Period, Frequency and Amplitude

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 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.

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.

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, \(\frac{\mathrm N}{\mathrm m}\).

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.

$$\mathrm{Amplitude}=\left|A\right|$$

$$\mathrm{Period}=\frac{2\pi}{\left|{\displaystyle\frac{2\pi}T}\right|}=T$$