Space and Time Invariance (Classical Wave Equation)

AI Thread Summary
The discussion centers on the invariance of the classical wave equation, specifically comparing the second-order wave equation, which is space and time invariant, to the first-order equation, which is not. The second-order equation, ∂²y/∂t² = v² ∂²y/∂x², maintains its form under transformations, indicating invariance. In contrast, the first-order equation, ∂y/∂t = -v ∂y/∂x, does not preserve its structure, demonstrating a lack of invariance. Participants suggest testing a simple wave function, y(x,t) = sin(x+vt), in both equations to illustrate the differences in behavior. Understanding these distinctions is crucial for grasping the fundamental principles of wave mechanics.
mess1n
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Hey, I've come across a part in my notes which I can't figure out. Essentially it says:

\frac{\partial^{2}y}{\partial t^{2}} = v^{2} . \frac{\partial^{2}y}{\partial x^{2}} is space and time invariant.

Whereas:

\frac{\partial y}{\partial t} = -v . \frac{\partial y}{\partial x} is not.

Why is this the case?

Cheers,
Andrew
 
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Try putting a simple wave like y(x,t) = sin(x+vt) into both equations and see what happens.

Guessing at equations is a standard way for physicists to explore a problem.
 

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