Problem 1 - University Physics with Modern Physics

Question

The speed of sound in air at \(20^{\circ} \mathrm{C}\) is 344 \(\mathrm{m} / \mathrm{s} .\) (a) What is the wavelength of a sound wave with a frequency of 784 \(\mathrm{Hz}\) , corresponding to the note \(\mathrm{G}_{5}\) on a piano, and how many milliseconds does each vibration take? (b) What is the wavelength of a sound wave one octave higher than the note in part (a)?

Step-by-Step Solution

Answer

(a) Wavelength is 0.439 m, each vibration takes 1.28 ms. (b) Wavelength is 0.219 m.

Step 1: Understand the Relationship Between Frequency, Wavelength, and Speed

The speed of a wave (\(v\)), frequency (\(f\)), and wavelength (\(\lambda\)) are related by the formula:\[v = f \times \lambda\]In this problem, the speed of sound (\(v\)) is given as 344 m/s, and we need to find the wavelength for \(f = 784 \text{ Hz}\).

Step 2: Calculate the Wavelength for Frequency 784 Hz

Rearrange the formula to solve for wavelength:\[\lambda = \frac{v}{f}\]Substitute the given values:\[\lambda = \frac{344 \, \text{m/s}}{784 \, \text{Hz}} = 0.4388 \, \text{m}\]So the wavelength of the sound wave is approximately 0.439 meters.

Step 3: Calculate the Period of the Wave

The period (\(T\)) of a wave is the reciprocal of the frequency:\[T = \frac{1}{f}\]Using the frequency 784 Hz:\[T = \frac{1}{784} \approx 0.00128 \, \text{s} = 1.28 \, \text{ms}\]Each vibration takes approximately 1.28 milliseconds.

Step 4: Calculate the Wavelength for a Note One Octave Higher

An octave higher frequency is double the original frequency. Therefore, the frequency for one octave above 784 Hz is 1568 Hz:\[f_{\text{octave}} = 2 \times 784 \text{ Hz} = 1568 \text{ Hz}\]Using the formula for wavelength:\[\lambda_{\text{octave}} = \frac{v}{f_{\text{octave}}} = \frac{344 \, \text{m/s}}{1568 \, \text{Hz}} \approx 0.2194 \, \text{m}\]The wavelength of the sound wave one octave higher is approximately 0.219 meters.

Key Concepts

Speed of Sound

In wave physics, the speed of sound is a key concept. It refers to how fast sound waves travel through a medium, such as air. At 20°C, the speed of sound in air is typically 344 meters per second (m/s). This speed can vary slightly due to changes in factors like temperature and pressure. Warm air allows sound to move faster because the molecules are more energetic and can transmit sound more briskly. By knowing the speed of sound, we can calculate various properties of sound waves, such as wavelength and frequency. Indeed, the formula \(v = f \times \lambda\) links the speed of sound \(v\) with the frequency \(f\) and the wavelength \(\lambda\). Hence, if we know the frequency of a sound and the medium it travels through, we can find out the wavelength.

Frequency and Wavelength Relationship

The relationship between frequency and wavelength is one of the foundational principles in wave physics. Frequency refers to how many wave cycles pass a fixed point in a second. It is measured in hertz (Hz). Wavelength is the distance between successive crests or troughs of a wave.
This relationship can be described by the equation \(v = f \times \lambda\), where \(v\) is the speed of sound, \(f\) is the frequency, and \(\lambda\) is the wavelength.

• To find the wavelength, rearrange the formula: \(\lambda = \frac{v}{f}\).
• An increase in frequency results in a shorter wavelength, assuming the wave speed in the medium remains constant.
• This formula is useful for identifying the size of the sound wave based on its frequency, providing insights into the nature of the sound itself.

Therefore, for a note like \(\text{G}_5\) on a piano with a frequency of 784 Hz, knowing the speed of sound allows us to calculate its wavelength as approximately 0.439 meters.

Harmonics and Octaves

Understanding harmonics and octaves helps us grasp the concept of musical sound waves and their behaviors. An octave occurs when the frequency doubles. For instance, if a sound wave has a frequency of 784 Hz, an octave higher would be 1568 Hz. Doubling the frequency compresses the wavelength into half. As a result, the sound wave becomes shorter.
Harmonics are integral multiples of the fundamental frequency and enrich the timbre or quality of sound. They help extend and refine musical notes.

• Musically, harmonics above a fundamental frequency contribute to the sound's distinctiveness.
• An octave is a special harmonic ratio where the frequency is doubled.

When we calculate wavelengths for these octaves using \(\lambda = \frac{v}{f}\), the wavelength for the octave higher \(\text{G}_5\) is approximately 0.219 meters. Thus, knowing octaves and harmonics not only aids in the understanding of musical concepts but also in comprehending how sound waves function.