Calculating functions for wave problems

[into space]

Homework Statement

A wave travels along a string in the positive x-direction at 30 m/s. The frequency of the wave is 50 Hz. At x = 0 and t = 0, the wave velocity is 2.5 m/s and the vertical displacement is y = 4 mm. Write the function y(x,t) for the wave.

Homework Equations

velocity = (wavelength)(frequency) = (angular velocity)/(wave number)
angular velocity = 2*pi*frequency
What the function should look like:
y(x,t) = Asin(kx - wt + θ)
A = amplitude
k = wave number
w = angular velocity
t = time (s)
θ = initial phase shift

The Attempt at a Solution

Figuring out the wave number (k) as well as the wavelength and angular velocity was the easy part:
w = 2*pi*(50 Hz) = 100*pi
wavelength = velocity/frequency = (30 m/s)/(50 Hz) = .6 m
wave number (k) = w/v = (100*pi)/(30 m/s) = 10.47 m^-1

So far the function looks like:
y(x,t) = Asin(10.47(x) - 314.16(t) + θ)

The only variables left to find are the amplitude (A) and initial phase shift (θ). I tried plugging in the initial conditions the problem gave me, but I end up with:
y(x,t) = Asin(θ) = .004
differentiating the function y(x,t) = Asin(10.47(x) - 314.16(t) + θ) with respect to time gives the velocity equation, which is:
v(x,t) = -A(314.16)cos(10.47(x) - 314.16(t) + θ)

and plugging in the initial conditions for velocity gives:
-A(314.16)cos(θ) = 2.5

I don't know where to go from here to find amplitude and the initial phase shift. I have tried setting the velocity equation equal to zero which would give me a critical point (amplitude), but that didn't help much. I've also tried using a substitution method between these two equations:
y(x,t) = Asin(θ) = .004
-A(314.16)cos(θ) = 2.5

but that didn't get me anywhere either. Any help would be greatly appreciated!

 

[Mike Pemulis]

Divide one of the equations by the other. So we have,

A sinθ = .004
-314.16Acosθ = 2.5

Gives,

-(1/314.16)tanθ = .004/2.5

A is gone, so this can be solved for θ immediately, and then of course you can plug this back into get A.

 

[into space]

So the initial phase shift is -.466 rad and the amplitude is .0089 m. In order to get the correct initial phase shift from the book I have to add -.466 and 3.14 (pi) which yields 2.67 rads (this is the answer given in the book). Why do I have to add my answer in order to get the desired answer from the book?

The amplitude is correct as is, though. At any rate thanks for the help!

 

[Mike Pemulis]

The phase shift problem is due to the multivalued-ness of the arctan "function" -- there are many angles that give the same tangent, even within the same 2π "period." In particular, for every solution in the first quadrant there is one in the third, and for every solution in the second quadrant there is one in the fourth (which you can verify by thinking about the definition of the tangent function, or looking at a graph of tan(x)). This means that even if you know tanθ, you need additional information to find θ itself.

In this problem, we need to decide between θ = -.466 and 2.67, which, as you can check with a calculator, have the same tangent. So is the choice arbitrary? It would be if we didn't have more information, but in this problem we do. θ = 2.67 is correct because we can plug it back into the original two equations involving sine and cosine and get a true statement. But plugging in θ = -.466 doesn't work, as you can check.

Sorry I didn't notice this before. It's actually a common problem when solving for angles; arcsine and arccosine do this too. In general, evaluating an inverse trig function will give you a couple of possibilities, and you need to pick the right one using additional information from the problem. Unfortunately, calculators will blithely give you a single number like nothing is wrong, so you need to be alert.

 

[asteriatic]

I would suggest using the initial conditions for both displacement and velocity to find the amplitude and initial phase shift. You can use the fact that the velocity is the derivative of the displacement function to solve for these variables. This will give you a system of equations that can be solved simultaneously to find the values of A and θ. Alternatively, you can also graph the equations and use the points given by the initial conditions to find the amplitude and initial phase shift visually. Additionally, make sure to check your units and conversions to ensure that your final answer is in the correct units.