Waves Find the phase constant from the graph.

In summary, the phase constant for the equation x(t) = Acos(wt + Φ) or x(t) = Asin(wt + Φ) can be determined by looking at the derivative of x(t) at t=0 and the sign of sinΦ. Depending on the function used, the phase constant can be either -π/3 or π/6.
  • #1
firezap
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0

Homework Statement


http://i.imgur.com/TZcJkjH.jpg
Find the phase constant

Homework Equations


x(t) = Acos(wt + Φ)
x(t) = Asin(wt + Φ)
w = 2pi/T

The Attempt at a Solution


I see A = 20cm. T = 4s. f = 1/4 = 0.25Hz. w = 2pi/4 = 1.57Hz.
10 = 20cosΦ
0.5 = cosΦ
Φ = π/3, 5.24?
 
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  • #2
firezap said:
x(t) = Acos(wt + Φ)
x(t) = Asin(wt + Φ)
The phase constant depends on if you use the sine or cosine function (so the phase constant in your relative equations is not the same Φ).

firezap said:
Φ = π/3, 5.24?
If you use cosine, then yes, it would be π/3, but should it be positive or negative?
 
  • #3
sinΦ= 0.5. Φ=pi/6, 5pi/6
How many answers are there?
I'm guessing -π/3 since you asked.
 
  • #4
firezap said:
10 = 20cosΦ
0.5 = cosΦ
Φ = π/3, 5.24?

Use the expression with pi when you give the phase.

You can decide which phase angle to use from the derivative of x(t). If x(t)=A cos (ωt+Φ) what is the sign of the derivative at t=0? What is the sigh of sin(Φ)? Which quadrant does it mean for Φ?
 
  • #5
ehild said:
Use the expression with pi when you give the phase.

You can decide which phase angle to use from the derivative of x(t). If x(t)=A cos (ωt+Φ) what is the sign of the derivative at t=0? What is the sigh of sin(Φ)? Which quadrant does it mean for Φ?
v(t)=-Aωsin(ωt+Φ).
sign of derivative is negative. sign of sinΦ is positive. That means in quadrant 1 and 2.
 
  • #6
The sign of the derivative of x(t) is positive at t=0 in the plot! V(t) is positive, so what is the sign of of sinΦ??
 
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  • #7
ehild said:
The sign of the derivative of x(t) is positive at t=0 in the plot! V(t) is positive, so what is the sign of of sinΦ??
How do you know it's positive? It's clearly negative because of the negative sign in front of amplitude. sinΦ must be negative.
 
Last edited:
  • #8
firezap said:
sinΦ= 0.5. Φ=pi/6, 5pi/6
How many answers are there?
Infinite. But there is always a Φ with |Φ|≤π/2 so I would suggest using that one.

firezap said:
How do you know it's positive? It's clearly negative because of the negative sign in front of amplitude. sinΦ must be negative.
Ehild did not say that sinΦ was positive, she said that the derivative of x(t) is positive at x(0) (because the function is sloping upwards).
 
  • #9
ehild said:
The sign of the derivative of x(t) is positive at t=0 in the plot! V(t) is positive, so what is the sign of of sinΦ??
This is much easier to explain than the way I perviously thought about it!

I suppose sin(ωt+Φ) would be even easier, because you don't even need check the derivative (just check if x(0) is positive or negative)

Thanks for this perspective, I never thought about it in this way.
 
  • #10
firezap said:
How do you know it's positive? It's clearly negative because of the negative sign in front of amplitude. sinΦ must be negative.
Look at the graph. X(t) increases at t=0, so its derivative is positive.
 
  • #11
the answer is -π/3 for cos and π/6 for sin.
 

1. How do you find the phase constant from a wave graph?

The phase constant can be found by determining the horizontal shift of the wave from its original position on the x-axis. This shift is represented by the value of the phase constant, typically denoted as φ (phi).

2. What is the significance of the phase constant in a wave?

The phase constant is important because it indicates the starting point of the wave and helps determine its position in time. It also affects the amplitude and frequency of the wave.

3. Can the phase constant be negative?

Yes, the phase constant can be negative. It depends on the direction of the horizontal shift of the wave from its original position. A negative phase constant indicates a shift to the left, while a positive phase constant indicates a shift to the right.

4. How does the phase constant affect the wavelength of a wave?

The phase constant does not directly affect the wavelength of a wave. However, it can affect the position of the wave within a given period of time, which in turn can affect the wavelength.

5. What is the difference between phase constant and phase shift?

The phase constant and phase shift both refer to the horizontal shift of a wave from its original position. However, the phase constant refers to this shift in terms of a specific angle (in radians), while the phase shift refers to the shift in terms of a fraction of the wavelength.

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