Does any function of the form y=f(x-vt) represent a wave?

[randomgamernerd]

I just want to know does any random equation of the form y=f (x-vt) reppresent a wave?? If yes why?? And is this true for all cases or is there any condition?

 

[andrewkirk]

No it doesn't. Consider the case where v=1 and f is the identity function. Then the equation is simply y = x - t, which defines a surface that is simply a tilted plane, in the 3D space with axes on x, t and y. For any location x, the graph of y vs t is simply a straight line.

To be a wave there must be periodicity, meaning that, for any point (x',y',t') on the surface, there must be many other values of t such that (x',y',t) = (x',y',t'). The periodicity is typically achieved by the function f being a trig function, usually sin or cos.

 

[Chestermiller]

No it doesn't. Consider the case where v=1 and f is the identity function. Then the equation is simply y = x - t, which defines a surface that is simply a tilted plane, in the 3D space with axes on x, t and y. For any location x, the graph of y vs t is simply a straight line.

To be a wave there must be periodicity, meaning that, for any point (x',y',t') on the surface, there must be many other values of t such that (x',y',t) = (x',y',t'). The periodicity is typically achieved by the function f being a trig function, usually sin or cos.

Doesn't f(x-vt) satisfy the wave equation, where v is the wave velocity?

 

[nasu]

To be a wave there must be periodicity, meaning that, for any point (x',y',t') on the surface, there must be many other values of t such that (x',y',t) = (x',y',t'). The periodicity is typically achieved by the function f being a trig function, usually sin or cos.

Do you have a reference for the periodicity requirement?
A short pulse of sound is not a wave?

 

[I like Serena]

According to wave on wiki, there are a few more requirements than just satisfying the wave equation.

 

[vanhees71]

I'd define any solution of the wave equation to be a wave. How else should you define it?

I'm not sure I'd like the Wikipedia definition. Is a standing wave all of a sudden no wave anymore, because it doesn't transfer energy from one place to another?

 

[jbriggs444]

Do fans at a football stadium do "the wave" or "the sequenced stand-sit"?

 

[I like Serena]

I'd define any solution of the wave equation to be a wave. How else should you define it?

I'm not sure I'd like the Wikipedia definition. Is a standing wave all of a sudden no wave anymore, because it doesn't transfer energy from one place to another?

I'd like to define it the way it is generally used and understood.
And I believe that something that is 'still' is not generally understood to be a wave, even though it satisfies the wave equation.
As for a standing wave, it is definitely a wave, and it does transfer energy around.
For instance, the molecules in a string oscillate, converting elastic energy into kinetic energy, not to mention that they transfer energy to the air where another wave is generated.

 

[nasu]

No, it does not transfer energy from one point of the string to another point, as a traveling wave does.
Dissipation of energy is a different thing. And conversion from KE to PE does not mean transferring energy from one place to another.

That Wiki ling seem to confuse or overlap the concepts of wave and vibration (or oscillation). The vibration of a system shows both conversion between KE and PE as well as dissipation. But a vibration is not a wave.

In the end, there is no right or wrong in a definition. But it should be some consistency, both internal and with the general understanding. Otherwise communication becomes difficult.

 

[vanhees71]

Well, anything that solves the wave equation is a wave. That seems to be the most simple and most common definition of what a wave is. After all it's only a word, and you have to clearly define what you are talking about for any case anyway.

 

[sanpokhrel]

No, it does not transfer energy from one point of the string to another point, as a traveling wave does.
Dissipation of energy is a different thing. And conversion from KE to PE does not mean transferring energy from one place to another.

That Wiki ling seem to confuse or overlap the concepts of wave and vibration (or oscillation). The vibration of a system shows both conversion between KE and PE as well as dissipation. But a vibration is not a wave.

In the end, there is no right or wrong in a definition. But it should be some consistency, both internal and with the general understanding. Otherwise communication becomes difficult.

Is it the case that the energy remain in the same point being confined between nodes or energy is transferred but the superposition makes the vertical displacement 0 at node.

 

[vanhees71]

Yes! Exactly: In a standing wave (e.g., a string on a violine or guitar) each point of the continuum just oscillates around its "equilibrium point", and in the nodes nothing moves at all. As with any oscillator, of course the potential and kinetic energy is exchanged keeping the total energy conserved (provided there's no dissipation, but that goes beyond the pure wave equation anyway).

 

[Charles Link]

It might be useful to write out the form of a standing wave mathematically: In the simplest standing wave we have two sinusoidal waves traveling in opposite directions: $y(x,t)=A [ \cos(\frac{2 \pi}{\lambda}(x-vt))+\cos(\frac{2 \pi}{\lambda}(x+vt))]=2A \cos(\frac{2 \pi}{\lambda} x) \cos(\frac{2 \pi}{\lambda} vt)$. Thereby a standing wave consists of two waves traveling in opposite directions, and is not the simplest form of a wave. $\\$ A standing wave is a wave, by one definition, and by another definition it would be two waves.

 

[nasu]

What are the two definitions?

 

[Charles Link]

What are the two definitions?

Maybe it's not completely obvious...
1) If it satisfies the wave equation, it is a wave. Also, just by its name alone (standing wave), it is necessarily a wave. $\\$ 2) If I have one solution, the left traveling portion of the standing wave, it is a wave, by definition (1). Likewise, the right-going portion of the standing wave is also a wave, by defintion (1). If I take one wave, and take another wave, by what I am calling definition (2), what I have is not a wave, but rather two waves.

 

[nasu]

So do you call definition 2? It is not clear from your post.