Periodic Wave

A periodic phenomenon is one that occurs at regular intervals. Periodic phenomena are all around us, from the rising of the sun every morning to the repeating structure of the DNA in our cells, it's clear that nature loves to repeat itself!

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What is the frequency of a wave travelling at \(v=200\,\mathrm{m}\,\mathrm{s}^{-1}\) if it has a wavelength of \(\lambda=0.5\,\mathrm{m}\)?

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The frequency \(f\) and time period \(T\) of a periodic wave are inversely related to each other. True or False?

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What is the frequency of a periodic wave if it makes 2 complete cycles in \(3\,\mathrm{s}\)?

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What is the definition of the angular frequency \(\omega\) of a periodic wave?

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Which equation relates the angular wavenumber \(k\) to the wavelength \(\lambda\) of a periodic wave?

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Consider a periodic wave, with angular frequency \(\omega\) and angular wavenumber \(k\), at a fixed time \(t\) with a maximum displacement at \(x=0\). Which formula describes how the displacement changes with distance from \(x=0\)? 

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Consider a snapshot of a sound wave with a wavelength \(\lambda=0.2\,\mathrm{m}\) and amplitude \(A=0.05\,\mathrm{m}\). What is the displacement of an air particle at distance of \(1\,\mathrm{m}\) away from a reference point \(x=0\) if the displacement is \(0\) at \(x=0\)?

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Which of these is not a difference between periodic waves and pulses?

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What is the equation for the displacement of a point as a function of time \(t\) on a periodic wave with amplitude \(A\) and time period \(T\), if the point has has zero displacement at \(t=0\)

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What is the frequency of a wave travelling at \(v=200\,\mathrm{m}\,\mathrm{s}^{-1}\) if it has a wavelength of \(\lambda=0.5\,\mathrm{m}\)?

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  • Mo

The frequency \(f\) and time period \(T\) of a periodic wave are inversely related to each other. True or False?

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  • Immunology
  • Cell Biology
  • Mo

What is the frequency of a periodic wave if it makes 2 complete cycles in \(3\,\mathrm{s}\)?

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  • Immunology
  • Cell Biology
  • Mo

What is the definition of the angular frequency \(\omega\) of a periodic wave?

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  • Immunology
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  • Mo

Which equation relates the angular wavenumber \(k\) to the wavelength \(\lambda\) of a periodic wave?

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  • + Add tag
  • Immunology
  • Cell Biology
  • Mo

Consider a periodic wave, with angular frequency \(\omega\) and angular wavenumber \(k\), at a fixed time \(t\) with a maximum displacement at \(x=0\). Which formula describes how the displacement changes with distance from \(x=0\)? 

Show Answer
  • + Add tag
  • Immunology
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  • Mo

Consider a snapshot of a sound wave with a wavelength \(\lambda=0.2\,\mathrm{m}\) and amplitude \(A=0.05\,\mathrm{m}\). What is the displacement of an air particle at distance of \(1\,\mathrm{m}\) away from a reference point \(x=0\) if the displacement is \(0\) at \(x=0\)?

Show Answer
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  • Immunology
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  • Mo

Which of these is not a difference between periodic waves and pulses?

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What is the equation for the displacement of a point as a function of time \(t\) on a periodic wave with amplitude \(A\) and time period \(T\), if the point has has zero displacement at \(t=0\)

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    Analyzing these phenomena scientifically is particularly elegant as we only have to understand a few characteristic features to get the full picture. In this article, we will apply this analysis to one of the fundamental phenomena in physics, periodic waves. By doing so we can find a full mathematical description of periodic waves from just a couple of key quantities, the wavelength, and the frequency of a wave.

    Definition of Periodic Waves

    In physics, waves are a type of energy transfer caused by an initial disturbance that is then propagated through space and time. The size of the disturbance defines the amplitude of the wave. If a wave has a continually repeating pattern, made up of cycles, such as particles oscillating around an equilibrium point, we call it a Periodic Wave. Periodic waves have several key quantities by which they can be characterized, depending on the time and distance between cycles. The characteristic time taken, in seconds \(\mathrm{s}\), for each cycle is known as the Time Period \(T\) of the wave. The number of cycles that occur within one second is known as the Frequency \(f\) of the wave, and is measured in Hertz \(\mathrm{Hz}\) equivalent to \(\mathrm{s}^{-1}\). By definition, the frequency and time period of a wave are inverses.

    \[f=\frac{1}{T}\]

    Periodic Waves Graph of a sine wave with the time period and amplitude annotated StudySmarterFig. 1 - A graph of a periodic wave, demonstrating how displacement from equilibrium varies over time. From the graph, we can see the amplitude \(A\) and time period \(T\) of the wave.

    Similarly, the characteristic distance taken for a cycle is known as the Wave Length \(\lambda\). The wavelength is related to the frequency of the wave by the velocity \(v\) with which the wave propagates through space.

    \[v=f\lambda=\frac{\lambda}{T}\]

    Periodic Waves Graph of a periodic wave with the wavelength and amplitude of the wave annotated StudySmarter Fig. 2 - The wavelength of a periodic wave is defined as the distance between two equivalent points of the wave.

    Sound Waves are a clear example of a periodic wave. An initial disturbance, such as someone clapping their hands, causes particles in the air to oscillate around their equilibrium position. These oscillations are then transferred through the air causing the sound to be heard away from the person's hands. The rate at which the air particles oscillate determines the frequency of the sound, which is how our ears characterize the 'pitch' of a sound.

    The speed of sound, which determines the relationship between a sound's frequency and wavelength, varies in different mediums. For example, in air the speed of sound is \(v=343\,\mathrm{m\,s}^{-1}\) so a sound with a frequency of \(f=300\,\mathrm{Hz}\) will have a wavelength of

    \[\lambda=\frac{v}{f}=\frac{343}{300}=1.14\,\mathrm{m}\]

    Transverse and Longitudinal Periodic Waves

    Periodic waves can be split into two main types of waves, depending on the direction of the displacement the wave causes. Longitudinal waves cause oscillations that are parallel to the direction of energy transfer. Some key examples of longitudinal are sound waves and stress waves within materials.

    On the other hand, many periodic waves are such that the direction of oscillation is perpendicular to the direction of energy transfer. For example, light is a type of electromagnetic radiation caused by electric and magnetic fields which oscillate at right angles to the direction the light travels in.

    Periodic Waves Graph of oscillating electric and magnetic fields forming a transverse electromagnetic wave StudySmarterFig. 4 - Electromagnetic waves are a type of transverse wave. The electric \(E\) and magnetic \(B\) fields oscillate perpendicularly to the direction the wave travels in \(z\).

    Pulse vs Periodic Wave

    Whilst the focus of this article is on periodic waves, its worth briefly looking at a-periodic waves known as pulses to emphasize the defining features of a periodic wave. Pulses are a very common type of energy transfer in physics, caused by sudden short disturbances which propagate as a short burst of energy. Whilst a pulse may have cycles like a periodic wave, pulses usually only contain one or two cycles such that we cannot properly define a wavelength or frequency of a pulse.

    Periodic Waves Graph showing a single pulse travelling leftward StudySmarterFig. 5 - We can model pulses as moving sharp peaks, note that a pulse contains no oscillations.

    For example, consider dropping a pebble into a pool. The disturbance caused by the pebble would lead to a ripple of a few water waves traveling outwards, however shortly after the pebble was dropped the water would return to equilibrium and no further waves would be produced. We say that the pebble produced pulses in the water, if instead the water was continuously disturbed, like with a wave machine, then the water waves would be periodic.

    The formula for Periodic Waves

    Given what we know about periodic waves so far, let's consider how we could best represent periodic waves using mathematical functions. As we have seen, the period of a wave is defined by its characteristic wavelength or time period. Consider a periodic wave, with an amplitude A, and a wavelength of \(\lambda\). This means we are looking for a function that satisfies the following conditions.

    \[\begin{align}f(x)&=f(x+\lambda)\\\max |f(x)|&=A\end{align}\]

    As you might be able to guess, the trigonometric sine and cosine functions are the functions we are looking for, given that they also satisfy similar periodicity conditions.

    \[\sin(x)=\sin(x+2n\pi),\,\cos(x)=\sin(x+2n\pi)\]

    As the sine and cosine functions can be made equivalent by adding a phase of \(\frac{\pi}{2}\), we simply choose which function we want, depending on the initial conditions of the wave. As we usually think of waves as starting with maximum displacement, we'll consider the cosine function.

    By smartly choosing scaling factors for the cosine function, we can satisfy the conditions needed

    \[\begin{align}f(x)&=A\cos\left(\frac{2\pi}{\lambda}x\right)\\\max|f(x)|&=A\\f(x+\lambda)&=A\cos\left(\frac{2\pi}{\lambda}\left(x+\lambda\right)\right)\\&=A\cos\left(\frac{2\pi}{\lambda}x+2\pi\right)\\&=A\cos\left(\frac{2\pi}{\lambda}x\right)\\&=f(x)\end{align}\]

    So a periodic wave, with amplitude \(A\) and wavelength \(\lambda\), at one fixed instant of time \(t\), is described by the function

    \[f(x)=A\cos\left(\frac{2\pi}{\lambda}x\right)\]

    In physics the quantity \(\frac{2\pi}{\lambda}\) is called the 'angular wavenumber' usually denoted \(k\).

    Applying the same thinking, we can find the function describing the oscillation of a single point on the wave over time. If the wave has a time period of \(T\) and amplitude \(A\), then the oscillations as a function of time are given by

    \[\begin{align}h(t)&=A\cos\left(\frac{2\pi}{T}t\right)\\&=A\cos\left(2\pi f t\right)\\&=A\cos\left(\omega t\right)\end{align}\]

    The quantity \(\omega=2\pi f\) is known as the angular frequency of the wave.

    Periodic Waves Examples

    Consider an oscillating electric field \(E(t)\) at a point \(x\), if the maximum value of the electric field is \(10\,\mathrm{N}\,\mathrm{C}^{-1}\) and the electric field oscillates with a frequency of \(f=124\,\mathrm{Hz}\), what will the value of the electric field be after \(0.3\,\mathrm{s}\)?

    Recall the wave equation given in the previous section.

    \[f(t)=A\cos\left(2\pi f t\right)\]

    From the question we know \(A=10\,\mathrm{N}\,\mathrm{C}^{-1}\) and that \(f=124\,\mathrm{Hz}\), plugging in these values gives

    \begin{align}f(t)&=10\,\mathrm{N}\,\mathrm{C}^{-1}\cos\left(2\pi\cdot124\,\mathrm{Hz}\cdot10\,\mathrm{s}\right)\\&=3.09\,\mathrm{N}\,\mathrm{C}^{-1}\end{align}

    What is the formula for the periodic electric field oscillations given below in Figure 6?

    Periodic Waves Example question graph of a periodic electric field-distant wave StudySmarterFig. 6 - The strength of an electric field oscillates as a periodic wave.

    Looking at the graph, we see that the wavelength of this periodic wave is \(4\,\mathrm{m}\) and its amplitude is \(4\,\mathrm{N}\,\mathrm{C}^{-1}\). Also, note that the wave begins at \(0\) field strength so we need to use the sine function.

    \[E(x)=4\sin\left(\frac{2\pi}{4}x\right)\,\mathrm{N}\,\mathrm{C}^{-1}=4\sin\left(\frac{\pi}{2}x\right)\,\mathrm{N}\,\mathrm{C}^{-1} \]

    Periodic Waves - Key takeaways

    • Periodic waves are waves with a repeating pattern made up of cycles that repeat over some interval.
    • The wavelength of a periodic wave is the spatial distance between each cycle of a wave, whilst the time period of a periodic wave is the time taken for one full cycle of a wave. These quantities can be used to characterize a periodic wave.
    • The time period \(T\) and frequency \(f\) of a wave are inversely related\[f=\frac{1}{T}\]
    • The frequency \(f\) and wavelength \(\lambda\) of a wave are related by the velocity \(v\) of the wave\[v=f\lambda\]
    • Pulses are an aperiodic type of wave, caused by short sudden disturbances, with only a few cycles meaning that they do not have a frequency or wavelength.
    • At a fixed instant of time the formula for a wave over space, with \(k=\frac{2\pi}{\lambda}\) the angular wavenumber and \(A\) the amplitude, is given by\[f(x)=A\cos\left(kx\right)\]
    • At a fixed position the formula for a wave over time, with \(\omega=2\pi f\) the angular frequency, is given by\[f(t)=A\cos\left(\omega t\right)\]

    References

    1. Fig. 1 - Time Period of Wave, StudySmarter Originals.
    2. Fig. 2 - Wavelength of Wave, StudySmarter Originals.
    3. Fig. 3 - Slinky Longitudinal Wave, StudySmarter Originals.
    4. Fig.4- Electromagnetic wave2 (https://commons.wikimedia.org/wiki/File:Electromagnetic_wave2.svg) by Francois~frwiki is licenced under CC SA-BY 4.0 (https://creativecommons.org/licenses/by-sa/4.0/deed.en)
    5. Fig. 5 - Pulse, StudySmarter Originals.
    6. Fig. 6 - Periodic Electric Wave, StudySmarter Originals.
    Frequently Asked Questions about Periodic Wave

    What are examples of periodic waves? 

    Water waves and all types of Electromagnetic waves are examples of periodic waves.

    What are periodic waves and their types? 

    Periodic waves are types of energy transfer where the disturbance has a continually repeating pattern made up of cycles. Periodic waves can be transverse, meaning the oscillations caused by the wave are perpendicular to the direction of energy transfer, and longitudinal waves where the oscillations are parallel to the direction of energy transfer.

    How to find the period of a wave? 

    The period of a wave defines how long it takes the wave to make one full cycle and is inversely related to the frequency of the wave.

    How can you generate a periodic wave?

    Periodic waves are caused by continuous repeated disturbances. For example, if you move a spring up and down in a repeated motion it will produce a periodic wave.

    What is a periodic wave function?

    A periodic wave function is any function f(t) defining a wave such that there is some period T such that f(t)=f(t+T).

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    What is the frequency of a wave travelling at \(v=200\,\mathrm{m}\,\mathrm{s}^{-1}\) if it has a wavelength of \(\lambda=0.5\,\mathrm{m}\)?

    The frequency \(f\) and time period \(T\) of a periodic wave are inversely related to each other. True or False?

    What is the frequency of a periodic wave if it makes 2 complete cycles in \(3\,\mathrm{s}\)?

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