Problem 89 - Fundamentals of Physics

Question

Two waves are described by $$ y_{1}=0.30 \sin [\pi(5 x-200 t)] $$ and $$ y_{2}=0.30 \sin [\pi(5 x-200 t)+\pi / 3] $$ where \(y_{1}, y_{2}\), and \(x\) are in meters and \(t\) is in seconds. When these two waves are combined, a traveling wave is produced. What are the (a) amplitude, (b) wave speed, and (c) wavelength of that traveling wave?

Step-by-Step Solution

Answer

(a) Amplitude = 0.30√3 m, (b) Wave speed = 40 m/s, (c) Wavelength = 0.4 m.

Step 1: Determine the Amplitude

To find the amplitude of the resulting wave, we use the formula for the amplitude of two interfering waves: \[A = 2a \cos \left( \frac{\Delta \phi}{2} \right)\]where \(a\) is the individual amplitude of each wave and \(\Delta \phi\) is the phase difference.For the given waves, \(a = 0.30\) m and \(\Delta \phi = \pi / 3\).Substituting these values:\[A = 2 \times 0.30 \cos \left( \frac{\pi/3}{2} \right) = 2 \times 0.30 \cos \left( \pi/6 \right)\]\(\cos(\pi/6) = \sqrt{3}/2\), so:\[A = 2 \times 0.30 \times \frac{\sqrt{3}}{2} = 0.30 \sqrt{3}\]

Step 2: Determine the Wave Speed

From the wave equations, the wave speed \(v\) can be determined using the equation:\[ v = \frac{\omega}{k} \]where \(\omega\) is the angular frequency and \(k\) is the wave number.For \(y_1\), we have \([5 \pi x - 200 \pi t]\), giving us \(k = 5\pi\) and \(\omega=200\pi\).Thus, the wave speed is:\[v = \frac{200\pi}{5\pi} = 40 \, \text{m/s}\]

Step 3: Determine the Wavelength

The wavelength \(\lambda\) is related to the wave number \(k\) by:\[\lambda = \frac{2\pi}{k}\]Given \(k = 5\pi\), we find:\[\lambda = \frac{2\pi}{5\pi} = \frac{2}{5} \, \text{m}= 0.4 \, \text{m}\]

Key Concepts

Amplitude Calculation

When dealing with wave interference, calculating the amplitude of the resulting wave is essential for understanding the overall effect of the interfering waves. If two waves have the same frequency but are out of phase, they can interfere constructively or destructively, affecting the resulting amplitude.

In the given exercise, the two waves are described by their equations and have identical initial amplitudes of 0.30 meters, but they are phase-shifted by an angle of \(\frac{\pi}{3}\).

To find the amplitude of the resulting wave, you use the formula:

• \(A = 2a \cos \left( \frac{\Delta \phi}{2} \right)\)

Here, \(a\) is the amplitude of the individual waves, and \(\Delta \phi\) is the phase difference.

Plugging in the provided values:
\(A = 2 \times 0.30 \cos \left( \frac{\pi/3}{2} \right)\)= \(2 \times 0.30 \times \frac{\sqrt{3}}{2}\)

Therefore, the amplitude of the combined wave, taking into account the interference, is \(0.30 \sqrt{3}\). It's important to understand this because the phase difference affects how the waves add together, either increasing or decreasing the resultant amplitude.

Wave Speed

The speed of a wave, often referred to as wave speed, is a crucial characteristic that indicates how fast a wave travels through a medium.

In the wave equations provided, we can find the wave speed using the formula:

• \(v = \frac{\omega}{k}\)

where \(\omega\) is the angular frequency, and \(k\) is the wave number.

In the given case, for the wave equation \(y_1 = 0.30 \sin [\pi(5x - 200t)]\):

• The wave number \(k\) is \(5\pi\). It represents how many wave cycles fit into a unit distance.
• The angular frequency \(\omega\) is \(200\pi\). It reflects how many cycles occur per unit time.

By substituting these values into the wave speed formula, you get:
\(v = \frac{200\pi}{5\pi} = 40 \, \text{m/s}\)

This shows the wave travels at a speed of 40 meters per second. Knowing the wave speed helps predict how the wave propagates across the medium.

Wavelength

The wavelength is a measure of the distance between consecutive crests (or troughs) of the wave. It is a fundamental aspect of wave behavior and is directly connected to other wave properties, such as wave number and frequency.

For the calculation of the wavelength, we use the relationship between wave number \(k\) and the wavelength \(\lambda\):

• \(\lambda = \frac{2\pi}{k}\)

In the given wave equation \(y_1 = 0.30 \sin [\pi(5x - 200t)]\), \(k\) is determined to be \(5\pi\).

Thus, substituting \(k\) into the wavelength formula gives:
\(\lambda = \frac{2\pi}{5\pi}\)
Simplifying this fraction, you find:
\(\lambda = \frac{2}{5} \, \text{m} = 0.4 \, \text{m}\)

This wavelength tells you that every 0.4 meters along the wave, a full wave cycle repeats. Understanding wavelength is vital for analyzing the spatial characteristics of wave phenomena.