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Two wires are stretched between two fixed supports and have the same length. One wire A there is a second-harmonic standing wave whose frequency is 660 Hz. However, the same frequency of 660 Hz is the third harmonic on wire B. (a) Is the fundamental frequency of wire A greater than, less than, or equal to the fundamental frequency of wire B? Explain. (b) How is the fundamental frequency related to the length L of the wire and the speed v at which individual waves travel back and forth on the wire? (c) Do the individual waves travel on wire A with a greater, smaller, or the same speed as on wire B? Give your reasoning.

Respuesta :

skyluke89

(a) Greater

The frequency of the nth-harmonic on a string is an integer multiple of the fundamental frequency, [tex]f_1[/tex]:

[tex]f_n = n f_1[/tex]

So we have:

- On wire A, the second-harmonic has frequency of [tex]f_2 = 660 Hz[/tex], so the fundamental frequency is:

[tex]f_1 = \frac{f_2}{2}=\frac{660 Hz}{2}=330 Hz[/tex]

- On wire B, the third-harmonic has frequency of [tex]f_3 = 660 Hz[/tex], so the fundamental frequency is

[tex]f_1 = \frac{f_3}{3}=\frac{660 Hz}{3}=220 Hz[/tex]

So, the fundamental frequency of wire A is greater than the fundamental frequency of wire B.

(b) [tex]f_1 = \frac{v}{2L}[/tex]

For standing waves on a string, the fundamental frequency is given by the formula:

[tex]f_1 = \frac{v}{2L}[/tex]

where

v is the speed at which the waves travel back and forth on the wire

L is the length of the string

(c) Greater speed on wire A

We can solve the formula of the fundamental frequency for v, the speed of the wave:

[tex]v=2Lf_1[/tex]

We know that the two wires have same length L. For wire A, [tex]f_1 = 330 Hz[/tex], while for wave B, [tex]f_B = 220 Hz[/tex], so we can write the ratio between the speeds of the waves in the two wires:

[tex]\frac{v_A}{v_B}=\frac{2L(330 Hz)}{2L(220 Hz)}=\frac{3}{2}[/tex]

So, the waves travel faster on wire A.

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