# Sinusodial Creation and Simple Harmonic Motion

The first premise to understand is that simple harmonic motion through time generates sinusoidal motion.

The following diagram will display the amplitude of the harmonic motion and for this we need to use the term A in our formula. We will also be using θ.

I have used the equation A sin θ where θ completes one cycle (degrees).
The axis displays values based on a unit circle with being interpreted as amplitude.
The x axis denotes degrees (θ) It then follows that:

When the angle θ is 0° or 180° then y = 0
sin 0° and sin 180° = y/A = 0

When the angle θ is 90° then y = 1
sin 90° = y/A = 1

When the angle θ is 270° then y = −1
sin 270° = y/A = −1

When constructing and working with sinusoids we need to plot our graph and define the axis.

I have chosen the y-axis for amplitude and the x-axis for time with phase expressed in degrees. However, I will later define the formulae that define the variables when we come to expressing the processes.

For now, a simple sine waveform, using each axis and defining them, will be enough.

I will create the y-axis as amplitude with a range that is set from -1 to +1.
y: amplitude

Now to create the x-axis and define its variables to display across the axis.

The range will be from -90 deg to 360 deg
x: time/phase/deg

The following diagram displays the axis plus the waveform and the simplest formula to create a sinusoid is y-sinx The diagram shows one cycle of the waveform starting at 0, peaking at +1 (positive), dropping to the 0 axis and then down to -1 (negative).

The phase values are expressed in degrees and lie on the x-axis. A cycle, sometimes referred to as a period, of a sine wave is a total motion across all the phase values.

I will now copy the same sine wave and phase offset (phase shift and phase angle) so you can see the phase values and to do this we need another simple formula and that is:
y=sin(x-t) where t (time/phase value) being a constant will, for now, have a value set to 0. This allows me to shift by any number of degrees to display the phase relationships between the two sine waves. The shift value is set at 90 which denotes a phase shift of 90 degrees. In essence, the two waveforms are now 90 degrees out of phase.

The next step is to phase shift by 180 deg and this will result in total phase cancelation. The two waveforms together, when played and summed, will produce silence as each peak cancels out each trough. Relevant content:

Frequency and Period of Sound

Total and Partial Phase cancellation

Digital Audio – Understanding and Processing

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