Akash sharma lo 2

BEATS

Orchestrating a Solution

FIRST YOU MUST UNDERSTAND INTERFERENCE

EXPLAINING INTERFERENCEThe interaction between two or more

wave functions

For Example: waves with differing phase constants

+

INTERFERENCEWhen mixing the two wave functions they interfere constructively and destructively in different times and

at different spaces

RESULTANT WAVEThe amplitude heard by you is the resultant wave of the two or more

waves functions interacting

Red: Wave Function 1 Blue: Wave Function 2 Purple: Resultant Wave

Here you can see wave functions

interacting

RESULTANT WAVE II

As you can see the resultant is the sum of the two amplitudes

BEATSBeats:

!Are the resultant variation in amplitude of two or more waves that are interacting

constructively or destructively. !!

However, beats are produced by waves out of frequency

RELATED TO FREQUENCY

The rate at which the amplitude varies is proportional to the frequency difference

*However, if frequency difference is too large than you hear two distinct tones rather than one tone which varies in intensity

EFFECT OF FREQUENCYFunctions out of phase, same frequency

Functions with different frequencies

RESULTANT WAVESResultant wave of functions with same frequency, but different phase

constant

Resultant wave of functions with different frequency

SEE THE DIFFERENCE?Resultant wave of differing phase

constant functions

Resultant wave of differing

frequency functions

SINUSOIDAL EFFECT

Amplitude vs time graph for the sum of the waves

EQUATION FOR RESULTANT AMPLITUDE

Variables: !

ω* = (ω1 + ω2)/2 Mean angular frequency !

Δω = (ω1 - ω2)/2 Angular frequency difference !

t = time !

*set X0 to 0

STotal(0,t) = 2smcos(ω*t)cos(Δωt)

EXAMPLE QUESTION

Exactly 5 minutes into a musical play you hear a peculiar tone that sounds really

good. The sound is made from the combined tones of a guitar and violin.

Being a rockstar, you want to know what tone that was to be able to incorporate

it into your next album. Also being a Physics Major you decide to use your combined knowledge of music and physics to find the tone’s amplitude.

What is the tone’s amplitude if you know that:

The guitar tone has a wave function of : 2cos(4x-3t)

and the violin has a wave function of 2cos(6x-9t)

Sm = nm

SOLUTIONYou want to put the equation in the form

STotal(0,t) = 2smcos(ω*t)cos(Δωt)

You know that S(x,t)= 2cos(4x-3t)

and S’(x,t)= 2cos(6x-9t)

So therefore ω = 3 Hz

and ω’ = 9 Hz

SOLUTIONTherefore find

!ω* = (ω1 + ω2)/2 (Mean angular frequency)

& Δω = (ω1 - ω2)/2 (Angular frequency difference)

ω* = (3+9)/2 = 6 Hz

!Δω = (3-9)/2

= -3 Hz

SOLUTION

At t = 5 minutes t = 5minutes * 60seconds/1minute

t = 300s

STotal(0,t) = 2smcos(ω*t)cos(Δωt)

STotal(0,t) = 2(2)cos(6(300))cos(-3(300))

The tone you heard had an amplitude of -0.263 nm

END OF BEATS LO

Graphing done at Desmos Online Graphing Calculator