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Chapter 16: Problem 47

One of the harmonic frequencies for a particular string under tension is \(325 \mathrm{~Hz}\). The next higher harmonic frequency is \(390 \mathrm{~Hz}\). What harmonic frequency is next higher after the harmonic frequency \(195 \mathrm{~Hz}\) ?

Short Answer

Expert verified
The next higher harmonic frequency after 195 Hz is 260 Hz.

Step by step solution

01

Identify the Harmonic Numbers

The problem provides two consecutive harmonic frequencies: \(325 \mathrm{~Hz}\) and \(390 \mathrm{~Hz}\). To find the harmonic numbers \(n\) and \(n+1\) corresponding to these frequencies, we use the formula for harmonic frequencies \(f_n = n \cdot f_1\), where \(f_1\) is the fundamental frequency.
02

Determine Frequency Difference

Calculate the difference between the two given frequencies: \(390 \mathrm{~Hz} - 325 \mathrm{~Hz} = 65 \mathrm{~Hz}\). This difference equals the fundamental frequency \(f_1\), since it is the difference between consecutive harmonics \((f_{n+1} - f_n = f_1)\).
03

Calculate Fundamental Frequency

From Step 2, we established that \(f_1 = 65 \mathrm{~Hz}\).
04

Calculate Harmonic Number for 195 Hz

To find what harmonic number corresponds to \(195 \mathrm{~Hz}\), use \(f_n = n \times 65\). Solving \(n = \frac{195}{65} = 3\). Thus, \(195 \mathrm{~Hz}\) is the 3rd harmonic.
05

Find the Next Higher Harmonic after 195 Hz

Since \(195 \mathrm{~Hz}\) is the 3rd harmonic, the next higher harmonic is the 4th harmonic. Calculate the 4th harmonic: \(f_4 = 4 \times 65 = 260 \mathrm{~Hz}\).

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Key Concepts

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

Fundamental Frequency
The fundamental frequency, often denoted as \( f_1 \), is the lowest frequency produced by a vibrating string or instrument. It serves as the base frequency upon which other harmonic frequencies build. Like a foundation in a building, the fundamental frequency determines the overall structure or sound profile of the object.

The equation used to calculate any harmonic frequency is \( f_n = n \cdot f_1 \), where \( n \) represents the harmonic number. This means each harmonic frequency is a multiple of the fundamental frequency:
  • First harmonic (fundamental frequency): \( f_1 \)
  • Second harmonic: \( 2 \times f_1 \)
  • Third harmonic: \( 3 \times f_1 \)
Understanding the fundamental frequency is crucial because it allows for the prediction and analysis of the entire harmonic sequence for a given system, such as a musical string or column of air.
Harmonic Number
Harmonic numbers label the sequence of frequencies generated by the harmonics of a vibrating object. The first harmonic corresponds to the fundamental frequency, while each subsequent harmonic frequency is a multiple of the fundamental frequency.

The harmonic number, \( n \), indicates how many times the fundamental frequency is multiplied to reach a particular harmonic frequency. For example, if \( n = 3 \), the frequency is the third harmonic, calculated with \( f_3 = 3 \times f_1 \). This concept allows us to
  • Identify the position of a frequency within the harmonic series
  • Determine the sequence of available harmonic frequencies
To find the next higher harmonic after a given frequency, simply increment the harmonic number by 1 and multiply it by the fundamental frequency. This rhythmic increase in frequency defines the sequence and allows musicians and engineers to thoroughly understand and manipulate sound.
Frequency Difference
The difference in frequency between consecutive harmonics provides valuable insights into the properties of the system. Often, this difference is precisely equal to the fundamental frequency itself.

When you have two consecutive harmonic frequencies, you can calculate this difference by subtracting the lower frequency from the higher one: \( f_{n+1} - f_n \). In a practical example:
  • Given two harmonic frequencies: \( 325 \) Hz and \( 390 \) Hz
  • The frequency difference is: \( 390 \text{ Hz} - 325 \text{ Hz} = 65 \text{ Hz} \)
This difference tells us that \( 65 \text{ Hz} \) is the fundamental frequency, reaffirming how it guides the spacing of all subsequent harmonics. Recognizing these differences can allow musicians to tune instruments accurately or acousticians to adjust for sound quality and clarity in various environments.

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Most popular questions from this chapter

A \(120 \mathrm{~cm}\) length of string is stretched between fixed supports. What are the (a) longest, (b) second longest, and (c) third longest wavelength for waves traveling on the string if standing waves are to be set up? (d) Sketch those standing waves.

A continuous traveling wave with amplitude \(A\) is incident on a boundary. The continuous reflection, with a smaller amplitude \(B\), travels back through the incoming wave. The resulting interference pattern is displayed in Fig. 16-51. The standing wave ratio is defined to be \(\mathrm{SWR}=\frac{A+B}{A-B}\) The reflection coefficient \(R\) is the ratio of the power of the reflected wave to the power of the incoming wave and is thus proportional to the ratio \((B / A)^{2} .\) What is the SWR for (a) total reflection and (b) no reflection? (c) For SWR \(=1.50\), what is \(R\) expressed as a percentage?

A sinusoidal wave of angular frequency \(1200 \mathrm{rad} / \mathrm{s}\) and amplitude \(3.00 \mathrm{~mm}\) is sent along a cord with linear density \(2.00 \mathrm{~g} / \mathrm{m}\) and tension \(1200 \mathrm{~N}\). (a) What is the average rate at which energy is transported by the wave to the opposite end of the cord? (b) If, simultaneously, an identical wave travels along an adjacent, identical cord, what is the total average rate at which energy is transported to the opposite ends of the two cords by the waves? If, instead, those two waves are sent along the same cord simultaneously, what is the total average rate at which they transport energy when their phase difference is (c) \(0,(\mathrm{~d}) 0.4 \pi \mathrm{rad}\), and \((\mathrm{e}) \pi \mathrm{rad} ?\)

A standing wave results from the sum of two transverse traveling waves given by $$ y_{1}=0.050 \cos (\pi x-4 \pi t) $$ and $$ y_{2}=0.050 \cos (\pi x+4 \pi t) $$ where \(x, y_{1}\), and \(y_{2}\) are in meters and \(t\) is in seconds. (a) What is the smallest positive value of \(x\) that corresponds to a node? Beginning at \(t=0\), what is the value of the (b) first, (c) second, and (d) third time the particle at \(x=0\) has zero velocity?

A standing wave pattern on a string is described by $$ y(x, t)=0.040(\sin 5 \pi x)(\cos 40 \pi t) $$ where \(x\) and \(y\) are in meters and \(t\) is in seconds. For \(x \geq 0\), what is the location of the node with the (a) smallest, (b) second smallest, and (c) third smallest value of \(x\) ? (d) What is the period of the oscillatory motion of any (nonnode) point? What are the (e) speed and (f) amplitude of the two traveling waves that interfere to produce this wave? For \(t \geq 0\), what are the \((\mathrm{g})\) first, \((\mathrm{h})\) second, and (i) third time that all points on the string have zero transverse velocity?
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