have the correct equation with the oscillator's position, no matter where it may have been located at \(t=0\).
What is the initial phase if the object's initial position and amplitude are equal to \(1\;\mathrm m\)?
To find the initial phase, we must use the following equation:
Consider the function \(x=3\sin\left(\pi t+\frac\pi2\right)\). What is the position of the object at \(t=0\)?
The following expressions yield the same outputs:
A negative initial phase will shift the function horizontally to the ___, while a positive initial phase will shift the function horizontally to the ___.
What type of functions are used to describe oscillating systems like spring-mass systems?
Consider the function \(f(x)=\sin\left(x-\frac\pi4\right)\). The sine function has shifted horizontally to the ___ by an amount of \(\frac\pi4\).
Achieve better grades quicker with Premium
Geld-zurück-Garantie, wenn du durch die Prüfung fällst
Review generated flashcards
Start learning or create your own AI flashcards
Team Phase Angle Teachers
In previous articles, we discussed the differential equation that describes oscillatory movement, particularly simple harmonic motion. We know the solution that satisfies the equation is expressed as
$$x=A\sin\left(\omega t+\phi_0\right).$$
Where \(A\) is the amplitude in meters \((\mathrm m)\), \(\omega\) is the angular frequency in radians per second \((\frac{\mathrm{rad}}{\mathrm s})\), and \(\phi_0\) is the initial phase in radians \((\mathrm{rad})\).
The phase angle is the angular component of a periodic wave, such that it is defined as the argument of the sine function, \(\omega t+\phi_0\). By choosing \(\phi_0\), we specify the oscillating object's initial position to make sure we have the correct equation with the oscillator's position, no matter where it may have been located at \(t=0\). We can restate the above equation in terms of the symbol \(\phi\) for the phase angle.
$$\begin{align*}\phi&=\omega t+\phi_0,\\x&=A\sin\left(\phi\right).\end{align*}$$
To determine the initial phase we use the following formula:
$$\phi_0=\sin^{-1}\left(\frac{x_0}A\right),$$
where \(A\) is the amplitude in meters \((\mathrm m)\) and \(x_0\) is the initial position of the object at \(t=0\) in meters \((\mathrm m)\).
A simple harmonic oscillator has an amplitude of \(3.0\;\mathrm{cm}\) and a frequency of \(4.0\;\mathrm{Hz}\). At time \(t=0\), its position is \(y=3.0\;\mathrm{cm}\). Where is it at time \(t=0.3\;\mathrm s\)?
The amplitude is \(A=0.03\;\mathrm m\) and the angular frequency is \(\omega=2\pi f=2\pi(4.0\;\mathrm{Hz})=8\pi\;{\textstyle\frac{\mathrm{rad}}{\mathrm s}}\). Now we can determine the initial phase,
\begin{align*}\phi_0&=\sin^{-1}\left(\frac{y_0}A\right),\\\phi_0&=\sin^{-1}\left(\frac{0.03\;\mathrm m}{0.03\;\mathrm m}\right),\\\phi_0&=\frac\pi2.\end{align*}
Now we know the position of the oscillator at any moment in time,
$$y(t)=0.03\sin\left(8\pi t+\frac\pi2\right).$$
We can find the position of the oscillator at time \(t=0.3\;\mathrm s\),
\begin{align*}y(0.3\;\mathrm s)&=(0.03\;\mathrm m)\sin\left((8\pi\;{\textstyle\frac{\mathrm{rad}}{\mathrm s}})(0.3\;\mathrm s)\;+\;\frac\pi2\;\mathrm{rad}\right),\\y(0.3\;\mathrm s)&=0.0093\;m.\end{align*}
The position of an oscillator is given by the equation:
$$y=(0.04\;\mathrm m)\sin\left((6\pi\;{\textstyle\frac{\mathrm{rad}}{\mathrm s}})t-\frac\pi2\;\mathrm{rad}\;\right).$$
Where is the oscillator at time \(t=0\)?
\begin{align*}y(0\;\mathrm s)&=(0.04\;\mathrm m)\sin\left((6\pi\;{\textstyle\frac{\mathrm{rad}}{\mathrm s}})(0\;\mathrm s)-\frac\pi2\;\mathrm{rad}\;\right),\\y(0\;\mathrm s)&=-0.04\;\mathrm m.\end{align*}
The initial phase will determine if a sine or cosine function is used to describe the position of the oscillating object. For example, if \(\phi_0=\frac\pi2\) we can use a cosine function instead of a sine function with the initial phase. This is due to the trigonometric identity, \(\sin\left(\frac\pi2+\theta\right)=\cos\left(\theta\right)\). The table below clarifies how the two expressions yield the same results at any time.
| Equation | \(t=0\) | \(t=\frac\pi{2\omega}\) |
| \(\sin\left(\omega t+\;\frac\pi2\right)\) | 1 | 0 |
| \(\cos\left(\omega t\right)\) | 1 | 0 |
As a side note, the phase angle plays a very important role in experimental physics, especially in electronics where there is a direct relationship between the voltage and sinusoidal functions. In electronics, the phase angle refers to the angular displacement between the voltage and current waveforms in an alternating current circuit.
We have covered the theoretical definition of the phase angle and the initial phase. How do we understand the impact of changing the initial phase of a sinusoidal function? It is easier to understand if we actually represent the sinusoidal functions in a graph.
Fig. 1 - Different examples of initial phases to visualize the impact of adjusting the initial phase of a sinusoidal function.
From the image above we see that at the initial value \(x=0\), \(f(0)=\sin\left(0\right)=0\). For the same sine function with an initial phase \(\phi_0=\frac{-\pi}4\), \(f(0)=\sin\left(0-\frac\pi4\right)=-\frac{\sqrt2}2\) and \(f(\frac\pi4)=\sin\left(\frac\pi4-\frac\pi4\right)=0\). We notice that the sine function has shifted horizontally to the right by an amount of \(\frac\pi4\). If we change the initial phase to \(\phi_0=-\pi\), we notice that the sine function shifts to the right by an amount of \(\pi\). We notice a pattern here, a negative initial phase will shift the function horizontally to the right, while a positive initial phase will shift the function horizontally to the left. This is visually represented in the figure below.
Fig. 2 - Sine function: case when the initial phase is zero.
Fig. 3 - Effect of a positive initial phase on a sinusoidal function.
Fig. 4 - Effect of a negative initial phase on a sinusoidal function.
Sign up for free to gain access to all our flashcards.
How to find phase angle?
To find the phase angle at a certain moment in time you must multiply the angular frequency by the time and add the sum of the initial phase: wt+initial phase.
What is phase angle formula?
To find the phase angle at a certain moment in time you must multiply the angular frequency by the time and add the sum of the initial phase: wt+initial phase.
What is Phase Angle?
The phase angle is the angular component of a periodic wave, such that it is defined as the argument of the sine function, wt+initial phase.
How do you describe phase angle?
To describe the phase angle at a certain moment in time you must multiply the angular frequency by the time and add the sum of the initial phase: wt+initial phase.
What is an example of Phase Angle?
For example, a wave with angular frequency of pi and an initial phase of pi over 2 at the initial moment of the oscillation when time is zero, will have a phase angle of pi over 2. Remember that the phase angle is given by wt+initial phase.
At StudySmarter, we have created a learning platform that serves millions of students. Meet the people who work hard to deliver fact based content as well as making sure it is verified.
Lily Hulatt is a Digital Content Specialist with over three years of experience in content strategy and curriculum design. She gained her PhD in English Literature from Durham University in 2022, taught in Durham University’s English Studies Department, and has contributed to a number of publications. Lily specialises in English Literature, English Language, History, and Philosophy.
Get to know Lily
Gabriel Freitas is an AI Engineer with a solid experience in software development, machine learning algorithms, and generative AI, including large language models’ (LLMs) applications. Graduated in Electrical Engineering at the University of São Paulo, he is currently pursuing an MSc in Computer Engineering at the University of Campinas, specializing in machine learning topics. Gabriel has a strong background in software engineering and has worked on projects involving computer vision, embedded AI, and LLM applications.
Get to know Gabriel
Explanations 
Flashcards 
StudySmarter AI 
Notes 
Study Plans 
Study Sets 
Exams 
Find a job 
Student Deals 
Magazine 
Mobile App 
