Problem 3

A string of length

L

fixed at both ends carries waves at speed

v

. Find the fundamental wavelength and frequency, and the frequency of the

n

th harmonic. A uniform string of length

L

has nodes at both fixed ends; consider mode number

n

, a positive integer. (a) Find wavelength

\lambda_n

and the number of interior nodes. (b) Find the fundamental frequency. (c) Find the frequency of mode

n

and explain its harmonic relation. Express each answer in terms of the supplied parameters, using SI units for any numerical evaluation. State the assumptions used, check dimensions at each result, and compare with the governing law to catch a sign error or a missing condition.

problems.proof.stepOf

problems.proof.prevStep problems.proof.nextStep

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

problems.proof.analysis

First isolate the parts of the system linked by the stated constraint and choose the governing law; this puts the quantities under a consistent convention. The idealizations exclude interactions not included in the model.

problems.proof.pitfall. Do not omit the sign convention or the model’s physical condition; check the limiting case before accepting a solution.