Precalculus

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Superposition of Harmonic Oscillations

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

Harmonic oscillation

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:

Relation between angular frequency, oscillation frequency and oscillation period

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

\[y\left( t \right) = A\sin \left( {\omega t + \alpha } \right) = A\left( {\cos \alpha \sin \omega t + \sin \alpha \cos \omega t} \right) = {C_1}\sin \omega t + {C_2}\cos \omega t,\]

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

\[\sqrt {C_1^2 + C_2^2} = A.\]

Then

\[{C_1}\sin \omega t + {C_2}\cos \omega t = A\left( {\frac{{{C_1}}}{A}\sin \omega t + \frac{{{C_2}}}{A}\cos \omega t} \right).\]

Take into account that

\[{\left( {\frac{{{C_1}}}{A}} \right)^2} + {\left( {\frac{{{C_2}}}{A}} \right)^2} = \frac{{C_1^2}}{{{A^2}}} + \frac{{C_2^2}}{{{A^2}}} = \frac{{C_1^2 + C_2^2}}{{{A^2}}} = \frac{{\cancel{{C_1^2 + C_2^2}}}}{{\cancel{{C_1^2 + C_2^2}}}} = 1.\]

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,

\[{C_1}\sin \omega t + {C_2}\cos \omega t = A\cos \alpha \sin \omega t + A\sin \alpha \cos \omega t = A\left( {\cos \alpha \sin \omega t + \sin \alpha \cos \omega t} \right) = A\sin \left( {\omega t + \alpha } \right).\]

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:

\[{y_1}\left( t \right) = {A_1}\sin \left( {\omega t + {\alpha _1}} \right),\]
\[{y_2}\left( t \right) = {A_2}\sin \left( {\omega t + {\alpha _2}} \right).\]

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

\[y\left( t \right) = {y_1}\left( t \right) + {y_2}\left( t \right).\]

Each of these oscillations can be written as

\[{y_1}\left( t \right) = {C_1}\sin \omega t + {C_2}\cos \omega t,\]
\[{y_2}\left( t \right) = {D_1}\sin \omega t + {D_2}\cos \omega t.\]

Then

\[y\left( t \right) = {y_1}\left( t \right) + {y_2}\left( t \right) = \left( {{C_1} + {D_1}} \right)\sin \omega t + \left( {{C_2} + {D_2}} \right)\cos \omega t.\]

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

\[y\left( t \right) = \cos \varphi \sin \omega t + \sin \varphi \cos \omega t = K\sin \left( {\omega t + \varphi } \right),\]

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

\[K = \sqrt {{{\left( {{C_1} + {D_1}} \right)}^2} + {{\left( {{C_2} + {D_2}} \right)}^2}} ,\]

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

\[\tan \varphi = \frac{{\sin \varphi }}{{\cos \varphi }} = \frac{{{C_2} + {D_2}}}{{{C_1} + {D_1}}}.\]

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

\[{y_1}\left( t \right) = A\sin {\omega _1}t = A\sin \left( {2\pi {f_1}t} \right),\]
\[{y_2}\left( t \right) = A\sin {\omega _2}t = A\sin \left( {2\pi {f_2}t} \right).\]

Using the sum-to-product formula

\[\sin \alpha + \sin \beta = 2\sin \frac{{\alpha + \beta }}{2}\cos \frac{{\alpha - \beta }}{2},\]

the superposition of two oscillations can be written in the form

\[y\left( t \right) = {y_1}\left( t \right) + {y_2}\left( t \right) = A\sin {\omega _1}t + A\sin {\omega _2}t = 2A\cos \left( {\frac{{{\omega _1} - {\omega _2}}}{2}t} \right)\sin \left( {\frac{{{\omega _1} + {\omega _2}}}{2}t} \right).\]

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

\[y\left( t \right) = {A_{\bmod }}\left( t \right)\sin \left( {\frac{{{\omega _1} + {\omega _2}}}{2}t} \right),\]

where

\[{A_{\bmod }}\left( t \right) = 2A\cos \left( {\frac{{{\omega _1} - {\omega _2}}}{2}t} \right) = 2A\cos \left[ {\left( {{f_1} - {f_2}} \right)\pi t} \right].\]

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

\[{f_\text{beat}} = \left| {{f_1} - {f_2}} \right| = 5\,\text{Hz},\]

and the beat period is

\[{T_\text{beat}} = \frac{1}{{{f_\text{beat}}}} = \frac{1}{5}\,\sec = 0.2\,\sec\]

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

beating phenomenon
Figure 1.

See solved problems on Page 2.

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