Trigonometric Graphs - the Sine Curve for Wave Motion

The principal characteristic of sine and cosine curves is their repetitive nature, that is, as x increases, the values of y re-occur regularly, at a specific frequency.  For the vibrating air particles which make a sound wave, this frequency determines the pitch of a musical note.  The x axis represents elapsed time, the value of y represents the displacement of the air particles from their starting stationery position. The simplest sort of sine curve equation is y = sin(x). This will give peaks and troughs when x is 90°  or p/2 radians, or all odd multiples of this value,  and the curve crosses the x-axis when x is zero or multiple of 180°  or p radians.  The wavelength is 2p radians.

In the demonstration, the formula used is y = a sin(bx + c) +d,  where a = 40 giving the amplitude of the wave, which is the y-value at the peaks and troughs. The amplitude determines the loudness of the musical note. Also b = 0.125,  this adjusts the frequency and wavelength.  Larger values of b give greater frequencies, and consequently, higher pitched notes. The wavelength is inversely proportional to the frequency, because for a sound wave, wavelength times frequency equals the speed of sound, which is constant at constant air temperature and pressure.  You can try adjusting coefficients a and b in the demonstration trigonometric chart, and observe the results.

For further information on sine curves for sound waves, go to MENU Theory -> Introduction -> Sound waves and Simple Harmonic Motion.    For an animated demonstration of the wave motion of a vibrating string, which is also a sine curve, go to MENU   Theory -> Pitch and Intervals -> Strings and Pipes


Trigonometric.htm     02 November 2006         MENU  Theory -> Maths and Physics -> Mathematical equations for music