- #1
fog37
- 1,568
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Hello Forum,
A traveling wave is something that travels and transfers energy from one location to another. There are many waves and many wave equations. There are also standing wave that are waves trapped within some cavity of boundaries (also called standing waves). A standing wave can be made by superposing two traveling waves moving in opposite directions. Two standing waves can form a traveling wave, etc.
That said, for one-dimensional traveling waves, introductory books show that the mathematical function describing a traveling wave must have an argument like ##(x-vt)## where the spatial and the time variables are combined into one argument. For example, ##sin(x- \omega t)## is a plane wave traveling along the x-direction. If the two variables , spatial ##x## and time ##t##, are separated, the wave is not traveling. Is that generally true?
For example, a function ##f(x,t)## that is separable,i.e. ##f(x,t) = g(x)*p(t)## cannot be a traveling wave but can be a standing wave. However, I then think of something like ##e^{i(x-\omega t)}= e^{ix}e^{-i\omega t}##, which also a traveling wave and is also separable...
What general observations can we make about the function that is supposed to represent a traveling wave? What about the spatial and time variables x and t?
Thanks!
A traveling wave is something that travels and transfers energy from one location to another. There are many waves and many wave equations. There are also standing wave that are waves trapped within some cavity of boundaries (also called standing waves). A standing wave can be made by superposing two traveling waves moving in opposite directions. Two standing waves can form a traveling wave, etc.
That said, for one-dimensional traveling waves, introductory books show that the mathematical function describing a traveling wave must have an argument like ##(x-vt)## where the spatial and the time variables are combined into one argument. For example, ##sin(x- \omega t)## is a plane wave traveling along the x-direction. If the two variables , spatial ##x## and time ##t##, are separated, the wave is not traveling. Is that generally true?
For example, a function ##f(x,t)## that is separable,i.e. ##f(x,t) = g(x)*p(t)## cannot be a traveling wave but can be a standing wave. However, I then think of something like ##e^{i(x-\omega t)}= e^{ix}e^{-i\omega t}##, which also a traveling wave and is also separable...
What general observations can we make about the function that is supposed to represent a traveling wave? What about the spatial and time variables x and t?
Thanks!