What Is Phase Constant in Wave Functions?

[PatreMagnus]

what is phase constant and how is possible to go about figuring it out in an unscaled graph that has no values associated with it.

 

[ehild]

Welcome to PF!

Be a bit more specific: what is the function ? Is it sine or cosine?

If it is f(x)=sin(wt+θ), the function would be zero at x=0 when θ = 0. If the phase constant is added, the function becomes zero sooner, and the plot shifts to negative direction. You know that a total wave corresponds to 2pi phase. Measure the shift of the zero with respect to the origin and divide it by the length of a period: it is the negative of the phase constant. You see from the plot that the shift is about 1/6 of the period in the negative direction so θ = 2pi/6.

ehild

 

[PatreMagnus]

I'm sorry about not giving enough details, as I have found out from the out set this forum has strict rules.
The image attached is from a workbook that I bought to provide a clearer understanding of the concepts I read. it is not an assignment.

I'm totally confused and frustrated with questions 19 and 20.

for question 19: it says that there're three waves traveling to the right, the first two shown at t=0 and the third at t=T/2
I think the answer is pi/6
reasoning being that for a sinusoidal wave, a description can be given by this formula.
D(x,t)=Asin(kx-wt+phi)
so, when x=0 and t=0
phi=(Sin(D(0,0)/A)^-1
From my understand a phase constant indicates how much of the wave to left of the origin is there, where by the answer is given in radians.
Therefore, for the first graph I say that the graph started when it was 30 deg into the graph.

As for the second graph I'm tempted to say the same thing, but the fact that it is reflected and doesn't look like a normal sine/cosine graph confuses the matter more.

For the third graph, I have my thoughts about it, but doesn't lead me to an answer.

For question 20: It says that A sinusoidal wave with wavelength is traveling along the x-axis. At t=0s the wave's phase at x=2 is pi/2

I'm thinking that the graph would look like a cosine graph and negatively reflected, and I'm also suspecting that there could be various answers.

 

[PatreMagnus]

@ehild

Does it matter which function I use when considering sinusoidal waves?

 

[ehild]

@ehild

Does it matter which function I use when considering sinusoidal waves?

sine and cosine are shifted by pi/2 with respect to each other. If you want the phase shift between two waves, use the same function sine or cosine for both.

ehild

Ps. I will try to digest your diagram. First turn it upside down...

 

[PatreMagnus]

@ehild

Done!
I didn't realize.

 

[ehild]

I'm sorry about not giving enough details, as I have found out from the out set this forum has strict rules.
The image attached is from a workbook that I bought to provide a clearer understanding of the concepts I read. it is not an assignment.

I'm totally confused and frustrated with questions 19 and 20.

for question 19: it says that there're three waves traveling to the right, the first two shown at t=0 and the third at t=T/2
I think the answer is pi/6
reasoning being that for a sinusoidal wave, a description can be given by this formula.
D(x,t)=Asin(kx-wt+phi)
so, when x=0 and t=0
phi=(Sin(D(0,0)/A)^-1
From my understand a phase constant indicates how much of the wave to left of the origin is there, where by the answer is given in radians.
Therefore, for the first graph I say that the graph started when it was 30 deg into the graph.

You need to give the phases in radians. The first answer is right, D is about half of the amplitude and the curve is shifted to the left by about 1/3 units. The period is about 4 unit, 1/3 unit corresponds to 2pi/12=pi/6=Φ₀.

As for the second graph I'm tempted to say the same thing, but the fact that it is reflected and doesn't look like a normal sine/cosine graph confuses the matter more.

At the equivalent points where the phases are equal or differ by integer multiple of 2pi not only the displacements are equal but the slopes to. At t=0, the second graph have negative slope. That corresponds a phase constant in the second quadrant.
D/A=sinΦ₀. What is the other angle with the same sine?

For the third graph, I have my thoughts about it, but doesn't lead me to an answer.

For question 20: It says that A sinusoidal wave with wavelength 2m is traveling along the x-axis. At t=0s the wave's phase at x=2 is pi/2

I'm thinking that the graph would look like a cosine graph and negatively reflected, and I'm also suspecting that there could be various answers.

Start with the form of the wave D(x,t)=Asin(kx-wt+phi). The phase is kx-wt+phi. Substitute the data for the wavelength, time, x. You have only one unknown, phi.


ehild

 

[PatreMagnus]

@ehild

so the second graph is pi-pi/6=5pi/6.
But that answer seems counter-intuitive for me.
so, it is not necessarily just the protruding bit that you take into account, you consider
where that bit is with respect to the displacement axis. Please explain it more
and with the trig reasoning.

The definition that I have of a phase is that it's about the oscillation that particle has done
with respect to an arbitrary point. Is that correct?

What about the third graph of question 19, that is what got my head going steamy.

 

[ehild]

I do not quite understand you.

A wave is periodic function both in time and space. At a certain point, it is periodic function of time. If the time is fixed, it is a periodic function of the spatial coordinate(s).A wave along a string can describe the motion of the tiny pieces of the string, which perform SHM, but taken a photo of the string, you get also a sinusoidal shape along the length of the string. A wave not necessarily means oscillating particles. It can describe the state of something extended. A sound wave, for example, describes how the pressure changes with respect to the normal pressure. The electromagnetic wave describes the change of electric field.

You have a periodic function and its argument is the phase. In one-dimensional waves the function is F(kx-wt+θ₀). F is periodic function of its argument, θ=kx-wt+θ₀. The argument is a linear function both x and t. θ₀ is the phase constant.

You need to know that w=2pi/T where T is the time period and k=2p/λ, where λ is the wavelength, the spatial period.

All three plots correspond to a wave with amplitude 2 and λ=4 units. So they can be written in the form D(x,t) = 2sin((2pi/4)x-wt+θ₀). The first two plots refer to t=0, so the functions are D(x,0) = 2((pi/2)x+θ₀). In the third case, t=T/2, so the phase is (pi/2)x-( 2pi/T)T/2+θ₀=(pi/2)x-pi+θ₀. The function is D(x,T/2)=2sin((pi/2)x-pi+θ₀). The plot is identical with the second one, it has the same apparent "phase constant" as the second one, but you have to add pi to it to get the phase constant of the original wave. ehild