MATH529: Mathematical methods for the physical sciences IIMarch 6, 2024

Answer the following problems (2 course points each). Present brief motivation of your solutions.

Motivation: Why do power lines hum?

A rod at rest and initial temperature $u(x,0)=0$ (K), of length $L=\pi$ (m), specific heat capacity $c=1$ (J/kg/K), and thermal diffusivity $k=1$ (m${}^2$/s) has thermally insulated ends held at fixed spatial positions. An electric current of intensity $j\left(t\right)=\sin (\omega t)$ is switched on at time $t=0$ (s) and dissipates heat throughout the rod according to Joule's law $q\left(t\right)=r{j}^2\left(t\right)$ (K/s). The rod experiences linear thermal dilatation leading to longitudinal displacements $dw=\gamma du$ (m). In accordance with Hooke's law, thermal dilatation produces a local acceleration $f(x,t)=\gamma {\left(u_{tt}\right)}_x$, where $u_{tt}$ is the second time derivative of the temperature.

Fundamental physical units: m - meter, s - second, kg - kilogram, K - Kelvin

Derived physical units: J = kg m${}^2$/s - Joule, W=J/s - Watt

Unforced heat equation $u_t=k{\nabla }^{2}u$, Unforced wave equation $w_{tt}=c^2{\nabla }^{2}w$

Alternating current frequency $\nu =60\mathrm{Hz}$, $\omega =2\pi \nu$.

In Octave 1, B-flat frequency: $$117 Hz, B frequency: 123 Hz. 1 Hz = 1/s.

Figure 1. Rod thermal transfer and wave motion

1. Write the initial, boundary value problem (IBVP) describing temperature evolution in the rod. Classify the resulting problem.

   Solution. This is a forced heat conduction problem. Thermally insulated implies no heat flux at ends, hence homogeneous Neumann conditions.

   ${\displaystyle \begin{array}{l}{u}_t=k{u}_{xx}+q,t>0,0<x<L(\mathrm{forced}\mathrm{heat}\mathrm{PDE})\\ u_x(0,t)=0,u_x(L,t)=0(\mathrm{Boundary}\mathrm{conditions})\\ u(x,0)=0(\mathrm{Initial}\mathrm{condition}).\end{array}}$

   The above is a linear, second-order, inhomogeneous PDE with homogeneous boundary and initial conditions.

2. Write the IBVP describing longitudinal elastic wave propagation in the rod. Classify the resulting problem.

   Solution. This is a forced wave propagation problem. Fixed ends implies homogeneous Dirichlet conditions.

   ${\displaystyle \begin{array}{l}{w}_t=c^2{w}_{xx}+f,t>0,0<x<L(\mathrm{forced}\mathrm{heat}\mathrm{PDE})\\ w(0,t)=0,w(L,t)=0(\mathrm{Boundary}\mathrm{conditions})\\ w(x,0)=0,w_t(x,0)=0(\mathrm{Initial}\mathrm{condition}).\end{array}}$

   The above is a linear, second-order, inhomogeneous PDE with homogeneous boundary and initial conditions.

3. Solve the heat IBVP.

   Solution. Express $u(x,t)$, $q\left(t\right)$ as series in the eigenfunctions $y_n\left(x\right)=\cos (nx)$ of the homogeneous Neumann condition Sturm-Liouville problem

   ${\displaystyle u(x,t)=\sum _{n=0}^{\infty }{a}_n\left(t\right)\cos (nx),q\left(t\right)=r{\sin }^2(\omega t)=r{\sin }^2(\omega t)\cdot \cos (0\cdot x).}$

   Obtain the ODEs for $a_n\left(t\right)$

   ${\displaystyle {\dot{a}}_0\left(t\right)=r{\sin }^2(\omega t)=\frac{r}{2}[1-\cos (2\omega t)]\Rightarrow a_0\left(t\right)=\frac{r}{2}[t-\frac{1}{2\omega }\sin (2\omega t)]}$ ${\displaystyle {\dot{a}}_n\left(t\right)=-k{n}^2{a}_n\left(t\right)\Rightarrow a_n\left(t\right)=A{e}^{-k{n}^{2}t},a_n\left(0\right)=0\Rightarrow A=0}$

   Solution is

   ${\displaystyle u(x,t)=\frac{r}{2}[t-\frac{1}{2\omega }\sin (2\omega t)]+\sum _{n=1}^{\infty }(e^{-k{n}^{2}t}-1)\cos (nx).}$

4. Solve the elastic wave IBVP.

   Solution. Use the above heat equation solution to obtain the forcing term

   ${\displaystyle f(x,t)=\gamma {\left(u_{tt}\right)}_x=\gamma n(1-k^2{n}^4{e}^{-k{n}^{2}t})\sin (nx).}$

   Express $w(x,t)$ and $f(x,t)$ as series in the eigenfunctions $z_n\left(x\right)=\sin (nx)$ of the homogeneous Dirichlet condition Sturm-Liouville problem

   ${\displaystyle w(x,t)=\sum _{n=1}^{\infty }{c}_n\left(t\right)\sin (nx),f(x,t)=\sum _{n=1}^{\infty }{d}_n\left(t\right)\sin (nx).}$

   Obtain the forcing coefficients

   ${\displaystyle d_n\left(t\right)=\gamma n(1-k^2{n}^4{e}^{-k{n}^{2}t}).}$

   Obtain coefficient ODEs

   ${\displaystyle {\ddot{c}}_n=-{(nc)}^2{c}_n+d_n.}$

   The homogeneous ODE solutions are

   ${\displaystyle c_n^{\left(h\right)}\left(t\right)=A_n\cos (nct)+B_n\sin (nct).}$

   The inhomogeneous particular ODE solution suggested by the $d_n$ forcing term is

   ${\displaystyle c_n\left(t\right)=c_n^{\left(h\right)}\left(t\right)+C_n+D_n{e}^{-k{n}^{2}t}.}$

   Replacing,

   ${\displaystyle k^2{n}^4{D}_n{e}^{-k{n}^{2}t}=-{(nc)}^2(C_n+D_n{e}^{-k{n}^{2}t})+\gamma n(1-k^2{n}^4{e}^{-k{n}^{2}t}).}$

   Identify coefficients of independent functions 1, $e^{-k{n}^{2}t}$

   ${\displaystyle 0=-{(nc)}^2{C}_n+\gamma n,k^2{n}^4{D}_n=-{(nc)}^2{D}_n-\gamma n{k}^2{n}^4\Rightarrow }$ ${\displaystyle C_n=\frac{\gamma }{n{c}^2},D_n=-\frac{\gamma n{k}^2{n}^4}{k^2{n}^4+{(nc)}^2}=\frac{\gamma k^2{n}^3}{k^2{n}^2+c^2}.}$

   Obtain solution

   ${\displaystyle w(x,t)=\sum _{n=1}^{\infty }{c}_n\left(t\right)\sin (nx),c_n\left(t\right)=A_n\cos (nct)+B_n\sin (nct)+\frac{n{c}^2}{\gamma }+\frac{\gamma k^2{n}^3}{k^2{n}^2-c^2}{e}^{-k{n}^{2}t}.}$

   Initial conditions specify $A_n,B_n$

   ${\displaystyle c_n\left(0\right)=0=A_n+\frac{n{c}^2}{\gamma }+\frac{\gamma k^2{n}^3}{k^2{n}^2-c^2}\Rightarrow A_n=-\frac{n{c}^2}{\gamma }-\frac{\gamma k^2{n}^3}{k^2{n}^2-c^2},}$
   ${\displaystyle {\dot{c}}_n\left(0\right)=0=nc{B}_n-\frac{\gamma k^3{n}^5}{k^2{n}^2-c^2}\Rightarrow B_n=\frac{\gamma k^3{n}^4}{c(k^2{n}^2-c^2)}}$

5. It has been observed that power lines hum at a pitch between B-flat and B. Propose an explanation.

   Solution. The thermal forcing term ${\sin }^2(\omega t)$ leads excitation at the double frequency $2\omega$ in accordance with the trigonometric identity

   ${\displaystyle {\sin }^2(\omega t)=1-\cos (2\omega t).}$

   Since the AC frequency is $\nu =60\mathrm{Hz}$, a sound at $2\nu \cong 120\mathrm{Hz}$ is heard.