Questions And Answers

PART-A QUESTIONS AND ANSWERS

1. What conditions are assumed in deriving the one dimensional wave equation?

See Page No. 3.13 LAJ

2. What are the various solutions of 

See Page No. 3.11

3. A string is stretched and fastened to two points / apart. Motion is started by displacing the string into the form y = yo sin pppppppp from which it is released at time t = 0. Formulate this problem as the boundary value problem.

Solution: The displacement function y (x, t) is the solution of the wave equation.

4. A string of length 27 stretched to a constant tension T, is fastened at both the ends and hence fixed. The mid point of the string is taken to a height b and then released from rest in that position.

It is desired to solve for transverse vibrations of the string. Write the governing equation and the corresponding conditions.

See Page No. 3.24

5. What is the constant a2 in the wave equation

6. State the suitable solution of the one dimensional heat equation

7. State the governing equation for one dimensional heat equation and necessary conditions to solve the problem.

Solution: The one dimensional heat equation is

where u (x, t) is the temperature at time t at a point distant x from the left end of the rod.

8. Write all variable separable solutions of the one dimensional heat equation u1 = άuxx

Solution :

9. Write down the diffusion problem in one-dimension as a boundary value problem in two different forms.

Solution :

10. State any two laws which are assumed to derive one dimensional heat equation.

Solution :

(i) Heat flows from higher to lower temperature.

(ii) The rate at which heat flows across any area is proportional to the area and to the temperature gradient normal to the curve. This constant of proportionality is known as the thermal conductivity (k) of the material. It is known as Fourier law of heat conduction.

11. Write any two solutions of the Laplace equation U_xx + U_yy involving exponential terms in x or y.

12. In steady state conditions, derive the solution of one dimensional  heat flow equation.

13. Write the boundary conditions and initial conditions for solving the vibration of string equation, if the string is subjected to initial displacement f (x) and initial velocity g(x).

14. Write down the governing equation of two dimensional steady state heat conduction.

15. The ends A and B of a rod of length 10 cm have their temperature kept at 20°C and 70°C. Find the steady state temperature distribution on the rod.

16. Write down the three possible solutions of Laplace equation in two-dimensions.

Solution :

17. Write down the partial differential equation governing the transverse vibrations of an elastic string.

18. Explain the various variables dominating the wave equation.

Solution: The wave equation is .

19. Classify the partial differential equation

20. Classify the partial differential equation

21. Classify the partial differential equation u_xx + 2u_xy + u_yy=0

Solution :

.

22. Classify the partial differential equation

Solution :

23. Classify the partial differential equation