Question

The voltage in a telegraph line is, under certain circumstances, given by $$ f(x, t)=A \sin \left(t-\frac{x}{v}\right)+A \sin \left(t-2 \tau+\frac{x}{v}\right) $$ where \(v\) is the velocity and \(t\) is the time for a wave to travel the length of the line. This represents the combination of a wave starting at one end and the reflection of the wave from the other end. What are the wavelengths and frequencies of the waves? Sketch the graphs of \(f(x, t)\) when \(A=2, v=3, \tau=1\), both as a function of \(t\) for \(x=2\) (over the time interval for which \(t\) is between 1 and 2 ) and as a function of \(x\) (between 0 and 3 ) when \(t=\frac{3}{2}\).

Short answer

The wavelength is 3, and the frequency is 1 Hz.

Step-by-step solution

Identify the given parameters and the equation

The given function is \(f(x, t) = A \sin \( t - \frac{x}{v} \) + A \sin \( t - 2\tau + \frac{x}{v} \)\). The provided constants are: \(A = 2\)\(v = 3\)\(\tau = 1\). This represents the combination of the initial wave and its reflection.

Determine the wave equation components

The general form of a sinusoidal wave is \(A \sin(\omega t - kx)\) where \(\omega\) is the angular frequency and \(k\) is the wave number. Comparing it to the given function, the expression for the first wave becomes \(\sin\left(t - \frac{x}{v}\right)\). This implies that the wave travels with velocity \(v\) and has phase \(t - \frac{x}{v}\).

Calculate the wavelength and frequency

From the relation \(\sin(t - \frac{x}{v})\), identify that the wavelength \(\lambda\) and wave number \(k\) are related as \(k = \frac{2\pi}{\lambda}\). Since \(k = \frac{2\pi}{v}\), the wavelength \(\lambda = v= 3\). The frequency \(f\) is related to the wavelength and velocity by \(f = \frac{v}{\lambda} = \frac{3}{3} = 1\ \text{Hz}\).

Graph the wave as a function of time

Set \(x = 2\) and plug into the equation: \[f(2, t) = 2\left( \sin\left(t - \frac{2}{3}\right) + \sin\left(t - 2(1) + \frac{2}{3}\right) \right)\]Plot this function within the interval \(1 \leq t \leq 2\).

Graph the wave as a function of position

Set \(t = \frac{3}{2}\) and plug into the equation: \[f(x, \frac{3}{2}) = 2\left( \sin\left(\frac{3}{2} - \frac{x}{3}\right) + \sin\left(\frac{1}{2} + \frac{x}{3}\right) \right)\]Plot this function within the interval \(0 \leq x \leq 3\).

Key concepts

Sinusoidal Waves

A sinusoidal wave is a mathematical representation of a wave that oscillates periodically. It can be described by the formula \(A \, \sin (\omega t - kx)\), where \(A\) is the amplitude, \(\omega\) is the angular frequency, and \(k\) is the wave number. Sinusoidal waves are fundamental in understanding wave behavior, as they describe how the wave moves and oscillates over time.

The given exercise deals with a telegraph line voltage that can be represented as a combination of two sinusoidal waves:

• The first part \(A \sin (t - \frac{x}{v})\) describes the original wave.
• The second part \(A \sin (t - 2\tau + \frac{x}{v})\) represents the reflected wave.

By combining these two parts, we understand how the superposition of waves happens, resulting in the overall function \(f(x, t)\).

Wave Frequency

Wave frequency describes how many times a wave cycle repeats per second. It's typically measured in Hertz (Hz). For sinusoidal waves, the frequency is an essential property as it determines the wave's periodic nature.

In the given problem, the wave function includes both an original and a reflected wave. To find the frequency, we use the relation between velocity \(v\), wavelength \(\lambda\), and frequency \(f\). From the calculation in the solution:

• Velocity \(v = 3\)
• Wavelength \(\lambda = 3\)

Therefore, the frequency can be found using the formula \(f = \frac{v}{\lambda} \), which gives us a frequency \(f = 1\text{Hz}\). This means the wave repeats its cycle once every second.

Wavelength Calculation

Wavelength \(\lambda\) is the distance between two consecutive points that are in phase, such as crests or troughs of the wave. It is an essential property that helps in understanding wave propagation.

In the exercise, wavelength calculation is crucial for analyzing the given wave function. According to the wave equation \(A \sin (t - \frac{x}{v})\), the wave number \(k\) and wavelength \(\lambda\) are related by \(k = \frac{2\pi}{\lambda}\).
Since the function \(\sin (t - \frac{x}{v})\) shows a wave traveling with velocity \(v\) and phase \(t - \frac{x}{v}\), comparing the given equation's components helps us find:

• \(v = 3\)
• \(k = \frac{2\pi}{v}\)

Thus, the wavelength can be calculated as \(\lambda = v = 3\). This indicates that the distance over which the wave's shape repeats is 3 units.

Telegraph Line Voltage

Telegraph line voltage represents the electric potential difference in a telegraph system. Understanding the voltage behavior helps in determining the nature of signal transmission over the line.

In the given exercise, the voltage is given by the function \(f(x, t) = A \sin (t - \frac{x}{v}) + A \sin (t - 2\tau + \frac{x}{v})\). Here, both components represent sinusoidal waves. The first term models the initial wave, while the second term accounts for the wave's reflection after hitting the other end of the telegraph line.

• Velocity \(v = 3\)
• Amplitude \(A = 2\)
• \(\tau = 1\)

This superposition principle explains how the initial voltage and its reflected counterpart combine, leading to complex wave behavior along the telegraph line. Analyzing this function helps in understanding signal transmission dynamics in communication systems.