Chapter 10, Problem 1RQ

To determine

Identify the incorrect form of the given waveform.

Answer to Problem 1RQ

The incorrect form of the given waveform is, “$\underset{\_}{\left(\text{b}\right)\text{ sin}\left(\beta z-\omega t-\frac{\pi }{2}\right)}$”.

Explanation of Solution

Description:

Write the expression of the given waveform:

$E_x=\text{cos}\left(\omega t-\beta z\right)$        (1)

Rewrite the expression in option (a) as follows:

$\begin{array}{c}{E}_x=\text{cos}\left[-\left(\omega t-\beta z\right)\right]\\ =\text{cos}\left(\omega t-\beta z\right)\left\{\because \cos \left(-\psi \right)=\cos \psi \right\}\end{array}$

As the obtained expression is equivalent to the expression in Equation (1), the expression in option (a) is the correct form of the given waveform.

Rewrite the expression in option (b):

$\begin{array}{c}{E}_x=\text{sin}\left(\beta z-\omega t-\frac{\pi }{2}\right)\\ =\text{sin}\left[-\left(\frac{\pi }{2}+\omega t-\beta z\right)\right]\\ =-\text{sin}\left(\frac{\pi }{2}+\omega t-\beta z\right)\left\{\because \text{sin}\left(-\psi \right)=-\sin \psi \right\}\\ =-\text{sin}\left[\frac{\pi }{2}-\left(\beta z-\omega t\right)\right]\end{array}$

Simplify the expression as follows:

$\begin{array}{c}{E}_x=-\text{cos}\left(\beta z-\omega t\right)\left\{\because \sin \left(\frac{\pi }{2}-\psi \right)=\cos \psi \right\}\\ =-\text{cos}\left[-\left(\omega t-\beta z\right)\right]\\ =-\text{cos}\left(\omega t-\beta z\right)\left\{\because \cos \left(-\psi \right)=\cos \psi \right\}\end{array}$

As the obtained expression is not equivalent to the expression in Equation (1), the expression in option (b) is the incorrect form of the given waveform.

Rewrite the expression in option (c):

$\begin{array}{c}{E}_x=\text{cos}\left(\frac{2\pi }{T}t-\frac{2\pi }{\lambda }z\right)\\ =\text{cos}\left(\omega t-\beta z\right)\left\{\begin{array}{l}\because \omega =\frac{2\pi }{T}\\ \because \beta =\frac{2\pi }{\lambda }\end{array}\right\}\end{array}$

As the obtained expression is equivalent to the expression in Equation (1), the expression in option (c) is the correct form of the given waveform.

As the real component of the given waveform is $\text{Re}\left[e^{j\left(\omega t-\beta z\right)}\right]$, the expression in option (d) is the correct form of the given waveform.

Rewrite the expression in option (e):

$\begin{array}{c}{E}_x=\text{cos}\left(\beta z-\beta ut\right)\\ =\text{cos}\left[\beta z-\beta \left(\frac{\omega }{\beta }\right)t\right]\left\{\because u=\frac{\omega }{\beta }\right\}\\ =\text{cos}\left[-\left(\omega t-\beta z\right)\right]\\ =\text{cos}\left(\omega t-\beta z\right)\left\{\because \cos \left(-\psi \right)=\cos \psi \right\}\end{array}$

As the obtained expression is equivalent to the expression in Equation (1), the expression in option (e) is the correct form of the given waveform.

Conclusion:

Thus, the incorrect form of the given waveform is, “$\underset{\_}{\left(\text{b}\right)\text{ sin}\left(\beta z-\omega t-\frac{\pi }{2}\right)}$”.