General solution of 1D vs 3D wave equations

In summary, the 1-dimensional wave equation has solutions of the form u(x ± ct), while the 3-dimensional wave equation has solutions of the form u(\mathbf{x},t) = f(\mathbf{x})e^{\pm i\omega t}. This is due to the change of variable \zeta = x + ct, \eta = x - ct that can be applied to the 1-dimensional case but not to the 3-dimensional case. Some people refer to the 3-dimensional wave equation as 1-dimensional after a change of variables, but this is a matter of semantics.
  • #1
yucheng
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TL;DR Summary
N/A
For the 1 dimensional wave equation,

$$\frac{\partial^2 u}{\partial x ^2} - \frac{1}{c^2}\frac{\partial ^2 u }{\partial t^2} = 0$$

##u## is of the form ##u(x \pm ct)##
For the 3 dimensional wave equation however,
$$\nabla ^2 u - \frac{1}{c^2}\frac{\partial ^2 u }{\partial t^2} = 0$$It appears that solutions need not be of the form ## u(\vec{k} \cdot \vec{r} - \nu t) ##, for instance spherical waves ## u = A/r \; \mathrm{exp}[-i(kr - vt)] ##

Am I right? Why is it so?
 
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  • #2
Conservation of energy. A spherical wave spatial expands so it’s magnitude needs to decrease.
 
  • #3
Frabjous said:
Conservation of energy. A spherical wave spatial expands so it’s magnitude needs to decrease.

But why doesn't it apply to the 1-dimensional case?
 
  • #4
yucheng said:
But why doesn't it apply to the 1-dimensional case?
Take a distance r from the origin. For a spherical wave, things are spread out on the surface of a sphere with surface area 4πr2. On a line, things are still located at a single point.
 
  • #5
yucheng said:
Summary: N/A

For the 1 dimensional wave equation,

$$\frac{\partial^2 u}{\partial x ^2} - \frac{1}{c^2}\frac{\partial ^2 u }{\partial t^2} = 0$$

##u## is of the form ##u(x \pm ct)##
For the 3 dimensional wave equation however,
$$\nabla ^2 u - \frac{1}{c^2}\frac{\partial ^2 u }{\partial t^2} = 0$$It appears that solutions need not be of the form ## u(\vec{k} \cdot \vec{r} - \nu t) ##, for instance spherical waves ## u = A/r \; \mathrm{exp}[-i(kr - vt)] ##

Am I right? Why is it so?

The change of variable [itex]\zeta = x + ct[/itex], [itex]\eta = x - ct[/itex] turns the 1D wave equation into [tex]
\frac{\partial^2 u}{\partial \zeta\,\partial \eta} = 0[/tex] with general soluton [tex]
u = f(\zeta) + g(\eta) = f(x + ct) + g(x-ct).[/tex] There is no equivalent of this change of variable in 2 or more spatial dimensions, so other types of solution are possible. In particular, solutions of the form [itex]u(\mathbf{x},t) = f(\mathbf{x})e^{\pm i\omega t}[/itex] are possible where [itex]f[/itex] satisfies [tex]
\nabla^2 f + \frac{\omega^2}{c^2}f = 0.[/tex] This equation separates in many coordinate systems, and only in cartesians do we get solutions of the form [tex]f(\mathbf{x}) = \int_{\|\mathbf{k}\| = \omega/c}A(\mathbf{k})e^{i\mathbf{k}\cdot \mathbf{x}}\,dS[/tex] so that [tex]
u(\mathbf{x},t) = \int_{\|\mathbf{k}\| = \omega/c}A(\mathbf{k})e^{i(\mathbf{k}\cdot \mathbf{x} \pm \omega t)} \,dS.[/tex] In 3D with spherical symmetry it can be shown that [itex]u(r,t)/r[/itex] satisfies the 1D wave equation, with general solution as above.
 
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  • #6
pasmith said:
1D wave equation
1D wave equation? Really?
 
  • #7
yucheng said:
1D wave equation? Really?
Oops let me quote it in context...

pasmith said:
In 3D with spherical symmetry it can be shown that u(r,t)/r satisfies the 1D wave equation
Should it really be called 1D? I mean after a change of variables from Cartesian to spherical, then taking the partial derivatives with respect to ##\theta## and ##\phi## as zero, we get an ODE, but...
 
  • #8
yucheng said:
Should it really be called 1D?
Some people do, some people don’t. You shouldn’t let semantics bother you.
 
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FAQ: General solution of 1D vs 3D wave equations

What is the difference between 1D and 3D wave equations?

The main difference between 1D and 3D wave equations is the number of dimensions they take into account. 1D wave equations only consider waves propagating in one direction, while 3D wave equations take into account waves propagating in three dimensions.

Why do we use 1D and 3D wave equations?

1D and 3D wave equations are used to model and understand the behavior of waves in different scenarios. 1D wave equations are commonly used in situations where the waves are only propagating in one direction, such as vibrations in a string. 3D wave equations are used in more complex scenarios where the waves are propagating in multiple directions, such as sound waves in a room.

How do we solve for the general solution of 1D and 3D wave equations?

The general solution of 1D and 3D wave equations can be solved using mathematical techniques such as separation of variables and Fourier series. These methods involve breaking down the equations into simpler parts and finding solutions for each part, which are then combined to form the general solution.

What are the applications of 1D and 3D wave equations?

1D and 3D wave equations have a wide range of applications in various fields such as physics, engineering, and seismology. They are used to understand and predict the behavior of waves in different systems, including mechanical, electromagnetic, and acoustic waves.

Are there any limitations to using 1D and 3D wave equations?

While 1D and 3D wave equations are powerful tools for understanding and modeling waves, they have some limitations. These equations assume ideal conditions and do not account for external factors such as friction, damping, and non-linear effects. In some cases, more complex equations or numerical simulations may be needed to accurately describe wave behavior.

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