Let's delve into the problem step by step. Here's how we tackle it:

(a) Length of the rope:

From the equation $y = (0.127 \, \text{m}) \sin(\pi x / 6.00) \sin(13.0 \pi t)$, we observe that $\sin(\pi x / 6.00)$ corresponds to the standing wave function. The wavelength $\lambda$ is twice the length of the rope for the second harmonic. Since $\lambda = 6.00 \, \text{m}$, the rope's length is: $L = \frac{\lambda}{2} = \frac{6.00}{2} = 3.00 \, \text{m}.$

(b) Speed of the waves:

The angular frequency $\omega = 13.0 \pi$ and the wave number $k = \frac{\pi}{6.00}$. Wave speed $v$ is given by: $v = \frac{\omega}{k} = \frac{13.0 \pi}{\pi / 6.00} = 13.0 \cdot 6.00 = 78.0 \, \text{m/s}.$

(c) Mass of the rope:

The tension $T = 198 \, \text{N}$, and wave speed $v$ relates to the mass per unit length $\mu$ as: $v = \sqrt{\frac{T}{\mu}} \implies \mu = \frac{T}{v^2}.$ $\mu = \frac{198}{(78.0)^2} = \frac{198}{6084} = 0.03254 \, \text{kg/m}.$ The total mass $m$ of the rope is: $m = \mu \cdot L = 0.03254 \cdot 3.00 = 0.0976 \, \text{kg}.$

(d) Period of oscillation for the third harmonic:

For the third harmonic, the frequency $f_3 = 3 \cdot f_1$, where $f_1 = \frac{v}{2L}$: $f_1 = \frac{78.0}{2 \cdot 3.00} = 13.0 \, \text{Hz}.$ $f_3 = 3 \cdot 13.0 = 39.0 \, \text{Hz}.$ The period $T$ is the reciprocal of frequency: $T = \frac{1}{f_3} = \frac{1}{39.0} = 0.02564 \, \text{s}.$

Final Answers:

(a) Length of rope: $3.00 \, \text{m}$
(b) Speed of waves: $78.0 \, \text{m/s}$
(c) Mass of rope: $0.0976 \, \text{kg}$
(d) Period of oscillation: $0.02564 \, \text{s}$

Do you want further clarifications on any of these calculations?
Here are some related questions for deeper understanding:

1. How is the wave equation derived for a standing wave?
2. Why does the second harmonic imply half the wavelength fits the rope's length?
3. What are the differences between harmonics in terms of wavelengths and frequencies?
4. How does tension affect wave speed?
5. What practical applications use these harmonic wave principles?

Tip: Always verify units in equations to ensure physical consistency!