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Tutorial sheet -5

Q1. Solve the following first order partial differential equations by separating

the variables:

(a) 0=∂

∂−

∂

∂

 y

u x

 x

u

(b) 02 =∂

∂−

∂

∂

 y

u y

 y

u x

Q2. A semi-infinite rectangular metal plate occupies the region ∞≤≤ x0 andb y ≤≤0 in the x-y plane. The temperature at the far end of the plate and along

its two long sides is fixed at 0°C. If the temperature of the plate at x=0 is alsofixed and is given by )( y  f   , fine the steady state temperature distribution

),( y xu of the plate. Hence find the temperature distribution if  0)( u y  f   = , where

0u is a constant.

Q3. Obtain a separable solution describing a particle confined to a box of side a

(ψ must vanish at the walls of the box). Show that the energy of the particle can

only take the quantized values

)(2

222

2

22

 z  y x nnnma

 E  ++=π   , where x

n ,  yn and  z n are integers.

Q4.The wave equation describing the transverse vibrations of a stretched

membrane under tension ‘T’ and having a uniform surface density  ρ is

2

2

2

2

2

2

)(t 

u

 y

u

 x

uT 

∂

∂=

∂

∂+

∂

∂ ρ 

Find a separable solution appropriate to a membrane stretched on a frame of 

length ‘a’ and width ‘b’, showing that the natural angular frequencies of such a

membrane are given by

)(2

2

2

222

b

m

a

nT +=

 ρ 

π ω  ,

Where ‘n’ and ‘m’ are any positive integers

Q5. Denoting the three terms of  2∇ in spherical polars by 2

r ∇ ,2θ  ∇ ,

2

φ ∇ in an

obvious way, evaluate ur 2

∇ , etc. For the two functions given below and verify

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that, in each case, although the individual terms are not necessarily zero their 

sum 2∇ u is zero. Identify the corresponding values of l and m.

(a)

2

1cos3)(),,(

2

3

2 −+=

θ φ θ 

r 

 B Ar r u  

(b) θ θ φ θ  ir 

 B Ar r u expsin)(),,(

2+=

Q6. A circular disc of radius ‘a’ is heated in such a way that its perimeter  a= ρ   

has a steady temperature distribution φ 2cos B A+ , where  ρ  and are plane

  polar coordinates, A and B are constants. Find the temperature ),( φ  ρ T   

everywhere in the region a< ρ  .

Q7. Prove that the potential for  a< ρ  associated with a vertical split cylinder of radius ‘a’, the two halves of which 0(cos >φ  and )0cos <φ  are maintained at

equal and opposite potentials ±V, is given by

φ  ρ 

π φ  ρ  )12cos()(

12

)1(4),(

12

0 ++

−∑=

+∞

=n

an

V u

n

n

n

Q8. Two identical copper bars each of length ‘a’. Initially one is at C ο 

0 and the

other at C ο 100 ; they are then joined together end to end and thermally isolated.

Obtain in the form of a Fourier series an expression ),( t  xu for the temperatureat any point a distance x from the join at a later time‘t’. Bear in mind the heat

flow conditions at the free ends of the bar.

Taking a=0.5m estimate the time it takes for one of the free ends to attain a

temperature of  C ο 

55 . The thermal conductivity of copper is 3.8x102Jm-1K -1s-1,

and its specific heat capacity is 3.4x106Jm-3K -1.