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Substituting Periodic Fourier series expansion equation with standing wave equation

Greetings All

I can re-create a periodic signal using Fourier series expansion using sin and cos waves. But how can I adapt the equation so the equation will be outputted in the format of a standing wave equation.

formatting may cut off some of the information/question I've included a link to an image of the full question http://dl.dropbox.com/u/6576402/questions/sub_per_fou_series_expansion.jpg and as text file http://dl.dropbox.com/u/6576402/questions/ques1.txt

Note: I export the equation in a text format Here's an example/format of the equation that is currently exported:

aa= 0.0000000000000000277555756156289135105908+VERTOFFmain_1+VERTOFFaad_1
+((AMPmain_1+AMPaaa_1)*0.4330127018922191872718485683435574173927)*cos((FREQmain_1+FREQaab_1)*    1.0 *(t-(((-PHASEmain_1)+PHASEaac_1)/(FREQmain_1+FREQaab_1)))) 
+((AMPmain_1+AMPaaa_1)*0.4330127018922194648276047246326925233006)*sin((FREQmain_1+FREQaab_1)*    1.0 *(t-(((-PHASEmain_1)+PHASEaac_1)/(FREQmain_1+FREQaab_1)))) 
+((AMPmain_1+AMPaaa_1)*-0.4330127018922191872718485683435574173927)*cos((FREQmain_1+FREQaab_1)*    2.0 *(t-(((-PHASEmain_1)+PHASEaac_1)/(FREQmain_1+FREQaab_1)))) 
+((AMPmain_1+AMPaaa_1)*-0.0000000000000000000000000000000000000000)*sin((FREQmain_1+FREQaab_1)*    2.0 *(t-(((-PHASEmain_1)+PHASEaac_1)/(FREQmain_1+FREQaab_1))));

If your interested here are some of the values I use and what the variables mean:

VERTOFFmain_1=0; %Vertical offset
VERTOFFaad_1=0; %Vertical offset
AMPmain_1=1; %amplitude increase 
AMPaaa_1=0; %amplitude increase
FREQmain_1=1; %Frequency increase 
FREQaab_1=0; %Frequency increase 
PHASEmain_1=0; %phase shift
PHASEaac_1=0; %phase shift

Here's a link explaining what a standing wave is and the equation I want to substitute in: http://en.wikipedia.org/wiki/Standing_wave

Here's another example with more data points showing a simple Periodic Fourier series expansion of a sin wave:

aa= 0.0000000000000000055511151231257830102669+VERTOFFmain_1+VERTOFFaad_1 
+((AMPmain_1+AMPaaa_1)*0.1516614837922138359083135128457797691226)*cos((FREQmain_1+FREQaab_1)*    1.0 *(t-(((-PHASEmain_1)+PHASEaac_1)/(FREQmain_1+FREQaab_1)))) 
+((AMPmain_1+AMPaaa_1)*0.9575529230109556255712277561542578041553)*sin((FREQmain_1+FREQaab_1)*    1.0 *(t-(((-PHASEmain_1)+PHASEaac_1)/(FREQmain_1+FREQaab_1)))) 
+((AMPmain_1+AMPaaa_1)*-0.0226651931432310521641326772623870056123)*cos((FREQmain_1+FREQaab_1)*    2.0 *(t-(((-PHASEmain_1)+PHASEaac_1)/(FREQmain_1+FREQaab_1)))) 
+((AMPmain_1+AMPaaa_1)*-0.0697562918038196477787948879267787560821)*sin((FREQmain_1+FREQaab_1)*    2.0 *(t-(((-PHASEmain_1)+PHASEaac_1)/(FREQmain_1+FREQaab_1)))) 
+((AMPmain_1+AMPaaa_1)*-0.0186918925876187295986863290409019100480)*cos((FREQmain_1+FREQaab_1)*    3.0 *(t-(((-PHASEmain_1)+PHASEaac_1)/(FREQmain_1+FREQaab_1)))) 
+((AMPmain_1+AMPaaa_1)*-0.0366849047602344491281201044330373406410)*sin((FREQmain_1+FREQaab_1)*    3.0 *(t-(((-PHASEmain_1)+PHASEaac_1)/(FREQmain_1+FREQaab_1)))) 
+((AMPmain_1+AMPaaa_1)*-0.0176163417098331789856224816048779757693)*cos((FREQmain_1+FREQaab_1)*    4.0 *(t-(((-PHASEmain_1)+PHASEaac_1)/(FREQmain_1+FREQaab_1)))) 
+((AMPmain_1+AMPaaa_1)*-0.0242468142342565640134921522985678166151)*sin((FREQmain_1+FREQaab_1)*    4.0 *(t-(((-PHASEmain_1)+PHASEaac_1)/(FREQmain_1+FREQaab_1)))) 
+((AMPmain_1+AMPaaa_1)*-0.0171650216811894164303797793991179787554)*cos((FREQmain_1+FREQaab_1)*    5.0 *(t-(((-PHASEmain_1)+PHASEaac_1)/(FREQmain_1+FREQaab_1)))) 
+((AMPmain_1+AMPaaa_1)*-0.0171650216811893852053572118165902793407)*sin((FREQmain_1+FREQaab_1)*    5.0 *(t-(((-PHASEmain_1)+PHASEaac_1)/(FREQmain_1+FREQaab_1)))) 
+((AMPmain_1+AMPaaa_1)*-0.0169359868831254270493680280651460634544)*cos((FREQmain_1+FREQaab_1)*    6.0 *(t-(((-PHASEmain_1)+PHASEaac_1)/(FREQmain_1+FREQaab_1)))) 
+((AMPmain_1+AMPaaa_1)*-0.0123047147243315942860553136029011511710)*sin((FREQmain_1+FREQaab_1)*    6.0 *(t-(((-PHASEmain_1)+PHASEaac_1)/(FREQmain_1+FREQaab_1)))) 
+((AMPmain_1+AMPaaa_1)*-0.0168085612357091948587672902704071020707)*cos((FREQmain_1+FREQaab_1)*    7.0 *(t-(((-PHASEmain_1)+PHASEaac_1)/(FREQmain_1+FREQaab_1)))) 
+((AMPmain_1+AMPaaa_1)*-0.0085643897189793712076966158974755671807)*sin((FREQmain_1+FREQaab_1)*    7.0 *(t-(((-PHASEmain_1)+PHASEaac_1)/(FREQmain_1+FREQaab_1)))) 
+((AMPmain_1+AMPaaa_1)*-0.0167362490935154296922693362148493179120)*cos((FREQmain_1+FREQaab_1)*    8.0 *(t-(((-PHASEmain_1)+PHASEaac_1)/(FREQmain_1+FREQaab_1)))) 
+((AMPmain_1+AMPaaa_1)*-0.0054379369715433292786777030869416194037)*sin((FREQmain_1+FREQaab_1)*    8.0 *(t-(((-PHASEmain_1)+PHASEaac_1)/(FREQmain_1+FREQaab_1)))) 
+((AMPmain_1+AMPaaa_1)*-0.0166987131473295795369704563881896319799)*cos((FREQmain_1+FREQaab_1)*    9.0 *(t-(((-PHASEmain_1)+PHASEaac_1)/(FREQmain_1+FREQaab_1)))) 
+((AMPmain_1+AMPaaa_1)*-0.0026448163359797938198880729032680392265)*sin((FREQmain_1+FREQaab_1)*    9.0 *(t-(((-PHASEmain_1)+PHASEaac_1)/(FREQmain_1+FREQaab_1)))) 
+((AMPmain_1+AMPaaa_1)*-0.0083435243106618067754354228782176505774)*cos((FREQmain_1+FREQaab_1)*   10.0 *(t-(((-PHASEmain_1)+PHASEaac_1)/(FREQmain_1+FREQaab_1)))) 
+((AMPmain_1+AMPaaa_1)*-0.0000000000000000000000000000000000000000)*sin((FREQmain_1+FREQaab_1)*   10.0 *(t-(((-PHASEmain_1)+PHASEaac_1)/(FREQmain_1+FREQaab_1))));

Is their a way to adapt/substitute the Periodic Fourier series equation with the standing wave equation?

formatting may cut off some of the information/question I've included a link to an image of the full question http://dl.dropbox.com/u/6576402/questions/sub_per_fou_series_expansion.jpg and as text file http://dl.dropbox.com/u/6576402/questions/ques1.txt tia

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"formatting may cut off some of the information/question" - nothing was cut off; note the scroll bar at the bottom of your code sections... –  J. M. May 6 '11 at 17:18
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1 Answer

As far as I can tell you only have one variable, $t$, in your result. Though this is sometimes called an expansion in sine and cosines "waves", this is a somewhat unfortunate imprecision in terminology, since a "wave" in the narrower sense is something that varies over time and space. In particular, the concept of a "standing wave" only makes sense in time and space, since it refers to how the changes in the wave over time and over space relate to each other. So there's no such thing as expressing your functions of time that don't vary over space as standing waves.

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