Combining Harmonic Motions in One Dimension
We saw, in the previous section, that the solution to a second degree linear equation in one dimension of the form
can be expressed in a number of ways.
We also mentioned that where a system is linear, its displacement at any time is found by adding together the individual displacements due to each harmonic motion.
Let us consider a system moving under the influence of movement described by two expressions with a common frequency, ω
x 1 = A 1 exp j ( ω t + φ 1 ) , and
x 2 = A 2 exp j ( ω t + φ 2 ) .
x 1 and x 1 are complex.
We have shown these in the diagram below as two phasors, each rotating, in the complex plane in an anti-clockwise direction with angular velocity ω
The pale blue and purple lines are the position phasors as they would be represented individually. To add two phasors, one places the tail of one phasor A2 at the head of the other phasor A1. The resultant phasor is the line A. The resultant can be found too by placing the tail of A1 at the head of the other phasor A2. The order of the addition is immaterial. The length of A A is the resultant amplitude while the angle it makes with the real axis is the resultant phase angle φ .
The expression for the resultant is a simple harmonic vibration with the same frequency as the original harmonic vibrations.
x 1 + x 2 = A 1 exp j ( ω t + φ 1 ) + A 2 exp j ( ω t + φ 2 ) , or
x = A exp j ( ω t + φ ) , where
This solution can generalized where there are many harmonic vibrations all with the same frequency. In that case the amplitude and phase angle are
The real displacement is, of course, the projection of the complex variable x onto the real axis which is the sum of the projection of each x n.
Where the harmonic motions have different frequencies the analysis is similar.
x 1 = A 1 exp j ( ω 1 t + φ 1 ) , and
x 2 = A 2 exp j ( ω 2 t + φ 2 ) so that
x 1 + x 2 = A 1 exp j ( ω 1 t + φ 1 ) + A 2 exp j ( ω 2 t + φ 2 ).
We introduce the term Δ ω = ω 2 - ω 1 so that we can now write the linear superposition in the form
x = A 1 exp j ( ω 1 t + φ 1 ) + A 2 exp j ( ω 1 t + Δ ω t + φ 2 )
Inspection of the solution indicates that both the amplitude A and the phase φ vary with time, in other words, they are not constants. The time dependence of the amplitude produces 'beating', in which the listener hears a slower modulation particularly if the 'beat' frequency is small (no much greater than ten cycles per seond) which can be the case when the frequencies of the original harmonic motions are almost identical.
The final solution can be written as
x = A (t) exp j ( ω 1 t + φ (t) ) , where
We illustrate the result in the diagram below
The linear superposition generally results in non-simple 'harmonic' motion, shown in red.
The lower frequency, quasi-simple harmonic wave shown as a dotted blue line is a periodic amplitude wave that, when of low enough frequency, will be perceived as a 'beat'. In the general case the amplitude varies between ( A 1 + A 2 ) and ( A 1 - A 2 ) .
In situations where the original waves have identical phases and identical amplitudes the frequencies are almost identical, the beats would be particularly audible and the amplitude would vary periodically between twice the original amplitude and zero. These results can be shown by making the required substitutions A 1 = A 2 and φ 1 = φ 2 in the equations for A (t) and φ (t) that we have derived above.
In this case we may conclude that from two waves with identical phase and amplitude but frequencies ω 1 and ω 2 we will produce a resultant wave of average frequency ( ω 1 + ω 2 ) / 2 and amplitude varying with a frequency given by ( ω 1 - ω 2 )
Combining Harmonic Motions in Two Dimensions
This section is currently under construction - more material will be added shortly.