Chapter 15, Problem 3SP

A pipe that is open at both ends will form standing waves, if properly excited, with antinodes near both ends of the pipe. Suppose we have an open pipe 60 cm in length.

a.    Sketch the standing-wave pattern for the fundamental standing wave for this pipe. (There will be a node in the middle, and antinodes at either end.)

b.    What is the wavelength of the sound waves that interfere to form the fundamental wave?

c.    If the speed of sound in air is 340 m/s, what is the frequency of this sound wave?

d.    If the air temperature increases so that the speed of sound is now 358 m/s. by how much does the frequency change?

e.    Sketch the standing-wave pattern and find the wavelength and frequency for the next harmonic in this pipe.

To determine

Sketch the sound-wave pattern for this pipe.

Answer to Problem 3SP

The fundamental wave pattern in the pipe is drawn.

Explanation of Solution

Conclusion:

Therefore, the fundamental sound-wave pattern is drawn.

To determine

The wavelength of the sound wave.

Answer to Problem 3SP

The wavelength is $1.2\text{m}$.

Explanation of Solution

Given Info: The length of the pipe is $60\text{cm}$.

Write the formula to calculate the fundamental wavelength inside the pipe.

$\frac{\lambda }{2}=L$

Here,

$\lambda$ is the fundamental wavelength inside the pipe

L is the length of the pipe

Substitute $60\text{cm}$ for L in the above equation to calculate $\lambda$.

$\begin{array}{c}\frac{\lambda }{2}=60\text{cm}\\ \lambda =2\left(60\text{cm}\right)\left(\frac{1\text{m}}{100\text{cm}}\right)\\ =1.2\text{m}\end{array}$

Conclusion:

Therefore, the wavelength is $1.2\text{m}$.

To determine

The frequency of the sound wave.

Answer to Problem 3SP

The frequency of the sound wave is $283.3\text{Hz}$.

Explanation of Solution

Given Info: The wavelength of the sound wave is $1.2\text{m}$ and speed is $340\text{m}/\text{s}$.

Write the expression to calculate the speed of the sound wave.

$v=f\lambda$

Here,

$\lambda$ is the wavelength of the sound wave

f is the frequency of the sound wave

Substitute $340\text{m}/\text{s}$ for v and $1.2\text{m}$ for $\lambda$ in the above equation to calculate f.

$\begin{array}{c}f\left(1.2\text{m}\right)=340\text{m}/\text{s}\\ f=\frac{340\text{m}/\text{s}}{1.2\text{m}}\\ =283.3\text{Hz}\end{array}$

Conclusion:

Therefore, the frequency of the sound wave is $283.3\text{Hz}$.

To determine

The change in frequency of the sound wave.

Answer to Problem 3SP

The change in frequency of the sound wave is $\text{15}\text{Hz}$.

Explanation of Solution

Given Info: The wavelength of the sound wave is The longest possible wavelength is $1.2\text{m}$ and the new speed is $358\text{m}/\text{s}$.

Write the expression to calculate the speed of the sound wave.

$v^{\prime }=f^{\prime }\lambda$

Here,

$v^{\prime }$ is the new speed of the sound

$f^{\prime }$ is the new frequency of the sound wave

Substitute $358\text{m}/\text{s}$ for $v^{\prime }$ and $1.2\text{m}$ for $\lambda$ in the above equation to calculate $f^{\prime }$.

$\begin{array}{c}{f}^{\prime }\left(1.2\text{m}\right)=358\text{m}/\text{s}\\ f^{\prime }=\frac{358\text{m}/\text{s}}{1.2\text{m}}\\ =298.3\text{Hz}\end{array}$

Write the formula to calculate the frequency change for the sound wave.

$\Delta f=f^{\prime }-f$

Here,

$\Delta f$ is the frequency change

Substitute $298.3\text{Hz}$ for $f^{\prime }$ and $283.3\text{Hz}$ for f in the above equation to calculate $\Delta f$.

$\begin{array}{c}\Delta f=298.3\text{Hz}-283.3\text{Hz}\\ =\text{15}\text{Hz}\end{array}$

Conclusion:

Therefore, the change in frequency of the sound wave is $\text{15}\text{Hz}$.

To determine

Sketch the standing-wave pattern for the next harmonic and find wavelength and frequency.

Answer to Problem 3SP

The frequency of the second harmonic wave is $566.6\text{Hz}$ and wave length is $60\text{cm}$, the sketch of the standing-wave is drawn.

Explanation of Solution

For the second harmonic, the frequency is twice as that of the fundamental standing-wave and wavelength is half of that value of the fundamental standing wave. Therefore the frequency of the second harmonic wave is $566.6\text{Hz}$ and wavelength is $60\text{cm}$.

The sketch of the second harmonic wave is shown below.

Conclusion:

Therefore, the frequency of the second harmonic wave is $566.6\text{Hz}$ and wave length is $60\text{cm}$, the sketch of the standing-wave is drawn.