Chapter 13, Problem 3OQ

Rank the waves represented by the following functions from the largest to the smallest according to (i) their amplitudes, (ii) their wavelengths, (iii) their frequencies, (iv) their periods, and (v) their speeds. If the values of a quantity are equal for two waves, show them as having equal rank. For all functions, x and y are in meters and t is in seconds. (a) y = 4 sin (3x − 15t) (b) y = 6 cos (3x + 15t − 2) (c) y = 8 sin (2x + 15t) (d) y = 8 cos (4x + 20t) (e) y = 7 sin (6x + 24t)

To determine

Rank the waves in the decreasing order of their amplitudes.

Answer to Problem 3OQ

The order is (c) = (d) > (e) > (b) > (a).

Explanation of Solution

Write the general expression which denotes a sinusoidal wave travelling in +ve x-direction.

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

Here, $y$ is the displacement, $A$ is the amplitude, $k$ is the wave number, $\omega$ is the angular frequency, and $\theta$ is the initial phase.

Wave can be represented as cosine function also.

    $y=A\cos \left(kx-\omega t+\theta \right)$

Write the amplitude of $y=4\sin \left(3x-15t\right)$.

    $A=4$

Write the amplitude of $y=6\cos \left(3x+15t-2\right)$.

    $A=6$

Write the amplitude of $y=8\sin \left(2x+15t\right)$.

    $A=6$

Write the amplitude of $y=8\sin \left(4x+20t\right)$.

    $A=8$

Write the amplitude of $y=7\sin \left(6x-24t\right)$.

    $A=8$

Conclusion:

Therefore, the order is (c) = (d) > (e) > (b) > (a).

To determine

Rank the waves in the decreasing order of their wavelengths.

Answer to Problem 3OQ

The order is (c) > (a) = (b) > (d) > (e).

Explanation of Solution

From the general expression for wave in part (a), it is understood that the coefficient of $x$ denotes the value of $k$.

Write the relation between wavenumber and wavelength.

    $\lambda =\frac{2\pi }{k}$

Here, $\lambda$ is the wavelength.

It means that higher the value of $k$, lower will be the value of $\lambda$.

Conclusion:

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

Substitute $3$ for $k$ in the equation for $\lambda$.

    $\lambda =\frac{2\pi }{3}$

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

Substitute $3$ for $k$ in the equation for $\lambda$.

    $\lambda =\frac{2\pi }{3}$

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

Substitute $2$ for $k$ in the equation for $\lambda$.

    $\lambda =\frac{2\pi }{2}$

Consider $y=8\sin \left(4x+20t\right)$.

Substitute $4$ for $k$ in the equation for $\lambda$.

    $\lambda =\frac{2\pi }{4}$

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

Substitute $8$ for $k$ in the equation for $\lambda$.

$\lambda =\frac{2\pi }{6}$

Therefore, the order is (c) > (a) = (b) > (d) > (e).

To determine

Rank the waves in the decreasing order of their frequencies.

Answer to Problem 3OQ

The order is (e) > (d) > (a) = (b) = (c).

Explanation of Solution

Write the relation between linear and angular frequency.

    $f=\frac{2\pi }{\omega }$

Here, $f$ is the frequency and $\omega$ is the angular frequency.

The coefficient of $t$ gives the value of $\omega$. The above relation tells that equation with lowest value of $\omega$ will have largest value of $f$.

Conclusion:

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

Substitute $15$ for $\omega$ in the equation for $f$.

    $f=\frac{2\pi }{15}$

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

Substitute $15$ for $\omega$ in the equation for $f$.

    $f=\frac{2\pi }{15}$

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

Substitute $15$ for $\omega$ in the equation for $f$.

    $f=\frac{2\pi }{15}$

Consider $y=8\sin \left(4x+20t\right)$.

Substitute $20$ for $\omega$ in the equation for $f$.

    $f=\frac{2\pi }{20}$

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

Substitute $24$ for $\omega$ in the equation for $f$.

    $f=\frac{2\pi }{24}$

Therefore, the order is order is (e) > (d) > (a) = (b) = (c).

To determine

Rank the waves in the decreasing order of their periods.

Answer to Problem 3OQ

The order is (a) = (b) = (c) > (d) > (e).

Explanation of Solution

Write the relation between $f$ and period.

    $T=\frac{1}{f}$

Here, $T$ is the time period.

$\omega$. The above relation tells that equation with lowest value of $f$ will have largest value of $T$. The decreasing order of frequencies is (e) > (d) > (a) = (b) = (c). From the above relation, it can be tell that the decreasing according to $T$ is (a) = (b) = (c) > (d) > (e).

Conclusion:

Therefore, the order is (a) = (b) = (c) > (d) > (e).

To determine

Rank the waves in the decreasing order of their speeds.

Answer to Problem 3OQ

The order is (c) > (a) = (b) = (d) > (e).

Explanation of Solution

Write the equation to find speed from wave number and angular frequency.

    $v=\frac{\omega }{k}$

Conclusion:

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

Substitute $15$ for $\omega$ and $3$ for $k$ in the equation for $v$.

    $v=\frac{15}{3}$

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

Substitute $15$ for $\omega$ and $3$ for $k$ in the equation for $v$.

    $v=\frac{15}{3}$

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

Substitute $15$ for $\omega$ and $2$ for $k$ in the equation for $v$.

    $v=\frac{15}{2}$

Consider $y=8\sin \left(4x+20t\right)$.

Substitute $20$ for $\omega$ and $4$ for $k$ in the equation for $v$.

    $v=\frac{20}{4}$

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

Substitute $24$ for $\omega$ and $6$ for $k$ in the equation for $v$.

    $v=\frac{24}{6}$

Therefore, the order is(c) > (a) = (b) = (d) > (e).