Wave Formula Audits

Standing Harmonic Fit Audit

allowed wavelengths

A fixed string accepts only wavelengths that fit an exact number of half-wavelengths.

Example

A fixed string accepts only wavelengths that fit an exact number of half-wavelengths, then frequency follows from speed over wavelength. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

A fixed string only accepts fitting wavelengths

A string fixed at both ends must have nodes at both ends. The allowed wavelengths are the ones whose half-wavelength pieces exactly fill the string. Then frequency follows from wave speed over wavelength.

L=n λn2,fn=vλnL=n\,\frac{\lambda_n}{2},\quad f_n=\frac{v}{\lambda_n}
Allowed standing-wave shapesA fixed string with nodes at its ends and an allowed bulging standing-wave pattern.

The first harmonic fits one half-wavelength

The fundamental has one bulge and nodes at the two ends. Its wavelength is twice the string length, and its frequency is the lowest allowed value.

λn=2Ln=2⋅6 m1=12 m,Lfit=6 m,fn=1 Hz\lambda_n=\frac{2L}{n}=\frac{2\cdot6\ \text{m}}{1}=12\ \text{m},\quad L_{\text{fit}}=6\ \text{m},\quad f_n=1\ \text{Hz}
First harmonicOne half-wavelength fits the whole fixed string.

The second harmonic fits two half-wavelengths

The next allowed shape puts a node in the middle. The wavelength is shorter, so the same wave speed gives a higher frequency.

λn=2Ln=2⋅6 m2=6 m,Lfit=6 m,fn=2 Hz\lambda_n=\frac{2L}{n}=\frac{2\cdot6\ \text{m}}{2}=6\ \text{m},\quad L_{\text{fit}}=6\ \text{m},\quad f_n=2\ \text{Hz}
Second harmonicTwo half-wavelengths fit the fixed string.

The third harmonic fits three half-wavelengths

The third allowed shape has three bulges. The fit column still returns the same string length, so this is an allowed wavelength, not an arbitrary sketch.

λn=2Ln=2⋅6 m3=4 m,Lfit=6 m,fn=3 Hz\lambda_n=\frac{2L}{n}=\frac{2\cdot6\ \text{m}}{3}=4\ \text{m},\quad L_{\text{fit}}=6\ \text{m},\quad f_n=3\ \text{Hz}
Third harmonicThree half-wavelengths fit the fixed string.

Fit first, then compute frequency

Read across each row: the fit column returns the same length of string, and the frequency column rises as the fitting wavelength gets shorter.

nλnLfitfn112 m6 m1 Hz26 m6 m2 Hz34 m6 m3 Hz\begin{array}{c|c|c|c}n & \lambda_n & L_{\text{fit}} & f_n \\ \hline 1 & 12\ \text{m} & 6\ \text{m} & 1\ \text{Hz} \\ 2 & 6\ \text{m} & 6\ \text{m} & 2\ \text{Hz} \\ 3 & 4\ \text{m} & 6\ \text{m} & 3\ \text{Hz}\end{array}
waves Check the standing-wave fit first, then compute each frequency from speed over wavelength.