Problem 69 - Fundamentals of Physics

Question

Three sinusoidal waves of the same frequency travel along a string in the positive direction of an \(x\) axis. Their amplitudes are \(y_{1}, y_{1} / 2\), and \(y_{1} / 3\), and their phase constants are \(0, \pi / 2\), and \(\pi\), respectively. What are the (a) amplitude and (b) phase constant of the resultant wave? (c) Plot the wave form of the resultant wave at \(t=0\), and discuss its behavior as \(t\) increases.

Step-by-Step Solution

Answer

The resultant amplitude is \( \frac{\sqrt{31}}{6} y_1 \) and the phase constant is \( \tan^{-1}(3/4) \). As time increases, the wave maintains its shape but moves in the positive x-direction.

Step 1: Represent the Sinusoidal Waves Mathematically

The three sinusoidal waves can be mathematically represented as follows:1. First wave: \( y_1(x, t) = y_1 \sin(kx - \omega t) \)2. Second wave: \( y_2(x, t) = \frac{y_1}{2} \sin(kx - \omega t + \frac{\pi}{2}) \)3. Third wave: \( y_3(x, t) = \frac{y_1}{3} \sin(kx - \omega t + \pi) \)We have the amplitudes as given and the respective phase constants.

Step 2: Use the Superposition Principle

According to the superposition principle, the resultant displacement at any point on the string is the sum of the displacements due to each wave.\[ y(x, t) = y_1(x, t) + y_2(x, t) + y_3(x, t) \]

Step 3: Simplify the Sinusoidal Functions

First convert each sine function using trigonometric identities:1. \( y_1(x, t) = y_1 \sin(kx - \omega t) \)2. \( y_2(x, t) = \frac{y_1}{2} \cos(kx - \omega t) \), (since \( \sin(\theta + \frac{\pi}{2}) = \cos(\theta) \))3. \( y_3(x, t) = -\frac{y_1}{3} \sin(kx - \omega t) \), (since \( \sin(\theta + \pi) = -\sin(\theta) \))

Step 4: Add the Components

Combine the sine and cosine components to solve for the resultant wave:The resultant wave \( y(x, t) \) becomes:\[ y(x, t) = \left(y_1 - \frac{y_1}{3}\right) \sin(kx - \omega t) + \frac{y_1}{2} \cos(kx - \omega t) \]\[ y(x, t) = \frac{2y_1}{3} \sin(kx - \omega t) + \frac{y_1}{2} \cos(kx - \omega t) \]

Step 5: Calculate Resultant Amplitude and Phase

Convert to a single sinusoidal function:The amplitude \( R \) is calculated by:\[ R = \sqrt{\left(\frac{2y_1}{3}\right)^2 + \left(\frac{y_1}{2}\right)^2} \]\[ R = \sqrt{\frac{4y_1^2}{9} + \frac{y_1^2}{4}} \]\[ R = \frac{\sqrt{31}}{6} y_1 \]Phase constant \( \phi \) is found using:\[ \tan(\phi) = \frac{\frac{y_1}{2}}{\frac{2y_1}{3}} \]\[ \tan(\phi) = \frac{3}{4} \]\[ \phi = \tan^{-1} \left(\frac{3}{4}\right) \]

Step 6: Sketch and Discuss resultant wave at t=0

At \( t = 0 \), substitute into the resultant wave equation.\[ y(x, 0) = R \sin(kx + \phi) = \frac{\sqrt{31}}{6} y_1 \sin(kx + \tan^{-1}(\frac{3}{4})) \]As \( t \) increases, the wave travels along the positive x-direction, maintaining a consistent waveform but shifting spatial position.

Key Concepts

Amplitude

Understanding amplitude is key to analyzing wave behavior. In this exercise, we deal with three waves, each contributing a distinct amplitude to the resultant wave. Each wave's amplitude can be seen as its height from the equilibrium position. The original amplitudes provided are:

• First wave - Amplitude: \( y_1 \)
• Second wave - Amplitude: \( \frac{y_1}{2} \)
• Third wave - Amplitude: \( \frac{y_1}{3} \)

The superposition principle tells us to sum these effects to find the total amplitude. However, due to differences in the phase, these aren't directly added. Instead, they form a combination of sine and cosine terms, leading to the formula:\[R = \sqrt{\left(\frac{2y_1}{3}\right)^2 + \left(\frac{y_1}{2}\right)^2}\]The resultant amplitude \( R \) can be interpreted as the maximum extent of vibration in the combined wave, illustrating how amplitude from different waves can constructively or destructively interfere, leading to a new wave height.

Phase Constant

The phase constant shifts a wave along the x-axis. It demonstrates the starting point of a wave cycle. In this exercise:

• First wave - Phase constant: \( 0 \)
• Second wave - Phase constant: \( \frac{\pi}{2} \)
• Third wave - Phase constant: \( \pi \)

These phase constants mean the peaks and troughs of the waves start at different points. To find the phase constant of the resultant wave, we employ the tangent function:\[\tan(\phi) = \frac{\text{coefficient of cosine term}}{\text{coefficient of sine term}} = \frac{\frac{y_1}{2}}{\frac{2y_1}{3}}\]Solving this gives us:\[\phi = \tan^{-1} \left(\frac{3}{4}\right)\]This resultant phase constant \( \phi \) represents how far the combined wave's cycle is from the start, influencing where its peaks and nodes occur relative to the individual contributing waves.

Sinusoidal Waves

Sinusoidal waves are smooth, continuous waves that can be described by sine and cosine functions. They are key in modeling periodic oscillations commonly found in nature. The waves in our problem can be mathematically expressed and manipulated using trigonometric identities. Here, each wave is expressed as:

• Wave 1: \( y_1 \sin(kx - \omega t) \)
• Wave 2: \( \frac{y_1}{2} \cos(kx - \omega t) \) (identity used: \( \sin(\theta + \frac{\pi}{2}) = \cos(\theta) \))
• Wave 3: \( -\frac{y_1}{3} \sin(kx - \omega t) \) (identity used: \( \sin(\theta + \pi) = -\sin(\theta) \))

The principle of wave superposition allows us to add these waves, using their sine and cosine components to find a combined expression for the resultant wave. The preparation of sinusoidal expressions lays the groundwork for understanding the complex interactions and resultant waveforms when multiple sinusoidal waves overlap.