A stretched wire of length 1.0 m is clamped at both ends. It is plucked at its center as shown. The three longest wavelengths in the wire are (in meters):
A) 4, 2, 1
B) 2, 1, 0.5
C) 2, 0.67, 0.4
D) 1, 0.5, 0.33
E) 1, 0.67, 0.5

Question

A stretched wire of length 1.0 m is clamped at both ends. It is plucked at its center as shown. The three longest wavelengths in the wire are (in meters): 
A) 4, 2, 1 
B) 2, 1, 0.5 
C) 2, 0.67, 0.4 
D) 1, 0.5, 0.33 
E) 1, 0.67, 0.5

Quizplus · Accepted Answer

The fundamental frequency of the wire is given by $f_1 = \frac{v}{2L}$, where $v$ is the speed of sound in the wire and $L$ is the length of the wire. The wavelength of the fundamental mode is $2L$.

For the second harmonic, we have $f_2 = 2f_1$, and thus $\lambda_2 = \frac{v}{f_2} = \frac{1}{2}\lambda_1$.

For the third harmonic, we have $f_3 = 3f_1$, and thus $\lambda_3 = \frac{v}{f_3} = \frac{1}{3}\lambda_1$.

Using $L = 1$ m and assuming a speed of sound of 500 m/s (appropriate for a thin wire), we get $\lambda_1 = 2$ m, $\lambda_2 = 1$ m, and $\lambda_3 = 0.67$ m. Therefore, the answer is (C) 2, 0.67, 0.4.

A Stretched Wire of Length 1