Problem 93 - Fundamentals of Physics

Question

93 A traveling wave on a string is described by $$ y=2.0 \sin \left[2 \pi\left(\frac{t}{0.40}+\frac{x}{80}\right)\right] $$ where \(x\) and \(y\) are in centimeters and \(t\) is in seconds. (a) For \(t=0\), plot \(y\) as a function of \(x\) for \(0 \leq x \leq 160 \mathrm{~cm} .\) (b) Repeat (a) for \(t=0.05 \mathrm{~s}\) and \(t=0.10 \mathrm{~s}\). From your graphs, determine (c) the wave speed and (d) the direction in which the wave is traveling.

Step-by-Step Solution

Answer

Wave speed is 200 cm/s; the wave travels in the positive x-direction.

Step 1: Understand the given wave equation

The wave equation given is\[y=2.0 \, \sin \left[2 \pi\left(\frac{t}{0.40}+\frac{x}{80}\right)\right]\]where the amplitude of the wave is 2.0 cm. The equation inside the sine function represents the phase of the wave, which depends on both time \(t\) and position \(x\).

Step 2: Plot at t=0

For \(t=0\), our equation simplifies to:\[y = 2.0 \, \sin \left( \frac{2\pi x}{80} \right)\]We need to plot \(y\) as a function of \(x\) from \(0\) to \(160\) cm. This results in a sine wave with a full period (since \(160/80 = 2\), we have two cycles over this range).

Step 3: Plot at t=0.05 s

Substitute \(t=0.05\) s into the wave equation:\[y = 2.0 \, \sin \left[2\pi\left(\frac{0.05}{0.40}+\frac{x}{80}\right)\right]\]This changes the phase, shifting the wave slightly. Plot \(y\) as a function of \(x\) ranging from \(0\) to \(160\) cm, which shows the wave shifted compared to \(t=0\).

Step 4: Plot at t=0.10 s

Substitute \(t=0.10\) s into the wave equation:\[y = 2.0 \, \sin \left[2\pi\left(\frac{0.10}{0.40}+\frac{x}{80}\right)\right]\]Again, this shifts the wave further. Plot \(y\) as a function of \(x\) over the same range, showing the wave has moved further compared to \(t=0.05\) s.

Step 5: Determine the wave speed

The wave speed can be determined by analyzing how far the wave shifts over time. The wave equation is of the form:\[y = A \, \sin \left[2\pi \left( \frac{t}{T} + \frac{x}{\lambda } \right) \right]\]where \(T=0.40\) s is the period in this context, and \(\lambda = 80\) cm is the wavelength. The wave speed \(v\) is given by:\[v = \frac{\lambda}{T} = \frac{80}{0.40} = 200\text{ cm/s}\].

Step 6: Determine the wave direction

Observing how the wave shifts with increasing time shows the wave moving towards higher values of \(x\), indicating it is traveling in the positive \(x\)-direction.

Key Concepts

Sine Wave

A sine wave is a smooth, periodic oscillation that is an important concept in wave motion and physics. It is characterized by its amplitude, wavelength, frequency, and phase. The sine wave can be represented mathematically as a function of time and position, often using the sine function itself.Key characteristics of a sine wave include:

• **Amplitude**: This is the maximum displacement of the wave from its equilibrium position. In the given equation, the amplitude is 2.0 cm, which indicates the peak height of the wave.
• **Wavelength**: The distance over which the wave's shape repeats. For the equation in question, \( rac{2\pi x}{80} \) indicates a wavelength of 80 cm.
• **Frequency and Period**: Frequency is how often the wave oscillates in a unit of time, and the period is the reciprocal of frequency. The parameter \( T = 0.40 \text{s} \) in our equation denotes the period.
• **Phase**: This depends on both time (\( t \)) and position (\( x \)), and it determines the wave's initial state or position.

Understanding these aspects of a sine wave is crucial for analyzing any wave system, whether it's a string, sound, or even light.

Wave Speed

Wave speed refers to how fast a wave travels through a medium. It is determined by the wave's wavelength and period. For a wave traveling on a string, like the one described in the exercise, the wave speed can be calculated using the formula \( v = \frac{\lambda}{T} \), where \( \lambda \) is the wavelength and \( T \) is the period.In our example:

• The wavelength (\( \lambda \)) is 80 cm.
• The period (\( T \)) is 0.40 s.

Thus, the wave speed is calculated as follows:\[v = \frac{80 \text{ cm}}{0.40 \text{ s}} = 200 \text{ cm/s}\]This means that the wave travels at a speed of 200 centimeters per second along the string. Understanding wave speed is vital in various applications, such as sound waves in air or water waves in the ocean.

Wave Direction

Wave direction indicates the path along which the wave energy travels. In the context of the exercise, determining wave direction involves observing how the wave shifts with time.When we substitute different time values into the wave equation, it becomes evident that the wave moves towards increasing values of \( x \). For example:

• At \( t = 0 \), the wave is in its initial position.
• At \( t = 0.05 \text{s} \) and \( t = 0.10 \text{s} \), we notice that the wave has shifted to the right from its initial position.

This rightward shift, or movement towards greater \( x \) values, indicates that the wave travels in the positive \( x \)-direction. Understanding wave direction can be crucial in applications like predicting wave behavior on strings, water, or even electromagnetic waves.

Wave Equation

The wave equation provides a mathematical model for wave motion, expressing how displacement varies with time and position. In our exercise, the wave equation is given as:\[y = 2.0 \sin \left[ 2 \pi \left( \frac{t}{0.40} + \frac{x}{80} \right) \right]\]Here's a breakdown of its components:

• The **amplitude** (2.0 cm) defines the maximum height from the central position the wave can reach.
• The fractions inside the sine function represent the **phase**, accounting for both time (\( t \)) and position (\( x \)).
• The **period** (0.40 s) and **wavelength** (80 cm) are embedded in the phase components, influencing the wave's speed and frequency.

This equation is particularly important as it helps predict wave behavior at any point along the string. Recognizing how each part of the wave equation contributes to motion is fundamental in fields such as acoustics, optics, and mechanical waves.