Question

If a standing wave is vibrating in the fourth harmonic and the wavelength is $\lambda$, what is the length of the string.

• A. $\lambda$
• B. $\frac{\lambda}{2}$
• C. $2\lambda$
• D. $4\lambda$

Answer 1

Answer: (C)

$2\lambda

Answer 2

For a string fixed at both ends, a standing wave forms when the length of the string ($L$) is an integer multiple of half wavelengths. The formula relating the length of the string, the wavelength ($\lambda_n$), and the harmonic number ($n$) is given by:

$L = n \frac{\lambda_n}{2}$

where $n = 1, 2, 3, \ldots$ represents the harmonic number.

Given:

• The standing wave is in the fourth harmonic, so $n = 4$.
• The wavelength is given as $\lambda$, so $\lambda_4 = \lambda$.

Calculation: Substitute the given values into the formula:

$L = 4 \frac{\lambda}{2}$

$L = 2\lambda$

The length of the string is $2\lambda$.