Wave Equations in 1 dimension

BS Mechanical Engineering

Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din

1 Partial Differential Equations (PDEs)

Method of Separation of Variable (MSV) for Wave Equations

1. One-dimensional Wave Equation

The homogeneous partial differential equation that describes the vibrations of a vibrating string

will be discussed by using a well-known method called the method of separation of variables.

The most important feature of the method of separation of variables is that it reduces the partial

differential equation into a system of ordinary differential equations that can be easily handled.

1.1. Analysis of the Method:

The initial-boundary value problem that controls the vibrations of a string is given by

(1)

BCs (2)

IC (3)

The wave function is the displacement of any point of a vibrating string at position x at

time t. The method of separation of variables consists of assuming that the displacement

is identified as the product of two distinct functions and , where depends on the

space variable x and depends on the time variable t. This assumption allows us to set

(4)

Differentiating both sides of Eq. (4) twice with respect to t and twice with respect to x we obtain

(5)

(6)

Substituting Eq. (4) and Eq. (6) into Eq. (1) yields

(7)

Dividing both sides of Eq. (7) by gives

(8)

The left hand side of Eq. (8) depends only on t and the right hand side depends only on x. This

means that the equality holds only if both sides are equal to the same constant. Therefore, we set

(9)

The selection of in Eq. (9) is essential to obtain nontrivial solutions. However, we can easily

show that selecting the constant to be zero or will produce the trivial solution




The result (9) gives two distinct ordinary differential equations given by

(10)

(11)

To determine the function , we solve the second order linear ODE

(12)

The auxiliary of Eq. (12) is

There the solutions of Eq. (12) is

(13)

where and are constants. To determine the constants and we use the homogeneous

boundary conditions

The condition gives

(14)

Similarly the condition gives

(15)

Now we solve Eq. (13) along with the conditions given in Eq. (14) and Eq. (15)

and

We exclude since it gives the trivial solution Accordingly, we find

It is important to note that is excluded since it gives the trivial solution The

function associated with is

Consequently, the solution associated with must satisfy

(16)

Proceeding as before the general solution of Eq. (16) is given by




where and are constants.

Combining the results and we obtain the infinite sequence of product functions

Recall that the superposition principle admits that a linear combination of the functions

also satisfies the given equation and the boundary conditions. Therefore, using this

principle gives the general solution by

(17)

where the arbitrary constants , are as yet undetermined. The derivative of Eq. (17)

with respect to is

(18)

To determine , we substitute in Eq. (17) and by using the initial condition

we obtain

so that the constants can be determined in this case by using Fourier coefficients given by the

formula


we obtain

(19)

so that

(20)




Having determined the constants and , the particular solution follows immediately

upon substituting Eq. (19) and Eq. (20) into Eq. (17).

Note: If we have the ICs in term of sine or cosine function the we compare the coefficients of

sine or cosine function to fine the values of and

1.2. One-dimensional Wave Equation with Dirichlet BCs: In this section we

consider two types of Wave equation subject to different type of initial conditions, the general

form of these two types of problems given below:

Type 1:

BCs

ICs

Type 2:

BCs

ICs

Problem 1: Consider the One dimensional Wave equation subject to Dirichlet boundary

conditions as

(1)

BCs (2)

ICs (3)

According to Method of Separation of Variables (MSV) we assume the following trial solution

(4)


(5)

(6)


(7)


(8)

This means that the equality holds only if both sides are equal to the same constant. Therefore,

we set




(9)

The selection of in Eq. (9) is essential to obtain nontrivial solutions. The result (9) gives two

distinct ordinary differential equations given by

(10)

(11)


(12)



(13)

Similarly we have

(14)

where and are constants. To determine the constants and we use the

homogeneous boundary conditions

The condition gives

(15)


(16)


and


Therefore we have Eq. (13) is

Consequently, we have Eq. (14) is




Now, combining the results and we obtain the infinite sequence of product

functions

Using principle of super position the general solution by

(17)


with respect to is

(18)


we obtain


we obtain

and except

Substituting the values of s and s into Eq. (17), we have

(19)

This is our required solution.

Verification:




Since Eq. (19) is the solution of Eq. (1-3) if it satisfies hole system (PDE, BCs and ICs) therefore

differentiate Eq. (19) twice w.r.t and twice w.r.t we get

(20)

(21)

From Eq. (20) and Eq. (21) we observe that

Now for BCs substitute in Eq. (19) we have

for the Eq. (19), we have

Now for ICs substitute in Eq. (19) we have

putting into , we have

This implies that Eq. (19) is the solution of Eq. (1-3).


conditions as

(1)

BCs (2)

ICs (3)


(4)


(5)

(6)


(7)


(8)





we set

(9)



(10)

(11)


(12)



(13)

Similarly we have

(14)



The condition gives

(15)


(16)


and








functions


(17)


with respect to is

(18)


we obtain

and except


we obtain


(19)





Verification:



(20)

(21)







1.3. One-dimensional Wave Equation with Neumann BCs: In this section we

consider three types of Wave equation subject to different type of initial conditions, the general

form of these three types of problems given below:

Type 1:

BCs

ICs

Type 2:

BCs

ICs

Type 3:

BCs

ICs

Problem 1: Consider the One dimensional Wave equation subject to Neumann boundary

conditions as




(1)

BCs (2)

ICs (3)


(4)


(5)

(6)


(7)


(8)


we set

(9)



(10)

(11)


(12)



(13)

Similarly we have

(14)






The condition gives

(15)


(16)

Now we solve Eq. (13) along with the conditions given in Eq. (15) and Eq. (16), first we

differentiate Eq. (13), w.r.t we get

(13)

and





functions


(17)


with respect to is

(18)





we obtain


we obtained

and except


(19)


Verification:



(20)

(21)


Now for BCs substitute in we have

for the expression we have








conditions as

(1)

BCs (2)

ICs (3)


(4)


(5)

(6)


(7)


(8)


we set

(9)



(10)

(11)


(12)






(13)

Similarly we have

(14)



The condition gives

(15)


(16)



(13)

and





functions





(17)


with respect to is

(18)


we obtain

and except


we obtained


(19)


Verification:



(20)




(21)








conditions as

(1)

BCs (2)

ICs (3)


(4)


(5)

(6)


(7)


(8)


we set

(9)






(10)

(11)


(12)



(13)

Similarly we have

(14)



The condition gives

(15)


(16)



(13)

and








functions


(17)


with respect to is

(18)


we obtain

and except


we obtained

except





(19)


Verification:



(20)

(21)







1.4. One-dimensional Wave Equation with Dirichlet BCs (Use of Fourier

Coefficient): In this section we consider two types of Wave equation subject to different type

of initial conditions, the general form of these two types of problems given below:

Type 1:

BCs

ICs

Type 2:

BCs

ICs

where and are not trigonometric function in sine and cosine functions.





conditions as

(1)

BCs (2)

ICs (3)


(4)


(5)

(6)


(7)


(8)


we set

(9)



(10)

(11)


(12)



(13)

Similarly we have

(14)






The condition gives

(15)


(16)


and





functions


(17)


with respect to is

(18)


we obtain




The arbitrary constants are determined by using the Fourier coefficients method, therefore we

find

so that

This means that we can express by


we obtain


(19)


Verification:



(20)

(21)











conditions as

(1)

BCs (2)

ICs (3)


(4)


(5)

(6)


(7)


(8)


we set

(9)



(10)




(11)


(12)



(13)

Similarly we have

(14)



The condition gives

(15)


(16)


and





functions





(17)


with respect to is

(18)


we obtain


we obtain

The arbitrary constants are determined by using the Fourier coefficients method, therefore we

find

so that


(19)


Verification:






(20)

(21)







1.5. One-dimensional Wave Equation with Inhomogeneous Dirichlet BCs: In

this section we will consider the case where the boundary conditions of the vibrating string are

inhomogeneous. It is well known that the Method of Separation of Variables requires that the

equation and the boundary conditions are linear and homogeneous. Therefore, transformation

formulas should be used to convert the inhomogeneous boundary conditions to homogeneous

boundary conditions.

In this section we will discuss wave equations where Dirichlet boundary conditions are not

homogeneous.

In this first type of boundary conditions, the displacements and of a

vibrating string of length are given. We begin our analysis by considering the initial-boundary

value problem

(1)

BCs (2)

ICs (3)




To convert the inhomogeneous boundary conditions of the Eq. (2) to homogeneous boundary

conditions, we simply use the the following transformation formula

(4)

Differential Eq. (4) w.r.t. we get

(5)


(6)

Now differentiate twice Eq. (4) w.r.t we get

(7)

Eq. (6) and Eq. (7) implies that

For BCs, evaluate Eq. (4) at we obtained

similarly evaluate Eq. (4) at we get

For ICs, evaluate Eq. (4) at we obtained


The transformed initial boundary value problem (IBVP) is

BCs

ICs




In view of the above system, the method of separation of variables can be easily used in the

above system as discussed before. Having determined of the above system, the wave

function of Eq. (1-3) follows immediately upon substituting into Eq. (4).

Now, we consider two types of Wave equation subject to different type of initial conditions, the

general form of these two types of problems given below:

Type 1:

BCs

ICs

Type 2:

BCs

ICs

Note: Must use transformation in the end to convert the solution into


conditions as

(1)

BCs (2)

ICs (3)

To convert the inhomogeneous BCs into homogenous BCs we use the following transformation

here we notice that

The above transformation reduces to

(4)


(5)


(6)


(7)










(8)

BCs (9)

ICs (10)


(11)

Differentiating both sides of Eq. (11) twice with respect to t and twice with respect to x we

obtain

(12)

(13)


(14)


(15)





we set

(16)

The selection of in Eq. (16) is essential to obtain nontrivial solutions. The result (16) gives

two distinct ordinary differential equations given by

(17)

(18)


(19)



(20)

Similarly we have

(21)



The condition gives

(22)


(23)


and








functions


(24)


with respect to is

(25)


we obtain

except


we obtain


(26)




Substituting Eq. (26) into Eq. (4) we get


Verification:



(27)

(28)








conditions as

(1)

BCs (2)

ICs (3)


here we notice that





(4)


(5)


(6)


(7)







(8)

BCs (9)

ICs (10)


(11)





obtain

(12)

(13)


(14)


(15)


we set

(16)



(17)

(18)


(19)



(20)

Similarly we have

(21)



The condition gives

(22)





(23)


and





functions


(24)


with respect to is

(25)


we obtain





we obtain

except


Substituting the above expression into Eq. (4) we get

(26)


Verification:



(27)

(28)










1.6. One-dimensional Wave Equation with Inhomogeneous Neumann BCs: In

this section we will consider the case where the boundary conditions of the vibrating string are

inhomogeneous. It is well known that the Method of Separation of Variables requires that the

equation and the boundary conditions are linear and homogeneous. Therefore, transformation

formulas should be used to convert the inhomogeneous boundary conditions to homogeneous

boundary conditions.

In this section we will discuss wave equations where Neumann boundary conditions are not

homogeneous.

In this first type of boundary conditions, the displacements and of a

vibrating string of length are given. We begin our analysis by considering the initial-boundary

value problem

(1)

BCs (2)

ICs (3)

To convert the inhomogeneous boundary conditions of the Eq. (2) to homogeneous boundary

conditions, we simply use the the following transformation formula

(4)


(5)


(6)


(7)


(8)










BCs

ICs

In view of the above system, the method of separation of variables can be easily used in the

above system as discussed before. Having determined of the above system, the wave

function of Eq. (1-3) follows immediately upon substituting into Eq. (4).

Now, we consider two types of Wave equation subject to different type of initial conditions, the

general form of these two types of problems given below:

Type 1:

BCs

ICs

Type 2:

BCs

ICs

Note: Must use transformation in the end to convert the solution into





conditions as

(1)

BCs (2)

ICs (3)


here we notice that


(4)


(5)


(6)


(7)


(8)










(9)

BCs (10)

ICs


(11)


obtain

(12)

(13)


(14)


(15)


we set

(16)



(17)

(18)


(19)






(20)

Similarly we have

(21)



The condition gives

(22)


(23)



(24)

and





functions





(25)


with respect to is

(26)


we obtain

and except


we obtained



(27)


Verification:






(28)

(29)



for the expression , we have





conditions as

(1)

BCs (2)

ICs (3)


here we notice that


(4)


(5)





(6)


(7)


(8)







(9)

BCs (10)

ICs


(11)


obtain

(12)

(13)





(14)


(15)


we set

(16)



(17)

(18)


(19)



(20)

Similarly we have

(21)



The condition gives

(22)


(23)



(24)




and





functions


(25)


with respect to is

(26)


we obtain


we obtained




and except



(27)


Verification:



(28)

(29)



for the expression , we have




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Wave Equations in 1 dimension

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