What is the relationship between tension and number of loops in a standing wave?

[rvhockey]

To demonstrate standing waves, one end of a string is attached to a tuning fork with a frequency of 120 Hz. The other end of the string passes over a pulley and is connected to a suspended mass M. The value of M is such that the standing wave pattern has four "loops". The length of the string from the tuning fork to the point where the string touches the top of the pulley is 1.20 m. The linear density of the string is 1.0 x 10^-4 kg/m, and remains constant throughout the experiment.

a. Determine the wavelength of the standing wave.
b. Determine the speed of the transverse waves along the string.
c. The speed of the waves along the string increases with increasing tension in the string. Indicate whether the value of M should be increased or decreased in order to double the number of loops in the standing wave patterns. Justify your answer.
d. If a point on the string at an amplitude moves a total vertical distance of 4 cm during one complete cycle, what is the amplitude of the standing wave?




lambda_n = 2L/n
f_n = v/(n*2L)
v = f*lamba = sqrt(tension/(m/L)



For a, i knew since there were four loops, that it was fourth harmonic, so n=4, and 2L/4 = .6m
for b, i knew frequency was 120 Hz, and set that equal that to v/(4*2L) to get 1152 m/s.
for c, I'm a little stuck, and i know the wavelength would be smaller with a higher harmonic, and the frequency would stay the same, so the velocity would be smaller, so the tension should be smaller, but that doesn't seem right.
d was easy, amplitude is just half of the total displacement, or 2 cm.

 

[kuruman]

For (c) you know that the speed of propagation is $v=\sqrt{T/\mu}.~$You also know that $v=\lambda f$ so that $$\lambda =\frac{1}{f}\sqrt{T/\mu}.$$Should you increase or decrease $T=Mg$ to double $\lambda$ and by what factor?