Introduction to 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 = \[\frac {1} {0.02\Pi}\] = \[\frac {50} {\Pi}\].
FAQs on Amplitude, Period, Phase Shift and Frequency
1. What are amplitude, period, frequency, and phase shift in the context of a wave?
These are the four fundamental parameters that describe a simple harmonic wave:
- Amplitude (A): The maximum displacement or distance moved by a point on a vibrating body or wave from its equilibrium (or central) position. It represents the wave's intensity or energy.
- Period (T): The time it takes to complete one full cycle of the wave. It is measured in seconds.
- Frequency (f): The number of complete cycles that occur per unit of time. It is the reciprocal of the period (f = 1/T) and is measured in Hertz (Hz).
- Phase Shift (φ): A horizontal shift of the wave from its normal position. It indicates the starting position of the wave at time t=0.
2. What is the standard formula used to describe a wave, showing its amplitude, period, and phase shift?
The general equation for a sinusoidal wave, such as one following simple harmonic motion, is given by:
y(t) = A sin(ωt + φ)
In this formula:
- y(t) is the displacement at time t.
- A is the amplitude.
- ω (omega) is the angular frequency, related to the period T by the formula ω = 2π/T.
- φ (phi) is the phase shift, which determines the initial position of the wave.
3. How can you determine the amplitude, period, and frequency from a given wave equation like y = 10 sin(4πt + π/2)?
To find the parameters, you compare the given equation with the standard form y(t) = A sin(ωt + φ).
- Amplitude (A): The amplitude is the coefficient of the sine function. Here, A = 10.
- Angular Frequency (ω): The coefficient of time (t) is the angular frequency. Here, ω = 4π rad/s.
- Period (T): The period is calculated using the formula T = 2π/ω. So, T = 2π / 4π = 0.5 seconds.
- Frequency (f): Frequency is the reciprocal of the period, f = 1/T. So, f = 1 / 0.5 = 2 Hz.
- Phase Shift (φ): The phase shift is the constant added inside the sine function. Here, φ = π/2.
4. What is the relationship between frequency and period, and how does amplitude relate to them?
Frequency and period have a direct inverse relationship. Frequency (f) is the reciprocal of the period (T), expressed as f = 1/T. This means if the time taken for one cycle (period) increases, the number of cycles per second (frequency) decreases, and vice versa.
The amplitude (A), however, is independent of both frequency and period. Amplitude relates to the energy or intensity of the wave, while frequency and period relate to its timing. A wave can have a high amplitude with a low frequency, or a low amplitude with a high frequency.
5. In real-world examples like sound or light, what is the physical importance of amplitude and frequency?
Amplitude and frequency correspond to distinct physical properties that we can perceive:
- For Sound Waves: The amplitude determines the loudness or volume of the sound. A larger amplitude means a louder sound. The frequency determines the pitch of the sound; a higher frequency corresponds to a higher pitch (like a whistle), while a lower frequency corresponds to a lower pitch (like a bass drum).
- For Light Waves: The amplitude is related to the intensity or brightness of the light. A larger amplitude means brighter light. The frequency determines the colour of the visible light. Red light has a lower frequency than blue light, for instance.
6. Why does a positive phase shift, as in sin(ωt + φ), cause a shift to the left on a graph?
This apparent contradiction is resolved by considering the time at which the wave cycle 'starts' (i.e., when the argument of the sine function is zero). For the wave y = A sin(ωt + φ), the cycle starts when ωt + φ = 0. Solving for t, we get t = -φ/ω. If the phase shift φ is positive, the starting time t is negative. This means the wave has reached its starting point earlier than a wave with no phase shift, which corresponds to a shift to the left along the time axis.
7. Can the amplitude of a wave be a negative value? Explain the difference between displacement and amplitude.
No, the amplitude of a wave is always a positive value by definition. It is a scalar quantity that measures the maximum displacement from the equilibrium position.
The key difference lies here:
- Displacement (y): This is the instantaneous position of a point on the wave at any moment in time. It is a vector quantity and can be positive, negative, or zero, as the wave oscillates above and below its central axis.
- Amplitude (A): This is a single, positive value that defines the peak or maximum possible displacement. For example, in a wave oscillating between +5m and -5m, the displacement varies, but the amplitude is always 5m.
8. How does phase shift fundamentally differ from the period of a wave?
Phase shift and period describe two entirely different characteristics of a wave. It is a common misconception to mix them up. The difference is:
- The Period (T) defines the duration of one complete cycle. It answers the question, "How long does it take for the wave to repeat itself?". It determines the horizontal 'width' of one wave cycle.
- The Phase Shift (φ) defines the starting position of the wave at t=0. It answers the question, "Where is the wave located at the very beginning?". It determines the horizontal 'shift' of the entire wave graph left or right.
Essentially, period dictates the wave's repetition rate, while phase shift dictates its starting point in time.
