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Test: Wave Equation - Electronics and Communication Engineering (ECE) MCQ


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10 Questions MCQ Test - Test: Wave Equation

Test: Wave Equation for Electronics and Communication Engineering (ECE) 2024 is part of Electronics and Communication Engineering (ECE) preparation. The Test: Wave Equation questions and answers have been prepared according to the Electronics and Communication Engineering (ECE) exam syllabus.The Test: Wave Equation MCQs are made for Electronics and Communication Engineering (ECE) 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Wave Equation below.
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Test: Wave Equation - Question 1

Which of the following represents a wave equation?

Detailed Solution for Test: Wave Equation - Question 1

Wave equation is a generalised partial differential equation defining any mechanical wave.

where, 
A one-dimensional wave equation is given by: 
A two-dimensional wave equation is given by: 

Additional Information
The heat equation is given as: 

Test: Wave Equation - Question 2

 represents the equation for

Detailed Solution for Test: Wave Equation - Question 2

Concept:
Wave equation:
It is a second-order linear partial differential equation for the description of waves (like mechanical waves).
The Partial Differential equation is given as, 


For One-Dimensional equation,

where, A = α2, B = 0, C = -1
Put all the values in equation (1)
∴ 0 - 4(α2)(-1)
2 > 0.
So, this is a one-dimensional wave equation or vibration of stretched spring.
Additional Information

having A = α2, B = 0, C = 0
Put all the values in equation (1), we get 0 - 4(α2)(0) = 0, therefore it shows parabolic function.
So, this is a one-dimensional heat equation.

having A = 1, B = 0, C = 1
Put all the values in equation (1), we get 0 - 4(1)(1) = -4, therefore it shows elliptical function.
So, this is a two-dimensional heat equation.

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Test: Wave Equation - Question 3

The solution at (x, t) = (2, 1) of the partial differential equation,  subject to initial condition of u(x, 0) = 5x and  

Detailed Solution for Test: Wave Equation - Question 3

Concept:
 where -∞ < x < ∞ , t > 0 and c > 0.
Satisfying the conditions u(x, 0) = f(x) and  where f(x) = initial displacement and g(x) is the initial velocity.
The solution for the above equation satisfying the conditions is given by D-Alembert's formula i.e.

Calculation:
Given:

Initial condition u(x, 0) = 5x ⇒ f(x) and 
∴ the D-Alembert solution is  
Putting the values of x = 2, t = 1, c = 6 and g(x) = 1

f(x) = 5x
f(- 4) = -20 and f(8) = 40 and 

u(2, 1) = 10 + 1 ⇒ 11

Test: Wave Equation - Question 4

One dimensional wave equation is

Detailed Solution for Test: Wave Equation - Question 4

Concept:
Wave equation:
It is a second-order linear partial differential equation for the description of waves (like mechanical waves).
The Partial Differential equation is given as, 

For One-Dimensional equation,

where, A = α2, B = 0, C = -1
Put all the values in equation (1)
∴ 0 - 4 (α2)(-1)
2 > 0.
So, this is a one-dimensional wave equation.
Additional Information

having A = α2, B = 0, C = 0
Put all the values in equation (1), we get 
0 - 4(α2)(0) = 0, therefore it shows parabolic function.
So, this is a one-dimensional heat equation.

having A = 1, B = 0, C = 1
Put all the values in equation (1), we get 0 - 4(1)(1) = -4, therefore it shows elliptical function.
So, this is a two-dimensional heat equation.

Test: Wave Equation - Question 5

Consider a function u which depends on position x and time t. The partial differential Equation  is known as the:

Detailed Solution for Test: Wave Equation - Question 5

3-D heat equation is given as below

For 1 – D & without heat generation:

Where α ÷ thermal diffusivity.
Wave equation is given by:

Laplace equation:

Test: Wave Equation - Question 6

The solution at x = 1, t = 1 of the partial differential equation,  subject to initial condition of  is _____.

Detailed Solution for Test: Wave Equation - Question 6

Concept:
It is 1D wave equation in the partial differential equation given by- 
Where C2 = T/m, T = Tension in the elastic string, and M = mass per unit length
Calculation:
On comparing the above equation, we get that C = 1/5
f(x) = 3x, g(x) = 3
f (x + ct) = f (x + t/5) = 3 (x + t/5) = 3x + 3t/5
f (x - ct) = f (x - t/5) = 3 (x- t/5) = 3x - 3t/5

Test: Wave Equation - Question 7

A one-dimensional domain is discretized into N sub-domains of width Dx with node numbers i = 0, 1, 2, 3…………, N. If the time scale is discretized in steps of Dt, the forward-time and centered-space finite difference approximation at ith node and nth time step, for the partial differential equation  is

Detailed Solution for Test: Wave Equation - Question 7


 (using forward time finite difference approximation)
Also, 
(Using centered space finite difference approximation)

 can be represented as

Test: Wave Equation - Question 8

The partial differential equation  where c ≠ 0 is known as

Detailed Solution for Test: Wave Equation - Question 8

3-D heat equation is given as below

For 1 – D & without heat generation:

Where α ÷ thermal diffusivity.
Wave equation is given by:

Laplace equation:


​Poisson’s equation: 

Test: Wave Equation - Question 9

Which of the following represents the steady state behaviour of heat flow in two dimensions x – y?

Detailed Solution for Test: Wave Equation - Question 9

Concept:
The heat flow equation with constant thermal conductivity (k) is given by:

For steady state

∴ Equation in 2D is 

Test: Wave Equation - Question 10

The differential equation representing the heat equation is

Detailed Solution for Test: Wave Equation - Question 10

Concept:
In one dimensional cartesian system,

Δ u = f → This is the general form of Poisson equation, where Δ is the Laplacian operator and u and f are real or complex-valued functions. If f = 0 then it represents the Laplace equation.
In two dimensional cartesian system,

 is the general form of the heat equation, where t is the independent variable time, C is the diffusivity of the medium.
 this is the general form of a wave the equation, where t is the independent variable time, c is a fixed non-negative real coefficient.

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