(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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.

2. Relevant 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

3. 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!

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Calculating functions for wave problems

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