Equation - Wave Equation Derivation Question

In summary, the conversation discusses the derivation of the wave equation and its general solution in 1+1 dimensions. The correct solution is obtained by substituting u_1=x-c t and u_2=x+ c t and integrating successively. The general solution is f(t,x)=f_1(x-c t)+f_2(x+c t), with two arbitrary functions that are at least two times differentiable with respect to their arguments.
  • #1
alejandrito29
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equation -- Wave Equation Derivation Question

Hello, my teacher says that if, on a wave equation

[tex]f(x-ct)=f(e)[/tex] then
[tex]\partial_{ee}= \partial_{tt}- c^2 \partial_{xx}[/tex]

but i think that

[tex]\partial_{t}=\frac{\partial }{\partial e} \frac{\partial e}{\partial t}=-c\frac{\partial }{\partial e} [/tex]

and

[tex]\partial_{x}=\frac{\partial }{\partial e} \frac{\partial e}{\partial x}=\frac{\partial }{\partial e} [/tex]

then

[tex]\partial_{tt}- c^2 \partial_{xx}= c^2 \partial_{ee}- c^2 \partial_{ee}=0[/tex]

what is the correct?
 
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  • #2
Of course you are right. Indeed the general solution of the homogeneous wave equation in 1+1 dimensions is
[tex]f(t,x)=f_1(x-c t)+f_2(x+c t).[/tex]
You come to this conclusion by the substitution
[tex]u_1=x-c t, \quad u_2=x+ c t[/tex]
This gives through the chain rule
[tex]\partial_{u_1} \partial_{u_2}=\frac{1}{4}(\partial_x^2-\partial_t^2/c^2).[/tex]
This means that you can write the wave equation
[tex]\left (\frac{1}{c^2} \partial_t^2-\partial_x^2 \right ) f=0.[/tex]
as
[tex]\partial_{u_1} \partial_{u_2} f=0.[/tex]
This is very easy to integrate successively. Integrating with respect to [itex]u_1[/itex] first gives
[tex]\partial_{u_2} f = \tilde{f}_2'(u_2)[/tex]
and then
[tex]f=\tilde{f}_1(u_1)+\tilde{f}_2(u_2) = \tilde{f}_1(x-ct)+\tilde{f}_2(x+c t)[/tex]
with two arbitrary functions that are at least two times differentiable with respect to their arguments.
 

What is the wave equation?

The wave equation is a mathematical model that describes the behavior of waves in various physical systems. It is a second-order partial differential equation that relates the spatial and temporal variations of a wave.

What is the derivation of the wave equation?

The wave equation can be derived using the principles of classical mechanics and electromagnetism. The derivation involves applying Newton's second law and Maxwell's equations to a small element of a medium, resulting in the wave equation.

What is the significance of the wave equation?

The wave equation is of great importance in physics and engineering as it allows us to predict and understand the behavior of waves in various systems. It is used in fields such as acoustics, optics, electromagnetics, and many others.

What are the assumptions made in the derivation of the wave equation?

The derivation of the wave equation assumes that the medium in which the wave propagates is uniform and continuous, and that the wave is small enough to be considered a disturbance of the medium's equilibrium state. It also assumes that there are no dissipative forces acting on the medium.

Are there any real-world applications of the wave equation?

Yes, the wave equation has numerous real-world applications. It is used to study the behavior of sound waves in musical instruments, seismic waves in earthquakes, electromagnetic waves in communication systems, and many other phenomena. It also plays a crucial role in the development of technologies such as medical imaging and radar systems.

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