This set of Fourier Analysis and Partial Differential Equations Multiple Choice Questions & Answers (MCQs) focuses on “Derivation and Solution of Two-dimensional Wave Equation”.

1. Who discovered the one-dimensional wave equation?

a) Jean d’Alembert

b) Joseph Fourier

c) Robert Boyle

d) Isaac Newton

View Answer

Explanation: Jean-Baptiste le Rond d’Alembert (16 November 1717 – 29 October 1783) was a French mathematician, mechanician, physicist, philosopher, and music theorist. Until 1759 he was co-editor with Denis Diderot of the Encyclopédie. He was the person responsible for the discovery of wave equation.

2. Wave equation is a third-order linear partial differential equation.

a) True

b) False

View Answer

Explanation: The wave equation is a second-order linear partial differential equation which is developed for the description of waves (water waves, sound waves, seismic waves, light waves), acoustics, electromagnetics, and fluid dynamics.

3. In which of the following fields, does the wave equation not appear?

a) Acoustics

b) Electromagnetics

c) Pedology

d) Fluid Dynamics

View Answer

Explanation: The wave equation arises in fields like acoustics, electromagnetics, and fluid dynamics. Historically, the problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d’Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis Lagrange. In 1746, d’Alembert discovered the one-dimensional wave equation, and within ten years Euler discovered the three-dimensional wave equation.

4. The wave equation is known as d’Alembert’s equation.

a) True

b) False

View Answer

Explanation: D’Alembert’s formula for obtaining solutions to the wave equation is named after him. The wave equation is sometimes referred to as d’Alembert’s equation.

5. Which of the following statements is false?

a) Equations that describe waves as they occur in nature are called wave equations

b) The problem of having to describe waves arises in fields like acoustics, electromagnetics, and fluid dynamics

c) Jean d’Alembert discovered the three-dimensional wave equation

d) Jean d’Alembert discovered the one-dimensional wave equation

View Answer

Explanation: The one-dimensional wave equation was discovered by d’Alembert in 1746 and by 1756, the three-dimensional wave equation was discovered by Euler.

6. What is the order of the partial differential equation, \(\frac{∂z}{∂x}-(\frac{∂z}{∂y})^3=0\)?

a) Order-5

b) Order-1

c) Order-4

d) Order-2

View Answer

Explanation: The order of an equation is defined as the highest derivative present in the equation. Hence, in the given equation, \(\frac{∂z}{∂x}-(\frac{∂z}{∂y})^5=0,\) the order is 1.

7. The half-interval method in numerical analysis is also known as __________

a) Newton-Raphson method

b) Regula Falsi method

c) Taylor’s method

d) Bisection method

View Answer

Explanation: The Bisection method, also known as binary chopping or half-interval method, is a starting method which is used, where applicable, for few iterations, to obtain a good initial value.

8. Wave equation is a linear elliptical partial differential equation.

a) False

b) True

View Answer

Explanation: Wave equation is a linear second order hyperbolic partial differential equation whereas a Laplace equation is a linear elliptical partial differential equation.

9. Which of the following is the condition for a second order partial differential equation to be hyperbolic?

a) b^{2}-ac < 0

b) b^{2}-ac=0

c) b^{2}-ac>0

d) b^{2}-ac= < 0

View Answer

Explanation: For a second order partial differential equation to be hyperbolic, the equation should satisfy the condition, b

^{2}-ac>0.

10. Which of the following statements is true?

a) Hyperbolic equations have three families of characteristic curves

b) Hyperbolic equations have one family of characteristic curves

c) Hyperbolic equations have no families of characteristic curves

d) Hyperbolic equations have two families of characteristic curves

View Answer

Explanation: The canonical variables ξ and η for a hyperbolic pde satisfy the equations,

\(aζ_x+(b+\sqrt{b^2-ac}) ζ_y=0 \, and \, aη_x+(b+\sqrt{b^2-ac}) η_y=0 \)

The families of curves ξ = constant and η = constant are the characteristic curves. Hence, hyperbolic equations have two families of characteristic curves.

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