Chapter 16, Problem 3OQ

(a)

To determine

Rank the functions from largest to the smallest according to their amplitude.

Answer to Problem 3OQ

The ranking of functions from largest to the smallest according to their amplitude is $\underset{\_}{\left(\text{c}\right)=\left(\text{d}\right)>\left(\text{e}\right)>\left(\text{b}\right)>\left(\text{a}\right)}$.

Explanation of Solution

Write the general expression for a sinusoidal wave.

  $y=A\sin \left(kx-\omega t+\varphi \right)$                                                                                                (I)

Here, $A$ is the amplitude, $k$ is the wave number, $t$ is the time, $\omega$ is the angular frequency, and $\varphi$ is the phase of wave.

Consider wave function (a).

  $y=4\sin \left(3x-15t\right)$                                                                                                   (II)

Compare equation (I) and (II). The amplitude of the wave (a) is $A=4$, $k=3$, and $\omega =15$.

Consider wave function (b).

  $y=6\cos \left(3x+15t-2\right)$                                                                                            (III)

Compare equation (I) and (III). The amplitude of the wave (b) is $A=6$, $k=3$, and $\omega =15$.

Consider wave function (c).

  $y=8\sin \left(2x+15t\right)$                                                                                                  (IV)

Compare equation (I) and (IV). The amplitude of the wave (c) is $A=8$, $k=2$, and $\omega =15$.

Consider wave function (d).

  $y=8\cos \left(4x+20t\right)$                                                                                                  (V)

Compare equation (I) and (V). The amplitude of the wave (d) is $A=8$, $k=4$, and $\omega =20$.

Consider wave function (e).

  $y=7\sin \left(6x-24t\right)$                                                                                                 (VI)

Compare equation (I) and (VI). The amplitude of the wave (e) is $A=7$, $k=6$, and $\omega =24$.

Thus the raking of amplitude each wave from largest to the smallest is (c)=(d)>(e)>(b)>(a).

Conclusion:

Therefore, the ranking of wave functions from largest to the smallest according to their amplitude is $\underset{\_}{\left(\text{c}\right)=\left(\text{d}\right)>\left(\text{e}\right)>\left(\text{b}\right)>\left(\text{a}\right)}$.

(b)

To determine

Rank the functions from largest to the smallest according to their wavelength.

Answer to Problem 3OQ

The ranking of functions from largest to the smallest according to their wavelength is $\underset{\_}{\left(\text{c}\right)>\left(\text{a}\right)=\left(\text{b}\right)>\left(\text{d}\right)>\left(\text{e}\right)}$.

Explanation of Solution

Write the expression for wavelength.

  $\lambda =\frac{2\pi }{k}$                                                                                                                  (VII)

Conclusion:

Substitute, $3$ for $k$ in equation (VII) to obtain the wavelength of wave (a).

  $\begin{array}{c}\lambda =\frac{2\pi }{3}\\ =2.09\end{array}$

Substitute, $3$ for $k$ in equation (VII) to obtain the wavelength of wave (b).

  $\begin{array}{c}\lambda =\frac{2\pi }{3}\\ =2.09\end{array}$

Substitute, $2$ for $k$ in equation (VII) to obtain the wavelength of wave (c).

  $\begin{array}{c}\lambda =\frac{2\pi }{2}\\ =3.14\end{array}$

Substitute, $4$ for $k$ in equation (VII) to obtain the wavelength of wave (d).

  $\begin{array}{c}\lambda =\frac{2\pi }{4}\\ =1.57\end{array}$

Substitute, $6$ for $k$ in equation (VII) to obtain the wavelength of wave (e).

  $\begin{array}{c}\lambda =\frac{2\pi }{6}\\ =1.04\end{array}$

Thus, the ranking of wavelength from larger to smaller is (c)>(a)=(b)>(d)>(e).

Therefore, the ranking of functions from largest to the smallest according to their wavelength is $\underset{\_}{\left(\text{c}\right)>\left(\text{a}\right)=\left(\text{b}\right)>\left(\text{d}\right)>\left(\text{e}\right)}$.

(c)

To determine

Rank the functions from largest to the smallest according to their frequencies.

Answer to Problem 3OQ

The ranking of functions from largest to the smallest according to their frequency is $\underset{\_}{\left(\text{e}\right)>\left(\text{d}\right)>\left(\text{a}\right)=\left(\text{b}\right)=\left(\text{c}\right)}$.

Explanation of Solution

Write the expression for frequency.

  $f=\frac{\omega }{2\pi }$                                                                                                                 (VIII)

Conclusion:

Substitute, $15$ for $\omega$ in equation (VIII) to obtain the frequency of wave (a).

  $\begin{array}{c}f=\frac{15}{2\pi }\\ =2.4\end{array}$

Substitute, $15$ for $\omega$ in equation (VIII) to obtain the frequency of wave (b).

  $\begin{array}{c}f=\frac{15}{2\pi }\\ =2.4\end{array}$

Substitute, $15$ for $\omega$ in equation (VIII) to obtain the frequency of wave (c).

  $\begin{array}{c}f=\frac{15}{2\pi }\\ =2.4\end{array}$

Substitute, $20$ for $\omega$ in equation (VIII) to obtain the frequency of wave (d).

  $\begin{array}{c}f=\frac{20}{2\pi }\\ =3.2\end{array}$

Substitute, $24$ for $\omega$ in equation (VIII) to obtain the frequency of wave (e).

  $\begin{array}{c}f=\frac{24}{2\pi }\\ =3.8\end{array}$

Thus, the ranking of frequencies of wave from largest to smallest is (e)>(d)>(a)=(b)=(c).

Therefore, the ranking of functions from largest to the smallest according to their frequency is $\underset{\_}{\left(\text{e}\right)>\left(\text{d}\right)>\left(\text{a}\right)=\left(\text{b}\right)=\left(\text{c}\right)}$.

(d)

To determine

Rank the functions from largest to the smallest according to their period.

Answer to Problem 3OQ

The ranking of functions from largest to the smallest according to their period is $\underset{\_}{\left(\text{a}\right)=\left(\text{b}\right)=\left(\text{c}\right)>\left(\text{d}\right)>\left(\text{e}\right)}$.

Explanation of Solution

Write the expression for period.

  $T=\frac{1}{f}$                                                                                                                     (IX)

Conclusion:

Since frequency is inversely proportional to time period, the ranking will be the reverse order of the ranking in part (c).

Thus, the ranking of functions from largest to the smallest according to their period is $\left(\text{a}\right)=\left(\text{b}\right)=\left(\text{c}\right)>\left(\text{d}\right)>\left(\text{e}\right)$.

Therefore, the ranking of functions from largest to the smallest according to their period is $\underset{\_}{\left(\text{a}\right)=\left(\text{b}\right)=\left(\text{c}\right)>\left(\text{d}\right)>\left(\text{e}\right)}$.

(e)

To determine

Rank the functions from largest to the smallest according to their speed.

Answer to Problem 3OQ

The ranking of functions from largest to the smallest according to their speed is $\underset{\_}{\left(\text{c}\right)>\left(\text{a}\right)=\left(\text{b}\right)=\left(\text{d}\right)>\left(\text{e}\right)}$.

Explanation of Solution

Write the expression for speed.

  $v=f\lambda$                                                                                                                      (X)

Conclusion:

Substitute, $2.09$ for $\lambda$, and $2.4$ for $f$ in equation (X) to obtain the speed of wave (a).

  $\begin{array}{c}v=\left(2.09\right)\left(2.4\right)\\ =5.02\end{array}$

Substitute, $2.09$ for $\lambda$, and $2.4$ for $f$ in equation (X) to obtain the speed of wave (b).

  $\begin{array}{c}v=\left(2.09\right)\left(2.4\right)\\ =5.02\end{array}$

Substitute, $3.14$ for $\lambda$, and $2.4$ for $f$ in equation (X) to obtain the speed of wave (c).

  $\begin{array}{c}v=\left(3.14\right)\left(2.4\right)\\ =7.5\end{array}$

Substitute, $1.57$ for $\lambda$, and $3.2$ for $f$ in equation (X) to obtain the speed of wave (d).

  $\begin{array}{c}v=\left(1.57\right)\left(2.4\right)\\ =5.02\end{array}$

Substitute, $1.04$ for $\lambda$, and $3.8$ for $f$ in equation (X) to obtain the speed of wave (e).

  $\begin{array}{c}v=\left(1.04\right)\left(3.8\right)\\ =3.9\end{array}$

Thus, the ranking of speed of the wave from largest to smallest is (c)>(a)=(b)=(d)>(e).

Therefore, ranking of functions from largest to the smallest according to their speed is $\underset{\_}{\left(\text{c}\right)>\left(\text{a}\right)=\left(\text{b}\right)=\left(\text{d}\right)>\left(\text{e}\right)}$.