1. -none of the above
First of all, we have to notice that the speed of a wave in a material depends only on the properties of the material: therefore, if we use two strings made by the same material, the wave speed in the two strings will be the same.
Secondly, we can observe that the fundamental frequency of a wave on a string is given by
[tex]f_1 = \frac{1}{2L}\sqrt{\frac{T}{\mu}}[/tex]
where
L is the length of the string
T is the tension in the string
[tex]\mu=\frac{m}{L}[/tex] is the linear density of the string, with m being its mass
In this problem, the only difference between the two strings is in the thickness. Therefore, since the two strings have same length (L), same tension (T) and same linear density ([tex]\mu[/tex]), their frequency is the same.
And lastly, since the wavelength is given by the equation
[tex]\lambda=\frac{v}{f}[/tex]
where v is the wave speed and f is the frequency, since both v and f do not change, then the wavelength in the two strings is also the same.
2. wavelength and wave frequency
As before, the two strings are made of same material, so they have same wave speed.
The frequency instead is given by
[tex]f_1 = \frac{1}{2L}\sqrt{\frac{T}{\mu}}[/tex]
and since this time the two strings have different length L, then they also have different frequency.
And then, from the equation
[tex]\lambda=\frac{v}{f}[/tex]
we also conclude that if the frequency changes and the wave speed remains the same, then the wavelength must change as well.
