diff --git a/content/Fourier Series (lec 28-29).md b/content/Fourier Series (lec 28-29).md index ee0b5f0..04f861b 100644 --- a/content/Fourier Series (lec 28-29).md +++ b/content/Fourier Series (lec 28-29).md @@ -174,7 +174,7 @@ then: $$\bar{f}(x)=\frac{2}{\pi}+\frac{2}{\pi}\sum_{k=1}^\infty\left( \frac{1}{2k+1}-\frac{1}{2k-1} \right)\cos(2k\pi x)$$ Even with 10 terms, we get a pretty good approximation: ![fouriercosineofsin.png](drawings/fouriercosineofsin.png) -Here's a little script I wrote to generate an audible waveform of this Fourier series! +Here's a little script that generates an audible waveform of this Fourier series! diff --git a/content/Wave equations (lec 35).md b/content/Wave equations (lec 35).md new file mode 100644 index 0000000..b3f158a --- /dev/null +++ b/content/Wave equations (lec 35).md @@ -0,0 +1,49 @@ +#start of lec 35 +last problem of the course which we will finish today. +$\frac{ \partial^2 u }{ \partial t^2 }=\frac{ \partial^2 u }{ \partial x^2 }+tx, 0\leq x\leq \pi$ $t>0$ +thats a driven wave equation. tx is the source term. +$u(0,t)=u(\pi,t)=0 \quad t>0$ +$u(x,0)=\sin(x) \quad 0\leq x\leq \pi$ +$\frac{ \partial u }{ \partial t }(x,0)=5\sin(2x)-3\sin(5x)\quad 0\leq x\leq \pi$ +if any of the boundary conditions are non zero, then we have to split(?) into X and T. +in this case there's a $\sin(x)$ term so its a nonhomogenous equation (?) + +when $tx$ wasn't there in last problem we had the solution: +$u(t,x)=\sum_{n=1}^\infty \underbrace{ (a_{n}\cos(nt)+b_{n}\sin(nt)) }_{ u_{n}(t) }\sin(nt)$ notice $L=\pi$ +if we expand $tx$: +$tx=\sum_{n=1}^\infty h_{n}\sin(nx)$ (this is called a formal expansion, the two arent exactly equal due to the discontinuouity in tx.) +$h_{n}=\frac{2}{\pi}\int _{0} ^\pi tx\sin(nx)\, dx$ +![draw](drawings/2023-12-06-13.14.28.excalidraw.png) + +continuous between 0 and pi but on the edge ponts, the foureir sin series will converge to the midpoint of the two edge points. +$h_{n}=\frac{2t}{\pi}\int _{0}^\pi x\sin(nx)\, dx=- \frac{2t}{\pi n}\left( x\cos(nx)|_{0}^\pi-\int _{0}^\pi \cancel{ \cos(nx) }\, dx \right)$ +$h_{n(t)}=\frac{2t}{n}(-1)^{n+1}$ +$h(x,t)=tx=\sum_{n=1}^\infty h_{n}\sin(nx)=\sum_{n=1}^\infty \frac{2t}{n}(-1)^{n+1}\sin(nx)$ +$\frac{ \partial^2 u }{ \partial t^2 }=\sum_{n=1}^\infty u_{n}''(t)\sin(nx)$ +$\frac{ \partial^2 u }{ \partial x^2 }=\sum_{n=1}^\infty -u_{n}(t)n^2\sin(nx)$ +$\sum_{n=1}^\infty \underbrace{ \left( u_{n}''+n^2u_{n}+\frac{2t}{n}(-1)^n \right) }_{ =0 }\sin(nx)=0$ +$u''_{n}+n^2u_{n}=\frac{2}{n}(-1)^{n+1}t, \quad n=1,2,\dots$ +use #mouc (or laplace, but that'll take much longer.) +characteristic eq: +$r^2+n^2=0$ +$r_{1,2}=\pm in$ +$u_{n}^h(t)=a_{n}\cos(nt)+b_{n}\sin(nt)$ +$u_{n}^p(t)=At+B$ +$B=0$ because there's no constant term on the RHS +$A=\frac{2(-1)^{n+1}}{n^3}$ +$u_{n}(t)=a_{n}\cos(nt)+b_{n}\sin(nt)+\frac{2(-1)^{n+1}}{n^3}t$ +$u(x,t)=\sum_{n=1}^\infty(a_{n}\cos(nt)+b_{n}\sin(nt)+\frac{2(-1)^{n+1}}{n^3}t)\sin(nx)$ +This is the last problem I'll be solving in my career. This is the last time he's teaching math 201 :( or any course for that matter. +$u(x,0)=\sum_{n=1}^\infty a_{n}\sin(nx)=\sin(x)$ +$a_{1}=1, \quad a_{k}=0, \quad k=2,3,\dots$ +*audio recording started* +$\frac{ \partial u }{ \partial t }(x,0)=\sum_{n=1}^\infty\left( b_{n}n+\frac{2(-1)^{n+1}}{n^3} \right)\sin(nx)=5\sin(2x)-3\sin(5x)$ +coefficients =0 if $n\ne_{2},5$ +$\implies b_{n}=\frac{2}{n^4}(-1)^n, \quad n\ne_{2},5$ +$b_{2}=\frac{5}{2}+\frac{2(-1)^2}{2^4}$ +$b_{5}=-\frac{3}{5}+\frac{2(-1)^5}{5^4}$ +$u(x,t)=\cos(t)\sin(x)+\frac{5}{2}\sin(2t)\sin(2x)-\frac{3}{5}\sin(5t)\sin(5x)+2\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^3}\left( t-\frac{\sin(nt)}{n} \right)\sin(nx)$ +finished the solution +#end of lec 35 + +man I got teary eyed from this lecture. \ No newline at end of file diff --git a/content/_index.md b/content/_index.md index 080b50d..67b69f2 100644 --- a/content/_index.md +++ b/content/_index.md @@ -26,6 +26,8 @@ I have written these notes for myself, I thought it would be cool to share them. [Separation of variables & Eigen value problems (lec 26-28)](separation-of-variables-eigen-value-problems-lec-26-28.html) [Fourier series (lec 28-29)](fourier-series-lec-28-29.html) [Partial differential equations (lec 30-34)](partial-differential-equations-lec-30-34.html) (raw notes, not reviewed or revised yet.) +[Wave equations (lec 35)](wave-equations-lec-35.html) (raw notes, not reviewed or revised yet.) +
[How to solve any DE, a flow chart](Solve-any-DE.png) (Last updated Oct 1st, needs revision. But it gives a nice overview.) [Big LT table (.png)](drawings/bigLTtable.png) diff --git a/content/drawings/2023-12-06-13.14.28.excalidraw.md b/content/drawings/2023-12-06-13.14.28.excalidraw.md new file mode 100644 index 0000000..d2c24d3 --- /dev/null +++ b/content/drawings/2023-12-06-13.14.28.excalidraw.md @@ -0,0 +1,1258 @@ +--- + +excalidraw-plugin: parsed +tags: [excalidraw] + +--- +==⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== + + +# Text Elements +%% +# Drawing +```compressed-json +N4KAkARALgngDgUwgLgAQQQDwMYEMA2AlgCYBOuA7hADTgQBuCpAzoQPYB2KqATL + +ZMzYBXUtiRoIACyhQ4zZAHoFAc0JRJQgEYA6bGwC2CgF7N6hbEcK4OCtptbErHAL + +RY8RMpWdx8Q1TdIEfARcZgRmBShcZQUARm0ATm1YhJo6IIR9BA4oZm4AbXAwUDBS + +6HhxdAAzQIRPKn4yxhZ2LjQAFgAGRshm1k4AOU4xbljOhIBWHgB2MdiADh6IQmYA + +EQyoOu4qgjClkm4IAGEoUgSANU6qyXaoeYBVAdIYEiqAaQAlfTTSyCrCfD4ADKsG + +Ch0kuGwGkCPzKzFObAA1ggAOokdSjJbw0hIhAgmBgiSCDywyA4vySDjhPJoWJLNh + +wSFqGCjTrdIqQazKQmodm/CCYbjOHgAZiS806XViADZYrERTx5rEeBMliy0MKePE + 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file mode 100644 index 0000000..f0fb0a5 Binary files /dev/null and b/content/drawings/2023-12-06-13.14.28.excalidraw.png differ diff --git a/static/playaudio.js b/static/playaudio.js index 823a123..6b9411d 100644 --- a/static/playaudio.js +++ b/static/playaudio.js @@ -22,7 +22,7 @@ var cspec = document.getElementById('spectrum'), sliderFreq.oninput = function() { textFreq.innerHTML=this.value; - osc.frequency.value=this.value; + osc.frequency.value=this.value/2; } // Update the current slider value (each time you drag the slider handle) slider.oninput = function() { @@ -43,7 +43,7 @@ slider.oninput = function() { real[2*i] = (2/Math.PI) *( 1/(2*i+1)-1/(2*i-1) ); } - osc.frequency.value = textFreq.innerHTML; + osc.frequency.value = textFreq.innerHTML/2; //var imag= new Float32Array([0,0,0,0,0]); // sine //var real = new Float32Array(imag.length); // cos var customWave = context.createPeriodicWave(real, imag); // cos,sine @@ -181,7 +181,7 @@ function playSound(h){ //var real = new Float32Array(imag.length); // cos var customWave = context.createPeriodicWave(real, imag); // cos,sine osc.setPeriodicWave(customWave); - osc.frequency.value = textFreq.innerHTML; + osc.frequency.value = textFreq.innerHTML/2; osc.connect(masterGain); osc.start(); isPlaying = true;