update on today's lectures

doc/src/week42/week42.do.txt

@@ -2651,9 +2651,17 @@ Let us remind of this and recast it in terms of the mathematical operation of co
 !split
 ===== Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms) =====
 
+For problems with so-called harmonic oscillations, given by for example the following differential equation
+!bt
+\[
+m\frac{d^2x(t)}(dt^2}+\eta\frac{dx}{dt}+x(t)=F(t),
+\]
+!et
+where $F(t)$ is an applied external force acting on the system (often called a driving force), one can use the theory of Fourier transformations to find the solutions of this type of equations.
+
 If one has several driving forces, $F(t)=\sum_n F_n(t)$, one can find
 the particular solution to each $F_n$, $x_{pn}(t)$, and the particular
-solution for the entire driving force is
+solution for the entire driving force is then given by a series like
 
 !bt
 \begin{equation}
@@ -2661,13 +2669,16 @@ x_p(t)=\sum_nx_{pn}(t).
 \end{equation}
 !et
 
-This is known as the principal of superposition. It only applies when
+!split
+===== Principle of Superposition =====
+
+This is known as the principle of superposition. It only applies when
 the homogenous equation is linear. If there were an anharmonic term
 such as $x^3$ in the homogenous equation, then when one summed various
 solutions, $x=(\sum_n x_n)^2$, one would get cross
 terms. Superposition is especially useful when $F(t)$ can be written
 as a sum of sinusoidal terms, because the solutions for each
-sinusoidal (sine or cosine)  term is analytic, as we saw above.
+sinusoidal (sine or cosine)  term is analytic. 
 
 Driving forces are often periodic, even when they are not
 sinusoidal. Periodicity implies that for some time $\tau$
@@ -2683,6 +2694,9 @@ components in electric circuits are non-linear, e.g. diodes, which
 makes many wave forms non-sinusoidal even when the circuits are being
 driven by purely sinusoidal sources.
 
+!split
+===== Simple Code Example =====
+
 The code here shows a typical example of such a square wave generated using the functionality included in the _scipy_ Python package. We have used a period of $\tau=0.2$.
 
 !bc pycod
@@ -2718,6 +2732,9 @@ F(t)=\frac{f_0}{2}+\sum_{n>0} f_n\cos(2n\pi t/\tau)+g_n\sin(2n\pi t/\tau).
 \end{equation}
 !et
 
+!split
+===== Wrapping up Fourier transforms =====
+
 We can write down the answer for
 $x_{pn}(t)$, by substituting $f_n/m$ or $g_n/m$ for $F_0/m$. By
 writing each factor $2n\pi t/\tau$ as $n\omega t$, with $\omega\equiv
@@ -2745,11 +2762,14 @@ x_p(t)&=&\frac{f_0}{2k}+\sum_{n>0} \alpha_n\cos(n\omega t-\delta_n)+\beta_n\sin(
 \end{eqnarray}
 !et
 
+!split
+===== Finding the Coefficients =====
+
 Because the forces have been applied for a long time, any non-zero
 damping eliminates the homogenous parts of the solution, so one need
 only consider the particular solution for each $n$.
 
-The problem will considered solved if one can find expressions for the
+The problem is considered solved if one can find expressions for the
 coefficients $f_n$ and $g_n$, even though the solutions are expressed
 as an infinite sum. The coefficients can be extracted from the
 function $F(t)$ by