Resonance: predict allowed patterns in strings and air columns

MCAT content library / Chemistry & physics / Resonance in strings and air columns

Use boundary conditions to decide which standing-wave patterns are allowed, then predict how length and wave speed change resonant frequency.

About 16 minutes, including the string/pipe comparison and checks.

Resonance requires an allowed standing wave

A resonant system responds strongly when it is driven near one of its natural frequencies. Those frequencies are determined by wave speed and by the boundary conditions that the standing wave must satisfy.

For a string fixed at both ends, the endpoints are displacement nodes. For an air column open at an end, the air displacement can be large there, corresponding to a displacement antinode. At a closed end, air displacement must go to zero, creating a displacement node. Those endpoint rules determine which wavelengths fit.

Concept: boundary conditions select the pattern

For a string of length L fixed at both ends, the allowed wavelengths are:

λ_n = 2L/n, where n = 1, 2, 3, …

The resonant frequencies are therefore:

f_n = nv/(2L).

An open-open air column has the same wavelength pattern for displacement: both ends are antinodes, and the allowed frequencies are integer multiples of the fundamental.

A pipe closed at one end and open at the other must have a node at one end and an antinode at the other. Its fundamental fits one quarter wavelength into the pipe:

λ₁ = 4L, so f₁ = v/(4L).

For the ideal closed-open pipe, the allowed sequence contains odd harmonics: 1, 3, 5, … times the fundamental.

Worked application: compare a string and two air columns

An original experiment compares three systems, all with length 0.50 m. The string wave speed is 100 m/s. The sound speed in both air columns is 340 m/s.

The string and open-open pipe have the same fundamental wavelength because their endpoint patterns fit half a wavelength into the system. Their frequencies differ because their wave speeds differ. The closed-open pipe fits one quarter wavelength into the same length, doubling the fundamental wavelength relative to the open-open pipe and halving its fundamental frequency for the same sound speed.

Pattern: length and wave speed change frequency differently

For a given mode, resonant frequency is proportional to wave speed and inversely proportional to length. Doubling the string length while keeping tension and linear density unchanged halves every resonant frequency. Increasing the wave speed while length remains fixed raises the resonant frequencies.

On a stretched string, wave speed depends on tension and linear mass density. Increasing tension raises wave speed and therefore raises the resonant frequencies. In an air column, sound speed depends on the medium and conditions, so changing temperature can shift the resonance even if pipe length is unchanged.

Rule: a higher mode is not a faster wave

In one unchanged medium, different standing-wave modes generally share the same wave speed. A higher mode has a shorter wavelength and a higher frequency, satisfying v = fλ. The wave does not travel faster merely because the resonant frequency is higher.

This distinction matters in graphs. If the second harmonic of a string has twice the fundamental frequency and half the wavelength, the speed calculated from fλ remains the same.

Check your reasoning

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Continue learning

• Wave amplitude, frequency, and speed
• Sound intensity and attenuation
• Browse Chemistry & Physics MCAT lessons

Sources

1. AAMC. Content Category 4D.
2. OpenStax Physics. Standing Waves.