Amplitude Formula

Before going to learn what is amplitude formula, let us recall what is amplitude. The maximum displacement of any particle of a medium, from its state or a position of equilibrium, is called the amplitude. Amplitude is represented with 'A'. In a periodic function with a bounded range, the amplitude is half the distance between the minimum and maximum values. The amplitude is the height from the centerline to the peak or to the trough. Let us learn the amplitude formula along with a few solved examples.

What is Amplitude Formula?

Amplitude refers to the maximum change of a variable from its mean value. The amplitude formula helps in determining the sine and cosine functions. Amplitude is represented by A. The sine function (or) cosine function can be expressed as,

 x = A sin (ωt + ϕ)   or   x = A cos (ωt + ϕ)

Here,

• x = displacement of wave (meter)
• A = amplitude
• ω = angular frequency (rad/s)
• t = time period
• ϕ = phase angle

The amplitude formula is also expressed as the average of the maximum and minimum values of the sine or cosine function. We always take the absolute value of the amplitude.

Amplitude = (max + min) / 2

Examples Using Amplitude Formula

Example 1: y = 2sin(4t) is a wave. Find its amplitude.

Solution:

Given: equation of wave y = 2sin(4t)

Using amplitude formula,

x = A sin(ωt + ϕ)

On comparing it with the wave equation:

A = 2
ω = 4
ϕ = 0

Therefore, the amplitude of the wave = 2 units.

Example 2: The equation of a wave is given by x = 10sin(5πt+π) is a wave. Find its amplitude.

Solution: 
Given: equation of wave y = 10sin(5πt + π)

Now, using amplitude formula,

x=A sin⁡(ωt + ϕ)

On comparing it with the wave equation:

A = 10
ω = 5π
ϕ = π

Therefore, the amplitude of the wave = 10 units.

Example 3: If y = 6 cos (7t + 1) is a wave. Find its amplitude.

Solution: 

Given: equation of wave y = 6cos(7t + 1)

Using amplitude formula,

x= A cos (ωt + ϕ)

On comparing it with the wave equation:

A = 6
ω = 7
ϕ = 1

Therefore, the amplitude of the wave = 6 units.