Amplitude, Period, Phase Shift and Frequency

Periodic Functions: A function is said to be periodic if it repeats its values at regular intervals of time.  

Example - The trigonometric functions (like sine and cosine) are periodic functions, with period 2π.

Wave

A wave is propagating dynamic disturbance or sometimes called a change in the equilibrium of one or more quantities. Waves are usually created by an oscillating object which causes disturbance in the medium and this travels as waves through the medium. Each wave has two basic properties of amplitude and frequency. The frequency is the number of waves that we have in a unit of time. This is calculated by dividing the total number of waves created by the total amount of time. The amplitude is the height of the wave from top to bottom. It is usually calculated by measuring the distance of wave from crest to trough. Amplitude is sometimes called the size of the wave. Another property by which the wave can be defined is the wavelength. Wavelength is the distance covered by a single wave. This is measured by measuring the distance covered by a wave from crest to trough.

There are different types of waves in nature, the most common of those are light and sound waves. These waves are generally described by using the above properties. You must have heard about ultraviolet and infrared lights. Ultraviolet is the light that has a very low frequency and very high wavelength whereas infrared are waves that have a very high wavelength and very low frequency. These two types of waves are invisible to the normal human eye but play a major role in the discovery of many theories and objects in astrophysics. The sound also has three segments where we have ultrasonic sound, audible sound, and infrasonic sound. Ultrasonic sound has very high frequency and low wavelength whereas infrasonic sound has high amplitude and low frequency. Both sounds are not audible to the human ear. The audible range falls in the range of 20Hz to 20 kHz. And anything above and below is known as infrasonic and ultrasonic sound waves respectively.

Amplitude Formula

The Amplitude is the maximum height from the centerline to the peak (or to the trough). Another way to find amplitude is to measure the height from highest to lowest points and divide that by 2.

What is Amplitude in Physics?

Amplitude, in physics, can be defined as the maximum displacement or distance moved by a point on a vibrating body or wave measured from its equilibrium position.

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x = A sin (⍵t + φ)

x is the displacement in meter (m).

A is the amplitude in meter (m).

⍵ is the angular frequency in radians per second (radians/s).

T is time in second (s).

φ is the phase shift in radians.

Frequency

Frequency is the number of occurrences of a repeating event per unit of time.

Frequency is also related to the period.

The relation between amplitude and frequency is given by the formula.

Frequency = \[\frac {1} {Period}\]

Let’s understand this with a graph.

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In the above diagram, the sine function repeats 4 times between 0 and 1.

Hence, the frequency is 4, and the period is \[\frac {1} {4}\].

Phase Shift Formula

The Phase Shift is how far the function is shifted horizontally from the usual position.

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The Vertical Shift is how far the function is shifted vertically from the usual position.

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The generalized equation for a sine graph is given by:

y = A sin (B(x + C)) + D

Where

• A is amplitude.

• The period is \[\frac {2\Pi} {B}\].

• C is phase shift (positive to the left).

• D is a vertical shift.

Graph of the above equation is drawn below:

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Note: Here we are using radian, not degree. Full rotation means 2π radian.

Amplitude Period and Phase Shift Calculator

The amplitude period phase shift calculator is used for trigonometric functions which helps us in the calculations of vertical shift, amplitude, period, and phase shift of sine and cosine functions with ease. We have to enter the trigonometric equation by selecting the correct sine or the cosine function and clicking on calculate to get the results.

How to Find Time Period?

The time period is defined as the time that is taken for one complete cycle of vibration to pass a given point. It is denoted by T. Unit of time period is second.

The formula for the time period is \[\frac {2\Pi} {\omega }\] where ⍵ is the angular frequency.  

Solved Examples

1. y = 2 sin (4(x - 0.5)) + 3

Sol: We will compare the given equation with the standard equation then we will write the given value.

So amplitude A = 2

Period = \[\frac {2\Pi} {B}\]. Here B value is 4 . So Period = \[\frac {2\Pi} {4}\] = \[\frac {\Pi} {2}\]

Phase shift = (-0.5) means it will be shifted to the right by 0.5.

Vertical shift = 3 positive value indicates the centre line is y = +3

The graph is shown below:

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2. Find frequency of the equation y = 3 sin (100 (t + 0.01)) and draw the graph.

Sol: In this amplitude (A) value is 3.

Period = 2π/B here B value is 100. So Period = \[\frac {2\Pi} {100}\] = 0.02π

Phase shift = 0.01

We know that Frequency = \[\frac {1} {Period}\].

So frequency = \[\frac {1} {0.02\Pi}\] = \[\frac {50} {\Pi}\].