Phase constant of waves

In summary, The conversation is about determining the correct phase constant for a given equation and graph. The equation is y(x,t)=0.04sin(10πx-π/5t+ø) and the graph shows the particle at x=0, t=0, and y=0. The question is how to choose the correct value for ø. The attempt at a solution involves taking the partial derivative to find the transverse velocity and plugging in t=1 to determine the correct value. The correct answer is +π, but both +π and -π would result in the same sign for velocity. The key is to remember that cos(a+π)=cos(a-π)=-cos(a), so either value will
  • #1
grassstrip1
11
0

Homework Statement


I've attached the question where the graph can be found.
Essentially I have no problem determining A=0.04M K= 10π rad/m λ=0.2M ω=π/5 rad/s
I'm having trouble choosing what ø should be.


Homework Equations


y(x,t)= 0.04sin(10πx - π/5t +ø)


The Attempt at a Solution


Since the graph is for the particle at x=0 and t=0 and y=0
0=0.04sin(ø) solving for ø gives π, -π and 0

To try and determine the right phase constant I took the partial derivative to find the transverse velocity.
v(x,t)= (0.04)(-π/5)cos(10πx - π/5t +ø)
Since the particle has a positive velocity at t=1 , plugging in t=1 should give a positive velocity
Therefore ø can't = 0 since using 0 as a phase constant gives a negative velocity for t=1 and x=0
Problem is both plus and minus pi give the right velocity and I'm not sure how to pick the correct one. The correct answer is +π
I'd like to know in general as well how to pick the correct value. Thanks!
http://imageshack.com/a/img547/3680/2tjw.png [Broken]
 
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  • #2
You did it just right. Many (me too) intuitively fill in ##\phi##=0. But if you draw a sin(x), and pull it to the right, you see x=0 is moving down.
 
  • #3
Okay thank you! I'm just wondering how you can tell the answer is positive pi and not negative since both would give the same sign for velocity
 
  • #4
It does not make any difference if you pick ∏ or -∏.
cos(a+∏)=cos(a-∏)=-cos(a)
 
  • #5



The phase constant of a wave is a measure of the initial displacement of the wave at a specific point in space and time. It represents the starting point of the wave and is usually denoted by the symbol ø. In this case, ø is the angle at which the wave starts, and it is measured in radians.

To determine the correct value for ø, we need to consider the given information and equations. We know that the wave has an amplitude of 0.04m, a wave number of 10π rad/m, a wavelength of 0.2m, and an angular frequency of π/5 rad/s. We also know that the particle at x=0 and t=0 has a displacement of 0, which means that the initial phase of the wave at that point is also 0.

Using the given equation y(x,t)= 0.04sin(10πx - π/5t +ø), we can see that when t=0 and x=0, the value of ø must be 0 in order for the displacement to be 0. This eliminates the possibility of ø being -π, as that would result in a non-zero displacement at x=0 and t=0.

Next, we can consider the transverse velocity equation v(x,t)= (0.04)(-π/5)cos(10πx - π/5t +ø). We know that at t=1 and x=0, the particle has a positive velocity. Plugging in these values, we get v(0,1)= (0.04)(-π/5)cos(0+ø)= (0.04)(-π/5)cos(ø). Since we want the velocity to be positive, we need cos(ø) to be positive. This eliminates the possibility of ø being -π as well, as that would result in a negative value for cos(ø).

Therefore, the only valid value for ø is +π, as that would result in a positive value for cos(ø) and satisfy both the initial displacement and transverse velocity conditions. This means that the correct equation for the wave is y(x,t)= 0.04sin(10πx - π/5t +π).

In general, to determine the correct value for the phase constant of a wave, you need to consider the given information and equations and ensure that it satisfies the initial conditions and any other relevant conditions
 

1. What is the definition of phase constant?

The phase constant of a wave is a measure of the displacement of the wave at a specific point in time. It is represented by the Greek letter phi (ϕ) and is measured in radians.

2. How is the phase constant related to the wave function?

The phase constant is an important component of the wave function, which describes the overall behavior of a wave. It determines the starting phase of the wave and how it changes over time.

3. How is the phase constant calculated?

The phase constant can be calculated using the formula ϕ = 2π/λ * x, where λ is the wavelength of the wave and x is the displacement at a specific point in time.

4. What is the significance of the phase constant in wave interference?

The phase constant plays a crucial role in determining the interference pattern of waves. Waves with the same phase constant will exhibit constructive interference, while those with opposite phase constants will exhibit destructive interference.

5. Can the phase constant of a wave change over time?

Yes, the phase constant can change over time as the wave propagates through a medium. This change in phase constant can be due to factors such as changes in frequency or direction of the wave.

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