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<p>[QUOTE="Zeppo Shanski, post: 3348730, member: 100596"]A <font size="6">SINE</font> curve ... not sin.</p><p><br /></p><p><span style="color: #000000">A sine wave or sinusoid is a <a href="https://en.wikipedia.org/wiki/Curve" target="_blank" class="externalLink ProxyLink" data-proxy-href="https://en.wikipedia.org/wiki/Curve" rel="nofollow">mathematical curve</a> that describes a smooth periodic <a href="https://en.wikipedia.org/wiki/Oscillation" target="_blank" class="externalLink ProxyLink" data-proxy-href="https://en.wikipedia.org/wiki/Oscillation" rel="nofollow">oscillation</a>. A sine wave is a <a href="https://en.wikipedia.org/wiki/Continuous_wave" target="_blank" class="externalLink ProxyLink" data-proxy-href="https://en.wikipedia.org/wiki/Continuous_wave" rel="nofollow">continuous wave</a>. It is named after the function <a href="https://en.wikipedia.org/wiki/Sine" target="_blank" class="externalLink ProxyLink" data-proxy-href="https://en.wikipedia.org/wiki/Sine" rel="nofollow">sine</a>, of which it is the <a href="https://en.wikipedia.org/wiki/Graph_of_a_function" target="_blank" class="externalLink ProxyLink" data-proxy-href="https://en.wikipedia.org/wiki/Graph_of_a_function" rel="nofollow">graph</a>. It occurs often in pure and applied <a href="https://en.wikipedia.org/wiki/Mathematics" target="_blank" class="externalLink ProxyLink" data-proxy-href="https://en.wikipedia.org/wiki/Mathematics" rel="nofollow">mathematics</a>, as well as <a href="https://en.wikipedia.org/wiki/Physics" target="_blank" class="externalLink ProxyLink" data-proxy-href="https://en.wikipedia.org/wiki/Physics" rel="nofollow">physics</a>, <a href="https://en.wikipedia.org/wiki/Engineering" target="_blank" class="externalLink ProxyLink" data-proxy-href="https://en.wikipedia.org/wiki/Engineering" rel="nofollow">engineering</a>, <a href="https://en.wikipedia.org/wiki/Signal_processing" target="_blank" class="externalLink ProxyLink" data-proxy-href="https://en.wikipedia.org/wiki/Signal_processing" rel="nofollow">signal processing</a> and many other fields. Its most basic form as a function of time (t) is:</span></p><p><span style="color: #000000"><br /></span></p><p><span style="color: #000000">{\displaystyle y(t)=A\sin(2\pi ft+\varphi )=A\sin(\omega t+\varphi )}<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c7ac13130e7d34a143d729920378bb1902ce74f" class="bbCodeImage wysiwygImage" alt="" unselectable="on" /></span></p><p><span style="color: #000000">where:</span></p><ul> <li><span style="color: #000000">A = the <a href="https://en.wikipedia.org/wiki/Amplitude" target="_blank" class="externalLink ProxyLink" data-proxy-href="https://en.wikipedia.org/wiki/Amplitude" rel="nofollow">amplitude</a>, the peak deviation of the function from zero.</span></li> <li><span style="color: #000000">f = the <a href="https://en.wikipedia.org/wiki/Frequency" target="_blank" class="externalLink ProxyLink" data-proxy-href="https://en.wikipedia.org/wiki/Frequency" rel="nofollow">ordinary frequency</a>, the <a href="https://en.wikipedia.org/wiki/Real_number" target="_blank" class="externalLink ProxyLink" data-proxy-href="https://en.wikipedia.org/wiki/Real_number" rel="nofollow">number</a> of oscillations (cycles) that occur each second of time.</span></li> <li><span style="color: #000000">ω = 2πf, the <a href="https://en.wikipedia.org/wiki/Angular_frequency" target="_blank" class="externalLink ProxyLink" data-proxy-href="https://en.wikipedia.org/wiki/Angular_frequency" rel="nofollow">angular frequency</a>, the rate of change of the function argument in units of <a href="https://en.wikipedia.org/wiki/Radians_per_second" target="_blank" class="externalLink ProxyLink" data-proxy-href="https://en.wikipedia.org/wiki/Radians_per_second" rel="nofollow">radians per second</a></span></li> <li><span style="color: #000000">{\displaystyle \varphi }<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="bbCodeImage wysiwygImage" alt="" unselectable="on" /> = the <a href="https://en.wikipedia.org/wiki/Phase_(waves)" target="_blank" class="externalLink ProxyLink" data-proxy-href="https://en.wikipedia.org/wiki/Phase_(waves)" rel="nofollow">phase</a>, specifies (in <a href="https://en.wikipedia.org/wiki/Radians" target="_blank" class="externalLink ProxyLink" data-proxy-href="https://en.wikipedia.org/wiki/Radians" rel="nofollow">radians</a>) where in its cycle the oscillation is at t = 0.</span><ul> <li><span style="color: #000000">When {\displaystyle \varphi }<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="bbCodeImage wysiwygImage" alt="" unselectable="on" /> is non-zero, the entire waveform appears to be shifted in time by the amount {\displaystyle \varphi }<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="bbCodeImage wysiwygImage" alt="" unselectable="on" />/ω seconds. A negative value represents a delay, and a positive value represents an advance.</span></li> </ul></li> </ul><p><span style="color: #000000">The oscillation of an undamped spring-mass system around the equilibrium is a sine wave</span></p><p><span style="color: #000000">The sine wave is important in physics because it retains its wave shape when added to another sine wave of the same frequency and arbitrary phase and magnitude. It is the only periodic waveform that has this property. This property leads to its importance in <a href="https://en.wikipedia.org/wiki/Fourier_analysis" target="_blank" class="externalLink ProxyLink" data-proxy-href="https://en.wikipedia.org/wiki/Fourier_analysis" rel="nofollow">Fourier analysis</a> and makes it acoustically unique.</span>[/QUOTE]</p><p><br /></p>
[QUOTE="Zeppo Shanski, post: 3348730, member: 100596"]A [SIZE=6]SINE[/SIZE] curve ... not sin. [COLOR=#000000]A sine wave or sinusoid is a [URL='https://en.wikipedia.org/wiki/Curve']mathematical curve[/URL] that describes a smooth periodic [URL='https://en.wikipedia.org/wiki/Oscillation']oscillation[/URL]. A sine wave is a [URL='https://en.wikipedia.org/wiki/Continuous_wave']continuous wave[/URL]. It is named after the function [URL='https://en.wikipedia.org/wiki/Sine']sine[/URL], of which it is the [URL='https://en.wikipedia.org/wiki/Graph_of_a_function']graph[/URL]. It occurs often in pure and applied [URL='https://en.wikipedia.org/wiki/Mathematics']mathematics[/URL], as well as [URL='https://en.wikipedia.org/wiki/Physics']physics[/URL], [URL='https://en.wikipedia.org/wiki/Engineering']engineering[/URL], [URL='https://en.wikipedia.org/wiki/Signal_processing']signal processing[/URL] and many other fields. Its most basic form as a function of time (t) is: {\displaystyle y(t)=A\sin(2\pi ft+\varphi )=A\sin(\omega t+\varphi )}[IMG]https://wikimedia.org/api/rest_v1/media/math/render/svg/6c7ac13130e7d34a143d729920378bb1902ce74f[/IMG] where:[/COLOR] [LIST] [*][COLOR=#000000]A = the [URL='https://en.wikipedia.org/wiki/Amplitude']amplitude[/URL], the peak deviation of the function from zero.[/COLOR] [*][COLOR=#000000]f = the [URL='https://en.wikipedia.org/wiki/Frequency']ordinary frequency[/URL], the [URL='https://en.wikipedia.org/wiki/Real_number']number[/URL] of oscillations (cycles) that occur each second of time.[/COLOR] [*][COLOR=#000000]ω = 2πf, the [URL='https://en.wikipedia.org/wiki/Angular_frequency']angular frequency[/URL], the rate of change of the function argument in units of [URL='https://en.wikipedia.org/wiki/Radians_per_second']radians per second[/URL][/COLOR] [*][COLOR=#000000]{\displaystyle \varphi }[IMG]https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e[/IMG] = the [URL='https://en.wikipedia.org/wiki/Phase_(waves)']phase[/URL], specifies (in [URL='https://en.wikipedia.org/wiki/Radians']radians[/URL]) where in its cycle the oscillation is at t = 0.[/COLOR] [LIST] [*][COLOR=#000000]When {\displaystyle \varphi }[IMG]https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e[/IMG] is non-zero, the entire waveform appears to be shifted in time by the amount {\displaystyle \varphi }[IMG]https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e[/IMG]/ω seconds. A negative value represents a delay, and a positive value represents an advance.[/COLOR] [/LIST] [/LIST] [COLOR=#000000]The oscillation of an undamped spring-mass system around the equilibrium is a sine wave The sine wave is important in physics because it retains its wave shape when added to another sine wave of the same frequency and arbitrary phase and magnitude. It is the only periodic waveform that has this property. This property leads to its importance in [URL='https://en.wikipedia.org/wiki/Fourier_analysis']Fourier analysis[/URL] and makes it acoustically unique.[/COLOR][/QUOTE]
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