[size=100]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 sine(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.[/size]

[table][tr][td][size=100]1.[/size][/td][td][size=100][icon]https://tube.geogebra.org/images/ggb/toolbar/mode_slider.png[/icon][/size][/td][td][size=100]Create three sliders [/size][i]a_1, ω_1[/i][size=100] and [/size][i]φ_1[/i][size=100] using the default settings for sliders.[/size][br][/td][/tr][tr][td][br][/td][td][br][/td][td][size=100][u]Hints[/u][size=100]: The input [/size][i][code]_1 [/code][/i][size=100] produces an index [/size][sub][size=50]1[/size][/sub][size=100]. [br]In order to insert a Greek letter, place the cursor in the [i]Name[/i] text field and click on the appearing letter [/size][math]\alpha[/math][size=100] on the right side of the text field. This opens a list of Greek letters to choose from[/size][size=100].[/size][br][/size][/td][/tr][tr][td][size=100]2.[/size][/td][td][icon]https://wiki.geogebra.org/uploads/thumb/4/40/Menu_view_algebra.svg/120px-Menu_view_algebra.svg.png[/icon][/td][td][size=100]Enter the sine function [font=Courier New]g(x)= a_1 sin(ω_1 x + [i]φ[/i]_1)[/font].[br][/size][/td][/tr][tr][td][size=100]3.[/size][/td][td][size=100][icon]https://tube.geogebra.org/images/ggb/toolbar/mode_slider.png[/icon][/size][/td][td][size=100]Create three sliders [i]a_2, ω_2[/i] and [i]φ_2[/i], again using the default settings for sliders.[br][u]Hint[/u]: Sliders can be moved in the [i]Graphics View[/i] when the [i]Slider[/i] tool is activated.[/size][/td][/tr][tr][td][size=100]4.[/size][/td][td][icon]https://wiki.geogebra.org/uploads/thumb/4/40/Menu_view_algebra.svg/120px-Menu_view_algebra.svg.png[/icon][/td][td][size=100]Enter another sine function [font=Courier New]h(x)= a_2 sin(ω_2 x + φ_2)[/font].[/size][/td][/tr][tr][td][size=100]5.[/size][/td][td][icon]https://wiki.geogebra.org/uploads/thumb/4/40/Menu_view_algebra.svg/120px-Menu_view_algebra.svg.png[/icon][/td][td][size=100]Create the sum of both functions [font=Courier New]sum(x) = g(x) + h(x)[/font].[/size][/td][/tr][tr][td][size=100]6.[/size][/td][td][center][img]https://wiki.geogebra.org/uploads/thumb/d/db/Stylingbar_icon_graphics.svg/32px-Stylingbar_icon_graphics.svg.png[/img][/center][/td][td][size=100]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.[/size][/td][/tr][/table]

[size=100]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]. [/size][list=1][*][size=100]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? [u][br]Note[/u]: In this case the resulting tone has the maximal volume. [/size][/*][*][size=100]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.[/size][/*][/list][size=100][br][/size]