What is the velocity dependence in the equation of a standing wave?

[andyrk]

Say we have 2 equations of progressive wave as y₁=Asin(kx+ωt) and y₂=Asin(kx-ωt)

Where ω=kv, k=Wave Number, v=Wave velocity

These equations combine according to the principle of superposition as:
y₁+y₂=[2Asin(kx)]cos(ωt).

Now we know that a standing wave is called so because all the points on the wave are not translating, they are just oscillating about their mean position with different amplitudes. But if we look in the wave equation, we see that there is a cos(ωt) factor in the equation of a standing wave and since ω=kv, we can say that a standing wave has a velocity dependence also. But how is this possible?

 

[BruceW]

Is this a homework problem, or a question that you had? There are 3 parameters (omega,k,v) and we have the equation omega=kv, so there are two free parameters. If you think about a standing wave, try to match the 2 free parameters to properties the standing wave has.

 

[andyrk]

It is a question I had. You mean v=0? k can't be zero..

 

[andyrk]

Anybody there?

 

[ehild]

The wave-fronts, peaks for example, where kx-ωt=const, travel with the phase velocity v=dx/td=ω/k. The standing wave is product of two sine function, one depending on x, the other depending on t. X and t are not related. Nothing travels.

ehild

 

[andyrk]

The wave-fronts, peaks for example, where kx-ωt=const, travel with the phase velocity v=dx/td=ω/k. The standing wave is product of two sine function, one depending on x, the other depending on t. X and t are not related. Nothing travels.

ehild

Hmm, but the second of the two sine functions, though it involves t, but it also involves v as ω=kv. Means there is a term of wave velocity in the equation of a standing wave. Doesn't that signify the standing wave moves? It does not, obviously, but then what does the v here stand for? What does the v here represent in the equation of a standing wave?

 

[ehild]

Hmm, but the second of the two sine functions, though it involves t, but it also involves v as ω=kv. Means there is a term of wave velocity in the equation of a standing wave. Doesn't that signify the standing wave moves? It does not, obviously, but then what does the v here stand for? What does the v here represent in the equation of a standing wave?

If a quantity is related to some velocity it does not mean that the quantity moves.

Speed is defined only for traveling waves: Think of the water waves, you see the peaks - "wave-fronts" moving along.
The standing wave does not travel. The nodes are always at the same positions.

ω and k are two parameters, and y, the displacement from equilibrium, depends on both. v is a constant, the speed of any wave of certain kind in the given medium determined by some physical properties of the matter. The wave is a motion, periodic both in space and time. The time period is T=2pi/ω, the space period is λ=2π/k. The wavelength of a wave is connected to its frequency through the speed of wave. The wavelength is the distance the wave travels in one period, λ=vT. k is defined as k=2π/λ, so you can write ω/k=v. v is just a proportionality factor between wavelength and period, angular frequency ω and k.


ehild

 

[andyrk]

If a quantity is related to some velocity it does not mean that the quantity moves.

Speed is defined only for traveling waves: Think of the water waves, you see the peaks - "wave-fronts" moving along.
The standing wave does not travel. The nodes are always at the same positions.

ω and k are two parameters, and y, the displacement from equilibrium, depends on both. v is a constant, the speed of any wave of certain kind in the given medium determined by some physical properties of the matter. The wave is a motion, periodic both in space and time. The time period is T=2pi/ω, the space period is λ=2π/k. The wavelength of a wave is connected to its frequency through the speed of wave. The wavelength is the distance the wave travels in one period, λ=vT. k is defined as k=2π/λ, so you can write ω/k=v. v is just a proportionality factor between wavelength and period, angular frequency ω and k.


ehild

I am sorry but I am not convinced fully. Could you explain your water wave peak example more? Maybe it would help me visualise. And you said that v s just a proportianality factor, but it was not like this, the equation of a wave originated from the basic function y=f(x+vt)! Please explain! Thanks though for your previous replies!

 

[ehild]

You throw a pebble into still water, it makes some disturbance, traveling outward, in form of alternating crests and troughs. https://www.e-education.psu.edu/astro801/files/astro801/image/Lesson%203/700px-2006-01-14_Surface_wa.jpg Why?The motion of the water is governed by laws of Physics. Its density is unchanged, gravity pulls it downward and so on. At the place where the pebble hits the water, it forces the water level downward. As the water volume does not change, somewhere it had to move upward. The hydrostatic pressure of the higher column of water forces the water flow away...So the surface of water moves up and down and that motion extends farther and farther.
When a boat passes , you see waves again, now in form of maxima and minima forming rather straight lines. http://upload.wikimedia.org/wikipedia/commons/4/4f/Wake.avon.gorge.2boats.arp.750pix.jpg
The motion of water (its height at a certain distance and time) can be described with an equation containing gravity, density and some other physical properties. The solution of such wave equations is the form y(x,t) f(x-vt) for a "plane wave" with wavefronts perpendicular to the x axis. On wavefront, we mean those places where y =constant. Of course, it happens where x-vt=constant, say 0. So the position x of the wavefront changes with time, x=vt, the wave travels, and its velocity is v.
But the function f(x-vt) is solution of an equation, containing some parameters which characterize the water. v is a parameter that is derived from these parameters. It depends on g and depth of the water - quite complicated.
For a wave traveling along a string, the velocity is determined by the tension T and the linear density μ.
In case of light waves, the speed of the wave is c = 300000km/s in vacuum, and v=c/n in a medium of refractive index n.

This video shows waves in a pond , first a traveling one, which, reflected from the opposite wall, becomes a standing wave. But it is still the sum of two ordinary waves traveling in opposite directions. If the amplitudes are not quite equal, there is a traveling component .

ehild

 

[andyrk]

Is the 'v' in a standing wave a useless quantity?

 

[ehild]

Is the 'v' in a standing wave a useless quantity?

No. You need the speed of the wave as it connects k and ω, or frequency and wavelength. As an example, all musical instruments are based on standing waves, and you need to choose the size according to the speed of sound. The sound travels in air at speed 340 m/s, so the wavelength of 500 Hz sound is λ=v/f= 0.68 m. The air column in a open pipe resonates if its length is an integer times of half wavelength. You get that sound with a 34 cm pipe or with a 68 cm pipe, but the fundamental mode of the longer pipe has 1.36 m wavelength, which corresponds to a 250 Hz sound.
In case of a guitar string, the tension in the string and its linear density determines the speed in it, and the string vibrates at frequency, which wavelength is twice the length of the string.

http://hyperphysics.phy-astr.gsu.edu/hbase/waves/string.htmlehild

 

[BruceW]

Hmm, but the second of the two sine functions, though it involves t, but it also involves v as ω=kv. Means there is a term of wave velocity in the equation of a standing wave. Doesn't that signify the standing wave moves? It does not, obviously, but then what does the v here stand for? What does the v here represent in the equation of a standing wave?

There's two important 'features' of the standing wave. 1) wavelength - the distance between peaks. 2) period - the time it takes for the wave to go through one full oscillation (at constant x).

You have 3 parameters (omega,v,k) and the equation omega=kv, so that means you have a choice for which two of these parameters you want to use to describe the two 'features'. Then the 3rd parameter is given by the equation.

The natural choice is to use omega to describe 2PI/period and use k to describe 2PI/wavelength. And then v is given by the formula. But you don't have to do it this way. You could use v to describe wavelength/period and omega as 2PI/period, then k is given by the formula. So it is your choice really.

In other words, the physical meaning of 'velocity' for a standing wave is wavelength/period.

 

[andyrk]

Thanks all for the help!

 

[BruceW]

no worries!