What if the wave speed is not given for a string?

Calculate the wave speed using $v = \sqrt{\frac{T}{\mu}}$, where $T$ is the tension in the string and $\mu$ is the linear mass density.

How do I handle a tube open at one end and closed at the other?

Treat the closed end as a displacement node and the open end as a displacement antinode. Apply the asymmetric formula $f_n = \frac{n v}{4L}$ where $n$ can only be an odd integer.

Does the fundamental frequency change if the diameter of an open tube is large?

Yes. For wide tubes, you must apply an end correction. The effective length becomes $L_{ ext{eff}} = L + 0.6r$ per open end, where $r$ is the radius of the tube. Use $L_{ ext{eff}}$ in the wavelength formulas.

How to solve a standing wave problem

A standing wave problem requires matching the wave's spatial variation to the physical boundary conditions of the medium.

This method applies to strings, acoustic tubes, and electromagnetic cavities where displacement or pressure nodes and antinodes are constrained at specific physical locations.

The setup

Identify the medium length $L$ , the wave speed $v$ , and the boundary conditions at both ends. Boundaries are either fixed (nodes) or free (antinodes). Symmetric systems have identical boundaries at both ends (e.g., fixed-fixed), while asymmetric systems have different boundaries (fixed-free).

The steps

Assign boundary conditions. Fixed ends enforce a node; free ends enforce an antinode.
Express the wavelength $\lambda_n$ in terms of $L$ . For symmetric boundaries, $\lambda_n = \frac{2L}{n}$ for $n = 1, 2, 3, \dots$ . For asymmetric boundaries, $\lambda_n = \frac{4L}{n}$ for $n = 1, 3, 5, \dots$ .
Relate wavelength to frequency using the wave equation $v = f \lambda$ .
Substitute $\lambda_n$ to find the harmonic frequencies: $f_n = \frac{n v}{2L}$ (symmetric) or $f_n = \frac{n v}{4L}$ (asymmetric).
Evaluate the expression using the given numerical values.

Checking the result

Verify that asymmetric systems (like a tube closed at one end) only produce odd harmonics. Check that higher harmonics in a symmetric system are exact integer multiples of the fundamental frequency ( $f_n = n f_1$ ).

Common errors

Using even values of $n$ for asymmetric fixed-free boundaries is physically impossible and mathematically invalid. Another frequent error is confusing the harmonic number $n$ with the number of nodes; a fixed-fixed string in the $n$ -th harmonic has $n+1$ nodes.

Worked example

A string of length $L = 0.8 ext{ m}$ is fixed at both ends. The wave speed on the string is $v = 120 ext{ m/s}$ . Calculate the frequency of the third harmonic.

Boundary conditions: Fixed-fixed (symmetric). Harmonic number: $n = 3$ . Wavelength equation: $\lambda_3 = \frac{2L}{3}$ Substitute length: $\lambda_3 = \frac{2(0.8 ext{ m})}{3} = 0.5333 ext{ m}$ Frequency equation: $f_3 = \frac{v}{\lambda_3}$ Calculate frequency: $f_3 = \frac{120 ext{ m/s}}{0.5333 ext{ m}} = 225 ext{ Hz}$

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References: University Physics Volume 1 (OpenStax) · Fundamentals of Physics (Halliday, Resnick, Walker)