Sound waves can be mathematically represented as a combination of sine waves. Every musical tone is composed of several sine waves of form [i]y(t) = a sin(ω t + φ)[/i].[br][br]The amplitude [i]a[/i] influences the volume of the tone while the angular frequency [i]ω[/i] determines the pitch of the tone. The parameter [i]φ[/i] is called "phase" and indicates if the sound wave is shifted in time.[br][br]If two sine waves interfere, superposition occurs. This means that the sine waves amplify or diminish each other. We can simulate this phenomenon with GeoGebra in order to examine special cases that also occur in nature.
[table][tr id=SineFunction][td]1.[/td][td][icon]https://wiki.geogebra.org/uploads/thumb/4/40/Menu_view_algebra.svg/120px-Menu_view_algebra.svg.png[/icon][/td][td]Create the sine function [i]g(x)= a_1 sin(ω_1 x + φ_1).[/i][br][/td][/tr][tr id=SineFunction][td][/td][td][/td][td][u]Hints[/u]: The input [i][code]_1[/code][/i] produces an index [sub]1[/sub]. The [img]https://wiki.geogebra.org/uploads/thumb/d/dc/Keyboard_open.svg/16px-Keyboard_open.svg.png[/img] virtual keyboard provides Greek letters for you.[br][/td][/tr][tr id=SecondSineFunction][td]2.[/td][td][icon]https://wiki.geogebra.org/uploads/thumb/4/40/Menu_view_algebra.svg/120px-Menu_view_algebra.svg.png[/icon][/td][td]Create the sine function [i]h(x)= a_2 sin(ω_2 x + φ_2)[/i].[br][/td][/tr][tr id=SumFunctions][td]3.[/td][td][icon]https://wiki.geogebra.org/uploads/thumb/4/40/Menu_view_algebra.svg/120px-Menu_view_algebra.svg.png[/icon][/td][td]Create the sum of both functions by entering [code]s(x) = g(x) + h(x)[/code].[/td][/tr][tr id=MatchColors][td]4.[/td][td] [img]https://wiki.geogebra.org/uploads/thumb/d/db/Stylingbar_icon_graphics.svg/32px-Stylingbar_icon_graphics.svg.png[/img][/td][td]Use the [i]Style Bar[/i] in order to change the color of the three functions and their corresponding sliders so they are easier to identify.[/td][/tr][/table]
Examine the impact of the parameters on the graph of the sine functions by changing the values of the sliders. Set [i]a[sub]1[/sub] = 1, ω[sub]1[/sub] = 1 [/i]and [i]φ[sub]1[/sub] = 0[/i] and answer the following questions:[br]
For which values of [i]a[sub]2,[/sub] ω[sub]2[/sub][/i] and [i]φ[sub]2[/sub][/i] does the sum have maximal Amplitude, if [i]a[/i][sub][i]1[/i][/sub][i]=1, ω[/i][sub][i]1[/i][/sub] = 1 and [i]φ[sub]1[/sub][/i] = 0 ? [u][br]Note[/u]: In this case the resulting tone has the maximal volume. [br]
For which values of [i]a[sub]2[/sub], ω[sub]2[/sub][/i], and [i]φ[sub]2[/sub][/i] do the two functions cancel each other? [u][br]Note[/u]: In this case the tone cannot be heard any more.[br]