That the frequency of a wave can be given as follows:
f = 1/2L * sqrt (T/m)
where L = length of oscillating string , T = tension and m = mass per unit length. Because the frequency that you stated is the fundamental frequency.
You can just state the second harmonics frequency for the given wave in the question as
f2=L1μT
or
λ2=f21μT
where λ2= L. Identifying that the period of the 2 half-loop is the same, this implies that the frequency is the same, too.
I suspect that we can assume that the tension is the same in this situation because the mass per unit mass changes. If both tension and mass per unit mass change, I think the problem can be complicated to solve. We need to come back to solve the partial differential equation of the wave equation.
Assume that the tension is same, then the wavelength is proportional to 1/μ. Since the wavelength increases from A to B as shown, the mass per unit length decreases from A to B so A has a greater diameter.
First of all, you should write the frequency of the normal modes of the standing wave is
fn=2LnμT where n= 1, 2, 3…
if you want to write the general equation for the frequency instead of
You can just state the second harmonics frequency for the given wave in the question as
f2=L1μT
or
λ2=f21μT
where λ2= L. Identifying that the period of the 2 half-loop is the same, this implies that the frequency is the same, too.
I suspect that we can assume that the tension is the same in this situation because the mass per unit mass changes. If both tension and mass per unit mass change, I think the problem can be complicated to solve. We need to come back to solve the partial differential equation of the wave equation.
Assume that the tension is same, then the wavelength is proportional to 1/μ. Since the wavelength increases from A to B as shown, the mass per unit length decreases from A to B so A has a greater diameter.
So just to reiterate it is true that wavelength is inversely proportional to the square root of mass per unit length (when constants are taken out).
And mass per unit definitely has to lower in order for wavelength to increase as shown in the figure according to the relationship. Silly of me why I couldn't see that again in the first place. Hence diameter also goes up.
And also don't know why I also expressed wavelength to be written as 2L if it explicitly says the second harmonic so L in this case