To understand the universe, you must understand that everything can be described by waves, from the most complex things to everyday things like the color of the objects we observe. When light passes through a prism, it gets divided into different components that we see as colors. Each of these colors can be identified by its unique frequency. A color can have different intensities, as the intensity of the color is related to the amplitude of the wave. This means that there can be two waves with the same frequency, but with different amplitudes. In this article, we will learn about the amplitude, frequency, and period of an oscillation, as well as understand the relationship between them.
Visible light spectrum, displaying that different colors, can be identified by their unique frequency and period. We see the inverse relationship between the frequency and the period. The lower the frequency, the larger the period and vice versa, Wikimedia Commons, DrSciComm (CC BY-SA 3.0)
Period, Frequency, and Amplitude: Definitions
Period, frequency, and amplitude are important properties of waves. As we mentioned before, the amplitude is related to the energy of a wave.
The amplitude is the maximum displacement from the equilibrium position in an oscillation
The period is the time taken for one oscillation cycle. The frequency is defined as the reciprocal of the period. It refers to how many cycles it completes in a certain amount of time.
The period is the time taken for one oscillation cycle.
The frequency describes how many oscillation cycles a system completes in a certain amount of time.
For example, a large period implies a small frequency.
$$f=\frac1T$$
Where \(f\) is the frequency in hertz, \(\mathrm{Hz}\), and \(T\) is the period in seconds, \(\mathrm s\).
Period, Frequency, and Amplitude: Examples
To visualize these concepts experimentally, imagine you and your friend grabbing a rope by the ends and shaking it up and down such that you create a wave that travels through the rope. Let's say that in one second, the rope completed two cycles. The frequency of the wave would be \(2\;\frac{\mathrm{cycles}}{\mathrm s}\). The period would be the inverse of the frequency, so the period of the wave would be half a second, meaning it would take half a second to complete one oscillation cycle.
A student observing an oscillating block counts \(45.5\;{\textstyle\frac{\mathrm{cycles}}\min}\). Determine its frequency and period.
$$f=45.5\;{\textstyle\frac{\mathrm{cycles}}\min}\times\frac1{60}{\textstyle\frac\min{\mathrm s}}=0.758\;{\textstyle\frac{\mathrm{cycles}}{\mathrm s}}$$
$$f=0.758\;\mathrm{Hz}$$
$$T=\frac1f=\frac1{0.758\;\mathrm{Hz}}=1.32\;\mathrm s$$
The period for an object oscillating in simple harmonic motion is related to the angular frequency of the object's motion. The expression for the angular frequency will depend on the type of object that is undergoing the simple harmonic motion.
$$\omega=2\pi f$$
$$T=\frac{2\pi}\omega$$
Where \(\omega\) is the angular frequency in radians per second, \(\frac{\mathrm{rad}}{\mathrm s}\).
The two most common ways to prove this are the pendulum and the mass on a spring experiments.
The period of a spring is given by the equation below.
$$T_s=2\pi\sqrt{\frac mk}$$
Where \(m\) is the mass of the object at the end of the spring in kilograms, \(\mathrm{kg}\), and \(k\) is the spring constant that measures the stiffness of the spring in newtons per meter, \(\frac{\mathrm N}{\mathrm m}\).
A block of mass \(m=2.0\;\mathrm{kg}\) is attached to a spring whose spring constant is \(300\;{\textstyle\frac{\mathrm N}{\mathrm m}}\). Calculate the frequency and period of the oscillations of this spring–block system.
$$T=2\pi\sqrt{\frac mk}=2\pi\sqrt{\frac{2.0\;\mathrm{kg}}{300\frac{\mathrm N}{\mathrm m}}}=0.51\;\mathrm s$$
$$f=\frac1T=\frac1{0.51\;\mathrm s}=1.9\;\mathrm{Hz}$$
The period of a simple pendulum displaced by a small angle is given by the equation below.
$$T_p=2\pi\sqrt{\frac lg}$$
Where \(l\) is the length of the pendulum in meters, \(\mathrm m\), and \(\mathrm g\) is the acceleration due to gravity in meters per second squared, (\frac{\mathrm m}{\mathrm s^2}\).
Relationship between Period, Frequency, and Amplitude
The period, frequency, and amplitude are all related in the sense that they are all necessary to accurately describe the oscillatory motion of a system. As we will see in the next section, these quantities appear in the trigonometric equation that describes the position of an oscillating mass. It is important to note that the amplitude is not affected by a wave's period or frequency.
It is easy to see the relationship between the period, frequency, and amplitude in a Position vs. Time graph. To find the amplitude from a graph, we plot the position of the object in simple harmonic motion as a function of time. We look for the peak values of distance to find the amplitude. To find the frequency, we first need to get the period of the cycle. To do so, we find the time it takes to complete one oscillation cycle. This can be done by looking at the time between two consecutive peaks or troughs. After we find the period, we take its inverse to determine the frequency.
Displacement as a function of time for simple harmonic motion to illustrate the amplitude and period. Distance from \(x=0\) to \(x=a\) is the amplitude, while the time from \(t=0\) to \(t=t\) is the period, StudySmarter Originals
Period, Frequency, and Amplitude of Trigonometric Functions
Trigonometric functions are used to model waves and oscillations. This is because oscillations are things with periodicity, so they are related to the geometric shape of the circle. Cosine and sine functions are defined based on the circle, so we use these equations to find the amplitude and period of a trigonometric function.
$$y=a\;c\mathrm{os}\left(bx\right)$$
The amplitude will be given by the magnitude of \(a\).
$$\mathrm{Amplitude}=\left|a\right|$$
The period will be given by the equation below.
$$\mathrm{Period}=\frac{2\pi}{\left|b\right|}$$
The expression for the position as a function of the time of an object in simple harmonic motion is given by the following equation.
$$x=A\cos\left(\frac{2\pi t}T\right)$$
Where \(A\) is the amplitude in meters, \(\mathrm m\), and \(t\) is time in seconds, \(\mathrm s\).
From this equation, we can determine the amplitude and period of the wave.
$$\mathrm{Amplitude}=\left|A\right|$$
$$\mathrm{Period}=\frac{2\pi}{\left|{\displaystyle\frac{2\pi}T}\right|}=T$$
Period, Frequency and Amplitude - Key takeaways
- The period is the time taken for one oscillation cycle.
- The frequency is defined as the inverse of the period. It refers to how many cycles it completes in a certain amount of time, \(f=\frac1T\).
- The period of an object oscillating in simple harmonic motion is related to the angular frequency of the object's motion, \(T=\frac{2\pi}\omega\) and \(\omega=2\pi f\).
- The amplitude is the maximum displacement from the equilibrium position in an oscillation. It is an important property that is related to the energy of a wave. The amplitude is not affected by a wave's period or frequency. There can be two waves with the same frequency, but with different amplitudes.
- Trigonometric functions are used to model waves and oscillations, so we use these equations to find the amplitude and period, \(y=a\cos\left(bx\right)\). To determine the amplitude, \(\mathrm{Amplitude}=\left|a\right|\). To determine the period, \(\mathrm{Period}=\frac{2\pi}{\left|b\right|}\).
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