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18. Waves & Sound

Beats

Physics

18. Waves & Sound

Beats

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5:12 minutes

Problem 51

Textbook Question

(II) Two violin strings are tuned to the same frequency, 294 Hz. The tension in one string is then decreased by 2.5%. What will be the beat frequency heard when the two strings are played together? [Hint: Recall Eq. 15–2.]

Verified step by step guidance

Start by recalling the relationship between the frequency of a string and its tension. The frequency of a vibrating string is proportional to the square root of the tension, expressed as:

f ∝ \sqrt{T}

, where

f

is the frequency and

T

is the tension.

Let the initial frequency of both strings be

f_{1} = 294

Hz. When the tension in one string is decreased by 2.5%, the new tension becomes

T' = 0.975 T

, where

T

is the original tension.

Using the proportionality

f ∝ \sqrt{T}

, the new frequency of the string with reduced tension can be expressed as:

f_{2} = f_{1} × \sqrt{\frac{0.975}{1}}

Simplify the expression for

f_{2}

to find the new frequency of the string with reduced tension. This involves calculating the square root of 0.975 and multiplying it by 294 Hz.

The beat frequency is the absolute difference between the two frequencies, given by:

f

_beat = |f1 - f2|. Substitute the values of

f_{1}

and

f_{2}

to find the beat frequency.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Frequency

Frequency is the number of cycles of a periodic wave that occur in one second, measured in Hertz (Hz). In this context, both violin strings are tuned to a frequency of 294 Hz, meaning they vibrate 294 times per second. When two strings vibrate at slightly different frequencies, the resulting sound can produce beats, which are periodic variations in loudness.

Tension in Strings

The tension in a string affects its frequency of vibration; increasing tension raises the frequency, while decreasing tension lowers it. The relationship is described by the formula f = (1/2L)√(T/μ), where f is frequency, L is the length of the string, T is tension, and μ is the linear mass density. In this problem, a 2.5% decrease in tension will result in a lower frequency for the affected string.

Beat Frequency

Beat frequency is the difference in frequencies of two sound waves that are close in frequency, resulting in a periodic variation in sound intensity. It can be calculated by taking the absolute difference between the two frequencies. In this case, after adjusting the tension of one string, the beat frequency will be the difference between the original frequency of 294 Hz and the new frequency of the altered string.

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Related Practice

Textbook Question

Piano tuners tune pianos by listening to the beats between the harmonics of two different strings. When properly tuned, the note A should have a frequency of 440 Hz and the note E should be at 659 Hz. The tuner starts with the tension in the E string a little low, then tightens it. What is the frequency of the E string when she hears four beats per second?

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Textbook Question

A guitar string produces 3 beat/s when sounded with a 350-Hz tuning fork and 8 beat/s when sounded with a 355-Hz tuning fork. What is the vibrational frequency of the string? Explain your reasoning.

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Textbook Question

A tuning fork is set into vibration above a vertical open tube filled with water (Fig. 16–41). The water level is allowed to drop slowly. As it does so, the air in the tube above the water level is heard to resonate with the tuning fork when the distance from the tube opening to the water level is 0.125 m and again at 0.395 m. What is the frequency of the tuning fork?

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Textbook Question

The two sources of sound in Fig. 16–15 face each other and emit sounds of equal amplitude and equal frequency (305 Hz) but 180° out of phase. For what minimum separation of the two speakers will there be some point at which complete constructive interference occurs? (Assume T = 20°C)

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Textbook Question

A source emits sound of wavelengths 2.54 m and 2.72 m in air. How many beats per second will be heard? (Assume T = 20°C.)

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Textbook Question

(III) A source emits sound of wavelengths 2.54 m and 2.72 m in air. How far apart in space are the regions of maximum intensity?

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Textbook Question

(I) A piano tuner hears one beat every 2.0 s when trying to adjust two strings, one of which is sounding 340 Hz. How far off in frequency is the other string?

516

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Multiple Choice

A string emits an unknown sound. You strike a tuning fork which emits a sound at EXACTLY 300 Hz, and you hear a beat frequency of 20 Hz. You then tighten the string, increasing the tension in string. After you pluck the string and strike the tuning fork, you hear a new beat frequency of 30 Hz. What is the unknown frequency of the string, originally?

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