PHYSICS 312
End of term review problem set 1999:

Problem 1:
In which of the following cases can solutions be found by the method of separation of variables?

$$\frac{\partial^2u}{\partial x^2}+x\frac{\partial^2u}{\partial t^2}=0$$ (1)

$$\frac{\partial^2u}{\partial x^2}+t^2\frac{\partial^2u}{\partial t^2}=0$$ (2)

$$\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial t^2}=e^{-x}$$ (3)

$$\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}=e^{-x}+ e^{-y}$$ (4)

$$\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}-\frac{\partial^2u}{\partial t^2}= \frac{1}{\sqrt{x^2+y^2}}$$ (5)

In cases where the variables don't separate directly, transform the equations so that product solutions can be found. You don't need to solve the resulting ordinary differential equations. (A modified version of this problem was given again as question 3 of midterm review problem set 2001.)

Problem 2:
The differential equation

$$xy^{\prime\prime}+(1-x)y^\prime+py=0$$

(where p is a constant) is called Laguerre's equation.

a:

Find a power series solution of Laguerre's equation.

b:

When p is a positive integer or zero the power series breaks off after a finite number of terms. If normalized so that y(0)=p! the resulting polynomials are called Laguerre polynomials L_p(x). Find L₀(x),L₁(x) and L₂(x).

Problem 3:
The concentration c of a certain chemical in a column of water as a function of height z and time is given by the diffusion equation

$$D\frac{\partial^2c}{\partial z^2}=\frac{\partial c}{\partial t}$$

where D is the diffusivity. The boundary conditions are

$$\frac{\partial c}{\partial z}=0,\;\;for\;z=0,\;z=h$$

where h is the height of the water column. The initial concentration is

$$c(z)=c_0\sin^2(\frac{\pi z}{2h})$$

a:

Find the steady state concentration.

b:

Find the concentration as a function of height and time.

c:

Find the time taken for the concentration at any point to reach the mean of the initial and steady state values.
The trigonometric identity

$$\sin^2(x)=\frac{1}{2}(1-\cos(2x))$$

may prove handy.

(This problem was given again as question 3 of problem set 4 2001.)

Problem 4:
A vibrating membrane is shaped in the form of a square of side L and is weakly damped by frictional forces. Assume a differential equation

$$\frac{\partial u}{\partial t^2}+2f\frac{\partial u}{\partial ... ...partial^2u} {\partial x^2}+\frac{\partial^2u}{\partial y^2})=0$$

where f and c are constants

a

Solve the differential equation using a Fourier expansion assuming as boundary condition that the amplitude u is zero at the edges of the square and that initially

$$u(x,y,0)=g(x,y),\; \frac{d}{dt}u(x,y,t)\vert _{t=0}=0$$

where g(x,y) is a known function.

b

Will the modes of high frequency die out more rapidly than the low frequency modes?

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