What is u in the Wave Equation u(x,t)?

  • Context: High School 
  • Thread starter Thread starter coverband
  • Start date Start date
  • Tags Tags
    Wave Wave equation
Click For Summary

Discussion Overview

The discussion centers on the interpretation of the variable u in the wave equation u(x,t). Participants explore its meaning in various contexts, including physical waves and mathematical representations, while considering different dimensions and visualizations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants suggest that u(x,t) represents the displacement of a vibrating string at location x and time t.
  • Others propose a visualization of the wave equation as a 3D surface where x is the horizontal axis, t is the vertical axis, and u represents the height, although this visualization is contested.
  • A participant argues that the wave travels in the x direction only, suggesting that the equation should be adjusted to reflect this directional movement.
  • Another viewpoint posits that u represents the complex amplitude of the wave, indicating the wave field distribution in space.
  • There are discussions about the need for a Laplace equation to account for multidimensional wave propagation.

Areas of Agreement / Disagreement

Participants express differing interpretations of what u represents in the wave equation, with no consensus reached on its definitive meaning. Multiple competing views remain regarding its physical and mathematical implications.

Contextual Notes

Some claims depend on specific definitions of wave properties and may not account for all dimensions of wave behavior. There are unresolved mathematical representations and assumptions about the context in which u is applied.

coverband
Messages
170
Reaction score
1
In the wave equation u(x,t), what does u represent?
 
Physics news on Phys.org
Can you write out the equation?
 
If you are representing a vibrating string, then u(x,t) represents the displacement of the string at location x at time t.
 
yungman: u''(t)=u''(x)c^2

lckurtz: so visually its like a plane/surface in 3D (x,y,z) where x = x, y = t and z = u ...?
 
coverband said:
yungman: u''(t)=u''(x)c^2

lckurtz: so visually its like a plane/surface in 3D (x,y,z) where x = x, y = t and z = u ...?

I would visualize x to the right, u vertically, and motion in time. You can see a gif on my home page: http://math.asu.edu/~kurtz" .
 
Last edited by a moderator:
Anyone else so visually its like a plane/surface in 3D (x,y,z) where x = x, y = t and z = u ...? !
 
coverband said:
yungman: u''(t)=u''(x)c^2

lckurtz: so visually its like a plane/surface in 3D (x,y,z) where x = x, y = t and z = u ...?

I don't think so, From your equation, it gives the wave is traveling in x direction only. I assume t is time, x is the distance. I think you switch x and t also. It should be u"(x)-(c^2)u"(t)=0.

If you want direction in x y and z, you have to use Laplace equation to have PD of all three direction:
[tex]\nabla[/tex] [tex]^{2}[/tex]u - C[tex]^{2}[/tex][tex]\frac{d^{2}u}{dt^{2}}[/tex] = 0

I can't get the equation look right, I hope you get what I mean!


So I look at your equation as the function u(x,t) ( of x direction and time t ) travel in x direction with the magnitude of the function u(x,t). "c" is the velocity.

This is a one dimension spatial problem of x.
 
Last edited:
coverband said:
Anyone else so visually its like a plane/surface in 3D (x,y,z) where x = x, y = t and z = u ...? !
Your original equation did not have a "y" variable so you can think of z as either the height of a string in one-dimension or as the height of a "cylindrical" wave, extending infinitely in the y direction. In any case, your original "u" is the height of the wave if you are thinking of a physical wave, the intensity if you are thinking of a light wave, etc.
 
coverband said:
In the wave equation u(x,t), what does u represent?

Actually, u is complex amplitude of the wave, which denotes the amplitude of given spatial point and time. So u represents the wave field distribution in space.
 

Similar threads

Undergrad General solution of 1D vs 3D wave equations
  • · Replies 7 ·
Replies
7
Views
4K
Undergrad What defines a wave according to the wave equation?
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad FEM Method for the Wave Equation
  • · Replies 3 ·
Replies
3
Views
3K
MHB Solving the Wave Equation with Initial Data: Ex. 5 Solution and Derivation
  • · Replies 11 ·
Replies
11
Views
3K
Undergrad D'Alembert solution for the wave equation: question about the speeds
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Green's function for the wave equation
  • · Replies 4 ·
Replies
4
Views
3K
MHB Solving the Wave Equation: Finding $u(x,t)$
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Finding the time for the first shock for a quasilinear first order PDE
  • · Replies 4 ·
Replies
4
Views
4K
MHB Solving the Wave Equation with Nonzero Initial Velocity
  • · Replies 2 ·
Replies
2
Views
2K
High School Difference between a continuously differentiable function and a wave
  • · Replies 11 ·
Replies
11
Views
2K