Lesson Summary 

Recognize and implement algebraic formulas associated with wave dynamics and duty cycles. 


Cartesian coordinates - a system that specifies each point uniquely in a plane by a pair of numerical coordinates. These numbers represent the distances from the point to two fixed perpendicular directed lines. Each reference line is called an axis , and the point where they meet is its origin, usually at ordered pair (0, 0). 

Speed of Sound - The speed at which sound waves propagate through Earth's atmosphere.  Averaged to 343 m/s and designated by the letter c.

Wavelength - The distance over which the wave's shape repeats. Determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings. Designated by the Greek letter lambda (λ).

Period - The time over which a wave's shape repeats. Determined in a similar manner as wavelength but in the time domain. It is the reciprocal of frequency and is designated by an uppercase T.  SI unit is the second. 

Amplitude -  The maximum absolute value of the signal. Designated by an uppercase A.

Frequency - The number of cycles per unit time.  The SI unit for frequency is hertz (Hz), named after the German physicist Heinrich Hertz; 1 Hz means that an event repeats once per second. Designated by a lowercase f.

Hertz - The unit used to represent frequency (Hz)

Angular Frequency - A scalar measure of rotation rate. It refers to the angular displacement per unit time, or the rate of change of the phase of a sinusoidal waveform.

Pulse wave - A periodic wave that oscillates between a fixed minimum value and fixed maximum value, its amplitude.  Also known as a rectangular wave.  A pulse wave's harmonic content is contingent on its duty cycle. 

Square wave - A special pulse wave signal with a 50% duty cycle. Contains only odd harmonics, producing a similar timbre to saxophones and clarinets. Highlighted in the diagram below. 


Duty Cycle - The percentage of one period in which a signal is active.  


Pulse Width Modulation - A variation of a sounds duty cycle to achieve subtle timbrel changes. 



View and hear duty cycle changes in a pulse wave.


Jumper Cables

1 x Arduino Microcontroller


To hear the Pulse wave provided by our VCO match the settings in Figure 1. Notice how the sound changes as we turn the PWM (Pulse Width Modulation) knob.  This knob is changing the duty cycle of our Pulse wave. 


Figure 1. Pulse wave setting


For our wave shape viewing exercise we will be using an Arduino to speak to the program Processing.  The Arduino should already have the Standard Firmata sketch uploaded to it. For more detailed information on the Arduino uploading process visit their website

Open Waveshape.pde, ensure that the jumpers are connected in the configuration displayed in Figure 2. and press the run button.  We should have a window pop up similar to that of Figure 3.  Notice how wave shape changes as the PWM knob is turned. Waveshape.pde allows for the user to define what color in RGB, and line thickness (stroke) are used for the wave in the function drawWave.    The included function settings state that we are drawing a white line at thickness 1. drawWave(r, g, b, stroke).

 Waveshape.pde also allows the user to zoom in and out as well as pause the wave animation.  These can be controlled by pressing the arrow keys. 

↑  Zoom In

↓  Zoom Out

←  Pause Animation

→   Play Animation

Figure 2.



Figure 3. PWM shown in the Waveshape Processing program.


For more information on Waveshape.pde refer to the comments in Figure 4. 

  Figure 4. Screen capture


Processing is an open source programming environment and community.




The Arduino is a micro controller and programming environment for interactive systems.


What is the frequency in Hertz of a Square wave with a period T of .25 seconds. 


What is the duty cycle of a pulse wave with a period of 1 second and an active time of .25 seconds? What would this sound like and why?


Illustrate a square wave listing its duty cycle, amplitude, and frequency. Give a brief explanation of what it sounds like and its use in synthesizers.


Illustrate a pulse wave with a 75% duty cycle listing amplitude and frequency.   Illustrate your wave and give a brief explanation of what it sounds like and why.