can be solved by the displacement equation , or , or even . To show this, enter the displacement equation and then show that both sides of the differential equation are equal.

>f:=(x,t)->sin(x+a*t); >g:=(x,t)->sin(k*x)*cos(k*a*t); >h:=(x,t)->sin(x+a*t)+37.8; >diff(f(x,t),t,t)-a^2*diff(f(x,t),x,x); >diff(g(x,t),t,t)=a^2*diff(g(x,t),x,x); >diff(h(x,t),t,t)=a^2*diff(h(x,t),x,x);

- The function

denotes temperature at a depth and time where the seasonal variation of the surface temperature is

Where is the annual average surface temperature and is chosen such that the period is one year.**A)**- Enter the function
**B)**- Show that the function satisfies the surface condition (i.e. ).
**C)**- Show that the function satisfies that one-dimensional heat equation .
**D)**- Enter the following data into memory (Do this
**after**completeing parts A, B, and C, or you will need to re-initialize the constants).- The thermal conductivity of brick is .
- The average temperature of the surface temperature is degrees celsius, please enter as as the conductivity uses units of Kelvin.
- The surface area of a brick is

**E)**- Plot the surface temperature of a brick for one year (). To make the temperature axis celsius, plot the function minus 273.
**F)**- From your graph, what is the approximate temperature variation of the surface of the brick?
**G)**- Plot the temperature of the inside of the brick with dimensions six by six by twelve (i.e. depth = ). Remember to plot using celsius.
**H)**- From your graph, what is the temperature variation of the inside of the brick?

- A function is called
**harmonic**if it satisfies Laplaces's equation

Determine which of the following functions is harmonic.**A)****B)****C)****D)**

- Using a function from exercise 2 that you found satisfies the Laplace equation, answer the following without calculating the differential equation.
**A)**- Will the function plus satisfy the Laplace equation? Why or why not?
**B)**- Will the function plus satisfy the Laplace equation? Why or why not?
**C)**- Will the function plus satisfy the Laplace equation? Why or why not?

2007-01-22