The speed of sound in air at 20°C is 344 m/s. (a) What is the wavelength of a sound wave with a frequency of 784 Hz, corresponding to the note G5 on a piano, and how many milliseconds does each vibration take? (b) What is the wavelength of a sound wave one octave higher (twice the frequency) than the note in part (a)?

Verified step by step guidance

1. Step 1: Calculate the wavelength of the sound wave for the note G5. Use the formula for the speed of a wave, which is given by \( v = f \lambda \), where \( v \) is the speed of sound, \( f \) is the frequency, and \( \lambda \) is the wavelength. Rearrange the formula to solve for \( \lambda \): \( \lambda = \frac{v}{f} \).
2. Step 2: Determine the time period of one vibration for the note G5. The time period (T) is the reciprocal of the frequency. Use the formula \( T = \frac{1}{f} \) to find the time period in seconds, then convert it to milliseconds by multiplying by 1000.
3. Step 3: Calculate the wavelength of the sound wave for the note one octave higher than G5. Since one octave higher is twice the frequency of G5, use the new frequency \( f' = 2f \). Apply the same wave speed formula \( \lambda' = \frac{v}{f'} \).
4. Step 4: Simplify the expression for the new wavelength using the relationship between the original frequency and the new frequency. Since \( f' = 2f \), substitute into the wavelength formula to get \( \lambda' = \frac{v}{2f} \), which shows that the new wavelength is half of the original wavelength.
5. Step 5: Summarize the results, stating that the wavelength of the note one octave higher is half the wavelength of the original note, and the time period for each vibration of the original note in milliseconds.

Key Concepts

Wave Speed Formula

The speed of a wave is determined by the formula v = fλ, where v is the wave speed, f is the frequency, and λ (lambda) is the wavelength. This relationship shows that the speed of sound in a medium is constant, while the wavelength and frequency are inversely related. Understanding this formula is essential for calculating the wavelength of sound waves based on their frequency.

Frequency and Period

Frequency, measured in hertz (Hz), is the number of vibrations or cycles per second of a wave. The period is the duration of one complete cycle, and it is the inverse of frequency (T = 1/f). This concept is crucial for determining how long each vibration takes, which can be expressed in milliseconds for practical applications.

Octave in Music

An octave is a musical interval where the frequency of a note is doubled. For example, if a note has a frequency of 784 Hz, the note one octave higher will have a frequency of 1568 Hz. This concept is important for understanding how sound waves relate to musical notes and how their wavelengths change with frequency.