A musical note played on an instrument produces a fundamental frequency of 440\text{ Hz}. … — JAMB UTME

A musical note played on an instrument produces a fundamental frequency of $440\text{ Hz}$ . Given that the speed of sound in air is $330\text{ m/s}$ , what is the wavelength, in meters, of the sound wave produced in air corresponding to its second overtone?

Answer: 0.25 m

Answer

0.25 m

The fundamental frequency (

f_1 = 440\text{ Hz}

) is the first harmonic. The second overtone is the third harmonic, which has a frequency of

3 \times 440\text{ Hz} = 1320\text{ Hz}

. Substituting this into the wave speed equation

\lambda = \frac{v}{f}

gives

\lambda = \frac{330\text{ m/s}}{1320\text{ Hz}} = 0.25\text{ m}

Step-by-Step Solution

Determine the harmonic number for the second overtone

The second overtone is the third harmonic (

n = 3

)

Overtones are integer harmonics above the fundamental frequency (

n = 1

). Thus, the 1st overtone is

n = 2

and the 2nd overtone is

n = 3

Calculate the frequency of the second overtone

f_3 = 3 \times 440\text{ Hz} = 1320\text{ Hz}

The frequency of the

n

-th harmonic is

n

times the fundamental frequency.

Calculate the wavelength using the wave speed equation

\lambda = \frac{v}{f_3} = \frac{330\text{ m/s}}{1320\text{ Hz}} = 0.25\text{ m}

Wavelength is determined by dividing the speed of sound by the wave frequency.

Key Concept

Harmonics, Overtones, and Wave Equation

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