The lowest note emitted by a stretched string has a frequency of 40Hz. How many overtones are there between 40Hz and 180Hz?

Analysis

To solve this, we need to understand the relationship between the fundamental frequency, harmonics, and overtones in a stretched string.

Concept

1. Fundamental Frequency ($f_0$): This is the lowest frequency produced by the string. In this case, $f_0 = 40\text{ Hz}$.
2. Harmonics: These are integer multiples of the fundamental frequency ($1f_0, 2f_0, 3f_0, 4f_0, \dots$).
3. Overtones: These are the frequencies produced above the fundamental frequency. • The 1st overtone is the 2nd harmonic ($2 \times f_0$). • The 2nd overtone is the 3rd harmonic ($3 \times f_0$). • The $n$-th overtone is the $(n+1)$-th harmonic.

Step-by-Step Solution

• Step 1: Identify the frequencies of the harmonics. • Fundamental frequency ($f_0$): $40\text{ Hz}$ • 1st overtone (2nd harmonic): $2 \times 40\text{ Hz} = 80\text{ Hz}$ • 2nd overtone (3rd harmonic): $3 \times 40\text{ Hz} = 120\text{ Hz}$ • 3rd overtone (4th harmonic): $4 \times 40\text{ Hz} = 160\text{ Hz}$ • 4th overtone (5th harmonic): $5 \times 40\text{ Hz} = 200\text{ Hz}$

• Step 2: Determine which overtones fall between 40Hz and 180Hz. • The overtones are $80\text{ Hz}$, $120\text{ Hz}$, and $160\text{ Hz}$. • All three of these values ($80, 120, 160$) are greater than $40\text{ Hz}$ and less than $180\text{ Hz}$. • The next overtone ($200\text{ Hz}$) is greater than $180\text{ Hz}$, so we exclude it.

• Step 3: Count the overtones. • There are exactly 3 overtones in the specified range.

Conclusion

There are 3 overtones (80Hz, 120Hz, and 160Hz) between 40Hz and 180Hz.

Exam Tip

Always remember the "n+1" rule: the $n$-th overtone is always the $(n+1)$-th harmonic. If you are ever confused, just list the multiples of the fundamental frequency and count everything after the first one!