"If music be the food of love, play on. - William Shakespeare.
When students first learn about trigonometry, it is often very difficult to see beyond the basics of "sin", "cos" and "tan" to get a good appreciation for just how important the topic is in everyday life. Many aspects of our natural world can be defined and explained in terms of periodic functions, including temperature modelling, tidal measurements and the motion of springs and pendulums. Perhaps one of the most important (and enjoyable) applications of periodic functions, however, is sound.
Sound simply consists of travelling waves, involving variations in pressure through a solid, liquid or gas. The most fundamental properties of a basic sound wave are frequency, which affects the pitch of a sound, and amplitude, which affects its loudness. Different instruments (including the voice) produce subtle yet complex changes in the basic wave structure, which give the instruments their characteristic qualities. In this assignment, we will be using trigonometric functions to model basic sounds, and we will investigate how harmonies can be formed by combining multiple soundwaves.
For the following questions, you may need to research how a musical note can be represented by a trigonometric function. (This information can be found on the Internet or in your textbook). The questions in Part 1 refer to data files found on the Google Drive - you will need to locate the files corresponding to your individual student number.
Part 1 - Basics of Sound
QUESTION 1 (KAPS - D)
When played, a musical instrument produces a soundwave with a frequency of f vibrations per second and an amplitude of a. Using the unique values of f and a corresponding to your student number in the data file, answer the following questions:
i) Find the angular frequency, w, of this soundwave assuming a sinusoidal waveform.
ii) Write the equation of the waveform.
QUESTION 2 (KAPS - C)
When two musical notes are played, a soundwave is made up of two partial waves both having amplitude k (from the data table). The period of the fundamental is f1 and that of the overtone is f2 (from the data table).
i) Find the angular frequency, w, of both partial waveforms.
ii) Write the equation of the composite waveforms.
iii) Produce a graph of this waveform using a graphing program of your choice over the interval o seconds. Attach a printout of your graph.
This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.Question 1
(i) ω = 2πf = 2x3.14x59.5 - 373.85 is angular frequency
(ii) y = sin(ωt + φ) - 0.188sin(373.85t) where φ is different from zero...