Test: Wave Equation - Civil Engineering (CE) MCQ

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: 

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)
4α² > 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.

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

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)
4α² > 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.

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:

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

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

 can be represented as

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: 

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

For steady state

∴ Equation in 2D is 

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.