Chapter 11, Problem 74P

(a)

To determine

The three lowest standing wave frequencies.

Answer to Problem 74P

The three lowest standing wave frequencies is $f_2=600.0\text{Hz}$, $f_3=900.0\text{Hz}$, and $f_4=1.200\text{kHz}$.

Explanation of Solution

Write the equation for lowest standing wave frequency.

$f_n=\frac{nv}{2L}$ (I)

Here, $f_n$ is the fundamental frequency, $n$ is the harmonic number, $v$ is the speed of the sound wave, and $L$ is the length of the string.

Conclusion:

The first lowest fundamental frequency of the standing wave is.

$f_1=300.0\text{Hz}$

The second lowest fundamental frequency of the standing wave is.

$f_2=2f_1$ (II)

Substitute $300.0\text{Hz}$ for $f_1$ in equation (II).

$\begin{array}{c}{f}_2=2\left(300.0\text{Hz}\right)\\ =600.0\text{Hz}\end{array}$

The third lowest fundamental frequency of the standing wave is.

$f_3=3f_1$ (III)

Substitute $300.0\text{Hz}$ for $f_1$ in equation (III).

$\begin{array}{c}{f}_3=3\left(300.0\text{Hz}\right)\\ =900.0\text{Hz}\end{array}$

The fourth lowest fundamental frequency of the standing wave is.

$f_4=4f_1$ (IV)

Substitute $300.0\text{Hz}$ for $f_1$ in equation (IV).

$\begin{array}{c}{f}_4=4\left(300.0\text{Hz}\right)\\ =1200.0\text{Hz}\\ =1.200\times {10}^3\text{Hz}\left(\frac{{10}^{-3}\text{Hz}}{1\text{kHz}}\right)\\ =1.200\text{kHz}\end{array}$

The pattern for the each situation for the three lowest standing wave frequencies is.

$n$	pattern	$n{f}_1$
2	N   A   N   A   N	600.0Hz
3	N A N A N A N	900.0Hz
4	NANANANAN	1.200Hz

Therefore, the three lowest standing wave frequencies is $f_2=600.0\text{Hz}$, $f_3=900.0\text{Hz}$, and $f_4=1.200\text{kHz}$.

(b)

To determine

The four lowest standing wave frequencies, when the strings press light.

Answer to Problem 74P

The four lowest standing wave frequencies is $f_2=600.0\text{Hz}$, $f_4=1.200\text{kHz}$, $f_6=1.800\text{kHz}$, and $f_8=2.400\text{kHz}$.

Explanation of Solution

Write the equation for lowest standing wave for fundamental frequency.

$f_n=n{f}_1$ (V)

Here, $f_n$ is the fundamental frequency for $n^{\text{th}}$ harmonic, $n$ is the harmonic number, and $f_1$ is the fundamental frequency.

Conclusion:

The lowest fundamental frequency of the standing wave is.

$f_1=300.0\text{Hz}$

Here, the lowest frequency is starts from $f_2$ and only even harmonics are allowed.

The first lowest fundamental frequency of the standing wave is.

$f_2=2f_1$ (VI)

Substitute $300.0\text{Hz}$ for $f_1$ in equation (VI).

$\begin{array}{c}{f}_2=2\left(300.0\text{Hz}\right)\\ =600.0\text{Hz}\end{array}$

The second lowest fundamental frequency of the standing wave is.

$f_4=4f_1$ (VII)

Substitute $300.0\text{Hz}$ for $f_1$ in equation (VII).

$\begin{array}{c}{f}_4=4\left(300.0\text{Hz}\right)\\ =1200.0\text{Hz}\\ =1.200\times {10}^3\text{Hz}\left(\frac{{10}^{-3}\text{Hz}}{1\text{kHz}}\right)\\ =1.200\text{kHz}\end{array}$

The third lowest fundamental frequency of the standing wave is.

$f_6=6f_1$ (VIII)

Substitute $300.0\text{Hz}$ for $f_1$ in equation (VIII).

$\begin{array}{c}{f}_6=6\left(300.0\text{Hz}\right)\\ =1800.0\text{Hz}\\ =1.800\times {10}^3\text{Hz}\left(\frac{{10}^{-3}\text{Hz}}{1\text{kHz}}\right)\\ =1.800\text{kHz}\end{array}$

The fourth lowest fundamental frequency of the standing wave is.

$f_8=8f_1$ (IX)

Substitute $300.0\text{Hz}$ for $f_1$ in equation (IX).

$\begin{array}{c}{f}_8=8\left(300.0\text{Hz}\right)\\ =2400.0\text{Hz}\\ =2.400\times {10}^3\text{Hz}\left(\frac{{10}^{-3}\text{Hz}}{1\text{kHz}}\right)\\ =2.400\text{kHz}\end{array}$

The pattern for the each situation for the four lowest standing wave frequencies is.

$n$	pattern	$n{f}_1$
2	N    A    N    A    N	600.0Hz
4	N   A   N   A   N   A   N   A   N	1.200kHz
6	N  A  N  A  N  A  N  A  N  A  N  A  N	1.800kHz
8	NANANANANANANANAN	2.400kHz

Therefore, the four lowest standing wave frequencies is $f_2=600.0\text{Hz}$, $f_4=1.200\text{kHz}$, $f_6=1.800\text{kHz}$, and $f_8=2.400\text{kHz}$.

(c)

To determine

The four lowest standing wave frequencies, when the strings press hard.

Answer to Problem 74P

The four lowest standing wave frequencies is $f_1=600.0\text{Hz}$, $f_2=1.200\text{kHz}$, $f_3=1.800\text{kHz}$, and $f_4=2.400\text{kHz}$.

Explanation of Solution

The effective length of the string is now half of the original length.

Write the equation for lowest standing wave for fundamental frequency.

${f^{\prime }}_n=\frac{nv}{2\left(\frac{L}{2}\right)}$ (X)

Here, ${f^{\prime }}_n$ is the new fundamental frequency for $n^{\text{th}}$ harmonic.

Rearrange the equation (X).

${f^{\prime }}_n=\frac{nv}{L}$

Substitute equation (I) in above equation.

${f^{\prime }}_n=2f_n$ (XI)

Conclusion:

The first lowest fundamental frequency of the standing wave is.

$\begin{array}{c}{f^{\prime }}_1=2\left(300.0\text{Hz}\right)\\ =600.0\text{Hz}\end{array}$

The second lowest fundamental frequency of the standing wave is.

${f^{\prime }}_2=2f_2$

Substitute $600.0\text{Hz}$ for $f_2$.

$\begin{array}{c}{f^{\prime }}_2=2\left(600.0\text{Hz}\right)\\ =1.200\text{kHz}\end{array}$

The third lowest fundamental frequency of the standing wave is.

${f^{\prime }}_3=2f_3$

Substitute $900.0\text{Hz}$ for $f_3$.

$\begin{array}{c}{f^{\prime }}_3=2\left(900.0\text{Hz}\right)\\ =1800.0\text{Hz}\\ =1.800\times {10}^3\text{Hz}\left(\frac{{10}^{-3}\text{Hz}}{1\text{kHz}}\right)\\ =1.800\text{kHz}\end{array}$

The fourth lowest fundamental frequency of the standing wave is.

${f^{\prime }}_4=2f_4$

Substitute $1.200\text{kHz}$ for $f_4$.

$\begin{array}{c}{f^{\prime }}_4=2\left(1.200\text{kHz}\right)\\ =2.400\text{kHz}\end{array}$

The pattern for the each situation for the four lowest standing wave frequencies is.

$n$	pattern	${f^{\prime }}_n=2f_n$
1	N     A     N	600.0Hz
2	N    A    N    A    N	1.200kHz
3	N  A  N  A  N  A  N	1.800kHz
4	NANANANAN	2.400kHz

Therefore, the four lowest standing wave frequencies is $f_1=600.0\text{Hz}$, $f_2=1.200\text{kHz}$, $f_3=1.800\text{kHz}$, and $f_4=2.400\text{kHz}$.