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Chapter 17: Problem 20

Two pianos each sound the same note simultaneously, but they are both out of tune. On a day when the speed of sound is 343 m/s, piano A produces a wavelength of 0.769 m, while piano B produces a wavelength of 0.776 m. How much time separates successive beats?

Short Answer

Expert verified
The time between successive beats is approximately 0.25 seconds.

Step by step solution

01

Understand the Problem

We are given two pianos that both produce sound waves at slightly different wavelengths, resulting in audible beats due to interference. Our task is to find the time interval between these successive beats.
02

Find the Frequency of Each Piano

Use the formula for the speed of sound to find the frequency of each piano: \( v = f \times \lambda \), where \( v \) is the speed of sound, \( f \) is the frequency, and \( \lambda \) is the wavelength.- For Piano A: \[ f_A = \frac{v}{\lambda_A} = \frac{343 \, \text{m/s}}{0.769 \, \text{m}} \approx 446 \text{ Hz} \]- For Piano B: \[ f_B = \frac{v}{\lambda_B} = \frac{343 \, \text{m/s}}{0.776 \, \text{m}} \approx 442 \text{ Hz} \]
03

Calculate the Beat Frequency

The beat frequency is the absolute difference between the two frequencies: \( f_\text{beat} = |f_A - f_B| \).\[ f_\text{beat} = |446 - 442| = 4 \text{ Hz} \]
04

Determine the Time Interval Between Beats

The time interval between successive beats is the reciprocal of the beat frequency.\[ T = \frac{1}{f_\text{beat}} = \frac{1}{4 \, \text{Hz}} = 0.25 \, \text{seconds} \]
05

Conclusion: Time Between Successive Beats

The time interval separating successive beats when the two pianos sound is approximately 0.25 seconds.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Beat Frequency
When two sound waves of nearly the same frequency combine, they create a phenomenon known as beats. The beat frequency is the rate at which the loudness fluctuates due to this interference. Imagine hearing a pulsating sound where the volume rises and falls.

This happens because the waves sometimes align (constructive interference), making the sound louder, and sometimes cancel each other out (destructive interference), making the sound quieter. The beat frequency is calculated by taking the absolute difference between two frequencies:
  • Beat Frequency Formula: \(f_{\text{beat}} = |f_1 - f_2|\)
In our scenario, the beat frequency between the two pianos is 4 Hz, meaning the listener will hear 4 beats per second.
Wavelength
Wavelength is the distance between consecutive crests (or troughs) of a wave. In the context of sound waves, it directly affects the pitch of the sound. A shorter wavelength means a higher frequency and thus a higher pitch.

The relationship between the speed of sound (\(v\)), wavelength (\(\lambda\)), and frequency (\(f\)) can be described with the equation:
  • \(v = f \times \lambda\)
For the pianos, each produces its unique wavelength that contributes to the out-of-tune effect. With piano A producing a 0.769 m wavelength and piano B a 0.776 m wavelength, these small differences lead to different frequencies.
Frequency
Frequency is the number of times a wave's crests pass a point in one second. It is measured in Hertz (Hz), with one Hertz equalling one cycle per second. This concept is crucial in music, as it defines the pitch of a note.

From the formula \(v = f \times \lambda\), we can solve for frequency by rearranging it as \(f = \frac{v}{\lambda}\). For piano A and piano B:
  • Piano A: \(f_A = \frac{343 \text{ m/s}}{0.769 \text{ m}} \approx 446 \text{ Hz}\)
  • Piano B: \(f_B = \frac{343 \text{ m/s}}{0.776 \text{ m}} \approx 442 \text{ Hz}\)
These frequencies show slightly different pitches, resulting in a beat frequency.
Interference
Interference happens when two or more waves overlap, affecting how loud or soft a sound seems. With constructive interference, the waves combine to make a louder sound; with destructive interference, they cancel each other out, making the sound softer.

When two pianos play out of tune, the waves they produce interfere with one another. This interference creates the beats, an effect heard as periodic volume changes. We calculate how often these beats occur using the beat frequency, which shows the direct result of such interference.
Speed of Sound
The speed of sound is how fast a sound wave travels through a medium, such as air. It is generally around 343 meters per second (m/s) at room temperature. This speed influences how wavelengths translate into frequencies.

Given the formula \(v = f \times \lambda\), the speed of sound links directly to both the frequency and wavelength of a sound wave. In the original exercise, knowing the speed of sound was essential to determine each piano frequency. This illustrates how changing conditions like air temperature can affect the speed of sound, influencing how we perceive musical notes.

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Most popular questions from this chapter

Standing waves are set up on two strings fixed at each end, as shown in the drawing. The two strings have the same tension and mass per unit length, but they differ in length by 0.57 cm. The waves on the shorter string propagate with a speed of 41.8 m/s, and the fundamental frequency of the shorter string is 225 Hz. Determine the beat frequency produced by the two standing waves.

A piccolo and a flute can be approximated as cylindrical tubes with both ends open. The lowest fundamental frequency produced by one kind of piccolo is 587.3 Hz, and that produced by one kind of flute is 261.6 Hz. What is the ratio of the piccolo鈥檚 length to the flute鈥檚 length?

Two out-of-tune flutes play the same note. One produces a tone that has a frequency of 262 Hz, while the other produces 266 Hz. When a tuning fork is sounded together with the 262-Hz tone, a beat frequency of 1 Hz is produced. When the same tuning fork is sounded together with the 266-Hz tone, a beat frequency of 3 Hz is produced. What is the frequency of the tuning fork?

A thin 1.2-m aluminum rod sustains a longitudinal standing wave with vibration antinodes at each end of the rod. There are no other antinodes. The density and Young鈥檚 modulus of aluminum are, respectively, 2700 \(kg / m^{3}\) and \(6.9 \times 10^{10} \mathrm{N} / m^{2} .\) What is the frequency of the rod鈥檚 vibration?

A tube is open only at one end. A certain harmonic produced by the tube has a frequency of 450 Hz. The next higher harmonic has a frequency of 750 Hz. The speed of sound in air is 343 m/s. (a) What is the integer n that describes the harmonic whose frequency is 450 Hz? (b) What is the length of the tube?
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