Problem 1

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

\boxed{\lambda_1=2L,\quad f_1=\frac{v}{2L},\quad f_n=\frac{nv}{2L},\quad N_{\rm internal}=n-1}

problems.proof.analysis

The final expressions are collected in the order requested and must satisfy the equations used above. Checking dimensions, signs, and limiting conditions catches algebra errors or omitted constraints.