Standing waves are created in the four strings shown in Figure 25. All strings have the same mass per unit length and are under the same tension The lengths of the strings are given. Rank the frequencies of the oscillations, from largest to smallest

The order of the frequencies is  

The frequency of the standing wave in a string tied at both ends is given by:

where n is the mode of frequency

v is the velocity of the wave

and L is the length of the string.

Now the velocity of a wave in a string tied at both ends is given by

where T is the tension and μ is the mass per unit length.

Since T and μ are the same for all the strings, velocity will be the same for all.

Now to find the mode of frequency we can calculate the number of nodes (including the nodes at the ends) in the given figure and subtract by 1. Nodes are the point where the amplitude of the wave is zero.

Hence,

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The rank of the frequencies from largest to smallest is

The largest frequency of oscillation is given by the string in option D

The second largest frequency of oscillation is given by the string in option B

The third largest frequency of oscillation is given by the string in option A

The smallest frequency of oscillation is given by the string in option C

Explanation:

The given parameters are;

The mass per unit length of all string, m/L = Constant

The tension of all the string, T = Constant

The frequency of oscillation, f, of a string is given as follows;

Where;

T = The tension in the string

m = The mass of the string

L = The length of the string

n = The number of overtones

For the string in option A, the length, L = 27 cm, n = 3 we have;

For the string in option B, the length, L = 30 cm, n = 4 we have;

For the string in option C, the length, L = 30 cm, n = 3 we have;

For the string in option D, the length, L = 24 cm, n = 4 we have;

Therefore, we have the rank of the frequency of oscillations of th strings from largest to smallest given as follows;

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