Sine Waves

When a signal is plotted as a function of time, the shape created is called a wave shape or a waveform. When the shape repeats over and Over in time (equal intervals of time) the waveform is said to be periodic. When there is no such repetition the waveform is said to be non-periodic.

Figure 4c: Examples of Waveforms Oscillatory Motion

Oscillatory motion occurs when there is back and forth movement (e.g. vibration or swinging) or when there is motion in which a point repeatedly tracks itself over and over. Whenever there is oscillatory motion, a periodic signal and its associated waveform are produced. Basic examples would be a pendulum, a tuning fork or a turning wheel.

When a pen is placed at the tip and a paper moving at uniform speed placed at the back of a moving pendulum, an oscillatory waveform would be traced. The trace would have maximum excursion to left and right of the resting position of the pendulum. The shape of the waveform produced is called a SINE WAVE (see Figure 4a).

One way that might help us understand sine waves better is to consider the movement made by the rotation of a circle (figure 5). A straight horizontal line is drawn from the centre of the circle to the right-hand side until it touches the circumference (reference). Then we label this point where the circumference and the line meet, point B. When circle is rotated counter Clockwise, point B will move from its resting place (where its vertical value is zero) to some vertical value above zero (or below zero after passing the reference line) . The value depends on the position of the point on the circumference and the angle of rotation θ from 0º (reference or resting place). If we plot the vertical values against their angles of rotation a sine wave will be traced.

 

Figure 5: Standard Sine Wave

The arrow shows direction of rotation of point B from reference line X. Point B has moved from

B-0 to B-1 on the circumference through angle θ. The vertical value from X of point B is projected and plotted on the graph to the right of the circle. If all vertical values are plotted for angles 0 to 360, a sine wave is traced as shown above. The highest points +A and –A are called the maximum AMPLITUDE.

θ is the angle of rotation. Small (a) is the AMPLITUDE at any particular angle and it’s called the instantaneous amplitude. T is the period and t is the instantaneous time. When the circle has gone through one complete revolution which is equivalent to 360º, the sine wave is said to have completed one CYCLE. The time it takes to complete one cycle is called the PERIOD T i.e. a cycle is completed in T seconds. The number of cycles completed in one second is called FREQUENCY.

A standard sine wave starts with time t = 0 and amplitude (vertical value) of zero and increases in the positive direction. Any sine wave that starts at any value of amplitude other than zero at time t = 0 is said to be the standard sine wave shifted along the time axis. The amount of this shift is called the PHASE of the waveform (see Figure 6 on the following page).

Figure 6: Sine Wave with a Phase Shift of φ Degrees

Figures 5 and 6 include parameters found in the sine wave equation stated below:

                    a(t) = A sin(Ft + φ)

Where a(t) = instantaneous amplitude as a function of time. A = maximum amplitude

sin = sine function in degrees, F = frequency, t = time and φ = phase. These parameters are important because they will help us to understand types of MODULATION covered in Unit 4 (Chapter 5).