Question

Write an equation for a sinusoidal radio wave of amplitude 10 and frequency$\mathbf{600}\mathbf{}\mathbf{kilohertz}$. Hint: The velocity of a radio wave is the velocity of light,$\mathbf{c}\mathbf{=}\mathbf{3}\mathbf{\times }{\mathbf{10}}^{\mathbf{8}}\mathbf{}\mathbf{m}\mathbf{/}\mathbf{sec}$

Short answer

The equation for a sinusoidal radio wave is $\mathrm{y}=10\sin \left[1200\times {10}^3\mathrm{\pi }\left(\frac{\mathrm{x}}{3\times {10}^8}-\mathrm{t}\right)\right]$.

Step-by-step solution

Given data

The dimensions are given as follows:

Amplitude,$A=10.$ .

Frequency,$f=600kilohertz.$

Velocity,$c=3\times {10}^{8}m/sec.$

Concept of Sinusoidal sound wave

The trigonometric cosine function can be used to explain a sinusoid's unique functional form, and the most fundamental sinusoid can be written as$\mathbf{A}\mathbf{}\mathbf{sin}\mathbf{}\mathbf{(}\mathbf{}\mathbf{2}\mathbf{\pi }\mathbf{(}\mathbf{ft}\mathbf{+}\mathbf{\varphi }\mathbf{\left)}\mathbf{\right)}$

Calculation to find the dimension of the wave equation

Following is a standard wave equation, $y=A\sin \left(\frac{2\pi }{\lambda }(x-vt)\right)$

Time period is the time required to travel a distance of$\lambda$ .

So, $\left(dis\tan ce\right)=\left(speed\right)\left(time\right)$

$\lambda =vT$

Since,

$$\begin{gathered} T=\frac{1}{f} \\ \lambda =\frac{v}{f} \end{gathered}$$

Hence, $v=\lambda f.$

Put the value of the obtained dimensions in the sinusoidal wave equation

Now, solve the wave equation as follows:

$\mathrm{y}=\mathrm{Asin}\left(\frac{2\mathrm{\pi }}{\mathrm{\lambda }}(\mathrm{x}-\mathrm{vt})\right)\mathrm{y}=\mathrm{Asin}\left(\frac{2\mathrm{\pi v}}{\mathrm{\lambda }}\left(\frac{\mathrm{x}}{\mathrm{v}}-\mathrm{t}\right)\right)$

Since, $\mathrm{v}=\mathrm{\lambda f}.$

$$\begin{gathered} \mathrm{y}=\mathrm{Asin}\frac{2\mathrm{\pi }\left(\mathrm{\lambda f}\right)}{\mathrm{\lambda }}\left(\frac{\mathrm{x}}{\mathrm{v}}-\mathrm{t}\right) \\ \mathrm{y}=\mathrm{Asin}2\mathrm{\pi f}\left(\frac{\mathrm{x}}{\mathrm{v}}-\mathrm{t}\right) \end{gathered}$$

Now putting all the value in the above equation as follows:

$\mathrm{y}=10\sin \left[1200\times {10}^3\mathrm{\pi }\left(\frac{\mathrm{x}}{3\times {10}^8}-\mathrm{t}\right)\right]$