# Superposition of Harmonic Oscillations

Simple harmonic motion is a special kind of periodic motion that is represented by a sinusoidal function

In this formula, \(y\left( t \right)\) denotes any quantity that oscillates, \(A\) is the amplitude (maximum value) of oscillations, \(\omega\) is the angular frequency, and \(\alpha\) is the initial phase.

The angular frequency \(\omega\) is related to oscillation frequency \(f\) and oscillation period \(T\) by the following formula:

Using the sine addition identity, the function \(y\left( t \right)\) can be written in the form

where \({C_1} = A\cos \alpha ,\) \({C_2} = A\sin \alpha.\)

The opposite statement is also true. Any function of kind \({C_1}\sin \omega t + {C_2}\cos \omega t\) can be represented in the form \(A\sin \left( {\omega t + \alpha } \right).\) Indeed, let

Then

Take into account that

So there exists an angle \(\alpha \in \left[ {0,2\pi } \right]\) being the only one such that \(\cos \alpha = \frac{{{C_1}}}{A}\) and \(\sin \alpha = \frac{{{C_2}}}{A}.\) Hence,

In physics, there are examples where a body participates in two harmonic oscillations at the same time, and these oscillations may have the same or different frequencies.

Consider both of these cases in more detail.

## Addition of Two Harmonic Oscillations of the Same Frequency

Suppose that a body or a material point is involved in two oscillatory processes along the same line:

Simple harmonic oscillations obey the superposition principle, whereby the resulting oscillation is given by the sum

Each of these oscillations can be written as

Then

The resulting oscillation has the same frequency \(\omega\) and is written as

where \(K\) is the oscillation amplitude equal to

and \(\varphi\) is the initial phase, which is determined from the relation

## Addition of Two Harmonic Oscillations of Different Frequency

Now we consider the superposition of two sinusoidal oscillations of different frequency. For simplicity, assume that both oscillations have the same amplitude and the initial phase of both oscillations is zero, that is

Using the sum-to-product formula

the superposition of two oscillations can be written in the form

This is an oscillatory motion with angular frequency \({\frac{{{\omega _1} + {\omega _2}}}{2}}\) and amplitude \(2A\cos \left( {\frac{{{\omega _1} - {\omega _2}}}{2}t} \right).\)

We see that the amplitude of the resultant oscillation changes in time. It is called modulated amplitude \({A_{\bmod }},\) so

where

When two oscillations have close frequencies \({f_1} \approx {f_2},\) its superposition shows so-called beats with the difference frequency \(\left| {{f_1} - {f_2}} \right|.\)

For instance, if \({f_1} = 100\,\text{Hz},\) \({f_2} = 105\,\text{Hz},\) the beat frequency is

and the beat period is

A typical beats pattern is shown in Figure \(1.\)