1) 2.5 wavelengths
2) 0.208 m
3) 1731 Hz
Explanation:
1)
Standing waves are waves that do not propagate, but instead the particles of the medium just oscillate around a fixed position. Examples of standing waves are the waves produced on a string with fixed ends.
The points of a standing wave in which the amplitude of the oscillation is always zero are called nodes.
The two fixed ends of the string are two nodes. In this problem, we have a total of 6 nodes along the string: this means that there are 4 additional nodes apart from the two ends of the string.
Therefore, this also means that the string oscillate in 5 different segments.
One wavelength is equal to 2 segments of the oscillation: therefore, since here there are 5 segments, this means that the number of wavelengths that we have in this string is
[tex]n=\frac{5}{2}=2.5[/tex]
2)
The wavelength of a wave is the distance between two consecutive crests (or throughs) of the wave.
The wavelength of a standing wave can be also measured as the distance between the nth-node and the (n+2)-th node: so, basically, the wavelength in a standing wave is twice the distance between two nodes:
[tex]\lambda = 2 d[/tex]
where
[tex]\lambda[/tex] is the wavelength
d is the distance between two nodes
Here the length of the string is
L = 0.520 m
And since it oscillates in 5 segments, the distance between two nodes is
[tex]d=\frac{L}{5}=\frac{0.520}{5}=0.104 m[/tex]
And therefore, the wavelength is
[tex]\lambda=2d=2(0.104)=0.208 m[/tex]
3)
The frequency of a wave is the number of complete oscillations of the wave per second.
The frequency of a wave is related to its speed and wavelength by the wave equation:
[tex]v=f\lambda[/tex]
where
v is the speed
f is the frequency
[tex]\lambda[/tex] is the wavelength
In this problem:
v = 360 m/s is the speed of the wave
[tex]\lambda=0.208 m[/tex] is the wavelength
Therefore, the frequency is
[tex]f=\frac{v}{\lambda}=\frac{360}{0.208}=1731 Hz[/tex]
