You are playing a note that has a fundamental frequency of 466.16 Hz on a guitar string of length 59.9 cm. At the same time, your friend plays a fundamental note on an open organ pipe, and 4.0 beats per seconds are heard. The mass per unit length of the string is 1.85 g/m. Assume the velocity of sound is 343 m/s.

a) What are the possible frequencies of the open organ pipe?

b) When the guitar string is tightened, the beat frequency decreases. Find the original tension in the string.

c) What is the length of the organ pipe?

To answer these questions, we'll need to use the formulas and concepts from physics related to waves, including the wave equation, the formula for frequency of a vibrating string, and the formula for the fundamental frequency of an open organ pipe.

a) The formula for the frequency of a vibrating string is:

f = (1/2L) * sqrt(Tension / linear mass density)

Where:
- f is the frequency
- L is the length of the string
- Tension is the tension in the string
- linear mass density is the mass per unit length of the string

Given that the frequency of the guitar string is 466.16 Hz and its length is 59.9 cm (0.599 m), and the linear mass density is 1.85 g/m (0.00185 kg/m), we can rearrange the formula to solve for the tension:

Tension = (4L^2 * f^2 * linear mass density)

Substituting the given values:

Tension = (4 * (0.599 m)^2 * (466.16 Hz)^2 * 0.00185 kg/m)

Now we can calculate the tension using this equation.

b) To find the original tension in the string when the beat frequency decreases, we can use the formula for beat frequency:

Beat frequency = |f1 - f2|

Where:
- Beat frequency is the difference in frequency between the two source tones
- f1 and f2 are the frequencies of the two source tones

Given that the beat frequency is 4.0 beats per second and one of the frequencies is 466.16 Hz, we can rearrange the formula to solve for the other frequency:

|f2 - 466.16 Hz| = 4.0 beats per second

Now we can solve for f2 using this equation.

c) The formula for the fundamental frequency of an open organ pipe is:

f = (v / 2L)

Where:
- f is the frequency
- v is the velocity of sound
- L is the length of the pipe

Given that the velocity of sound is 343 m/s and the beat frequency is 4.0 beats per second, we can rearrange the formula to solve for the length of the organ pipe:

L = (v / (2f))

Now we can calculate the length of the organ pipe using this equation.