Two strings on a musical instrument are tuned to play at 392...

Two strings on a musical instrument are tuned to play at 392 Hz (G) and 494 Hz (B). ($a$) What are the frequencies of the first two overtones for each string? ($b$) If the two strings have the same length and are under the same tension, what must be the ratio of their masses ($m_G/m_B$)? ($c$) If the strings, instead, have the same mass per unit length and are under the same tension, what is the ratio of their lengths ($\ell_G/\ell_B$)? ($d$) If their masses and lengths are the same, what must be the ratio of the tensions in the two strings?

Key Concepts

Linear Mass Density and Its Role

The linear mass density (?) is defined as the mass per unit length of the string. It plays a crucial role in determining the vibrational frequency. For strings of the same length and under the same tension, a higher mass (or higher ?) results in a lower frequency, while a lower mass leads to a higher frequency. This relationship is critical when comparing strings that differ in mass while other properties remain constant.

Overtones

Overtones are the higher frequency vibrational modes of the string, occurring at integer multiples of the fundamental frequency. The first overtone corresponds to the second harmonic (n = 2), the second overtone to the third harmonic (n = 3), and so on. Their frequencies are essential for understanding the timbre and pitching of instrument sounds.

Relationships Between Frequency, Tension, Length, and Mass

The fundamental frequency equation illustrates how changes in tension, length, and mass affect the sound produced. For example, if two strings are under the same tension and have the same length, differing frequencies indicate differences in their linear mass densities (or total masses). Conversely, if the mass per unit length is constant but the frequencies differ, it implies that the lengths of the strings must differ inversely with the frequency. Similarly, if masses and lengths are identical, variations in frequency point to differences in the applied tensions. These proportionalities are key in designing and tuning musical instruments.

Vibrating String Harmonics

When a string vibrates, it does so in specific modes or harmonics. The fundamental frequency is the lowest frequency at which the string vibrates, and it corresponds to the mode where the string has two fixed nodes at its ends. Higher modes, called overtones, occur at integer multiples of the fundamental frequency (for ideal strings with fixed ends), meaning the nth harmonic has a frequency of n times the fundamental frequency.

Fundamental Frequency Equation

The frequency of a vibrating string is given by f = (n/(2L)) * ?(T/?), where f is the frequency of the nth harmonic, L is the length of the string, T is the tension, and ? is the linear mass density (mass per unit length). This equation highlights how the frequency depends not only on the length and tension but also on how mass is distributed along the string.

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